Newton’s laws of motion form the backbone of classical mechanics and are a central topic in A-Level Physics examinations. They provide the framework to analyse forces, predict motion, and solve problems involving everything from simple blocks to collisions and connected systems. This guide breaks down essential concepts, common applications, and exam tips to help you master the topic.
1. Newton’s First Law: The Law of Inertia | 牛顿第一定律：惯性定律
A body remains at rest or moves with constant velocity in a straight line unless acted upon by a net external force. This is often summarised as: if the resultant force is zero, the acceleration is zero.
除非受到合外力的作用，物体将保持静止或匀速直线运动状态。这常被总结为：若合力为零，则加速度为零。
∑F = 0 ⇔ a = 0 and v = constant
Inertia is the tendency of an object to resist changes in its state of motion. Mass is a measure of inertia — the greater an object’s mass, the harder it is to start or stop it moving. This law underpins concepts such as why passengers lurch forward when a car brakes suddenly.
In exam questions, you may need to apply equilibrium conditions: net force = 0 both horizontally and vertically. Free-body diagrams help visualise forces at rest or moving at constant speed.
2. Newton’s Second Law: Force, Mass and Acceleration | 牛顿第二定律：力、质量与加速度
The net force acting on an object is directly proportional to the rate of change of its momentum, and for constant mass, this reduces to the famous equation:
物体所受的合外力与其动量的变化率成正比，对于质量不变的情况，这归约为著名的方程：
Fnet = m a
Force is measured in newtons (N), mass in kilograms (kg), and acceleration in metres per second squared (m s⁻²). One newton is the force needed to give a 1 kg mass an acceleration of 1 m s⁻².
The direction of acceleration is always the same as the direction of the net force. This is a vector relationship: if multiple forces act, you must resolve or add them vectorially before applying F = m a.
加速度的方向始终与合力的方向相同。这是一个矢量关系：如果有多个力作用，必须在应用 F = m a 之前按矢量将其分解或相加。
A-Level questions often involve combining F = m a with suvat equations or calculating acceleration from a force diagram. Always draw a clear diagram and state a positive direction before writing equations.
A-Level 考题常将 F = m a 与匀加速运动方程结合，或要求从受力图中计算加速度。务必先画出清晰的示意图，并规定正方向，再列出方程。
3. Newton’s Third Law: Action and Reaction | 牛顿第三定律：作用力与反作用力
If body A exerts a force on body B, then body B exerts a force of equal magnitude but opposite direction on body A. These two forces are always of the same type, act on different bodies, and occur simultaneously.
若物体 A 对物体 B 施加一个力，则物体 B 会同时对物体 A 施加一个大小相等、方向相反的力。这两个力总是同一类型，作用在不同物体上，并且同时产生。
FA on B = −FB on A
A common misconception is that action–reaction pairs cancel each other out. They do not cancel because they act on different objects. For example, the Earth pulls you down with gravity, and you pull the Earth up with the same magnitude — but the effects differ because of vastly different masses.
This law is critical when analysing collisions, rocket propulsion, and interactions between connected bodies. Exam questions might ask you to identify third-law pairs in a given scenario — always check that the two forces act on different objects and are of the same nature.
4. Drawing and Interpreting Free-body Diagrams | 受力分析图的绘制与解读
A free-body diagram shows all the forces acting on a single object, drawn as arrows from the centre of mass. Each force should be labelled with its type (weight, tension, normal reaction, friction, etc.) and direction.
To draw correctly: isolate the object, draw all forces acting on it, do not include forces it exerts on other objects. For equilibrium or acceleration problems, the diagram helps write resolved components along perpendicular axes, usually horizontal/vertical or parallel/perpendicular to an inclined plane.
A classic A-Level mistake is omitting the normal reaction force or the tension in a string when it should be included. Practise drawing diagrams for blocks on inclines, objects on pulleys, and bodies in elevators.
5. Resolving Forces and Vector Components | 力的分解与矢量分量
Forces at angles must be resolved into perpendicular components, typically using trigonometry. For a force F at angle θ to the horizontal:
带角度的力必须分解为相互垂直的分量，通常使用三角函数。对于与水平面成 θ 角的力 F：
Fhorizontal = F cos θ, Fvertical = F sin θ
Choosing a suitable pair of axes simplifies calculations. For an inclined plane, it is usually best to resolve along and perpendicular to the plane, so that the normal force and friction are along these axes, while weight gets components parallel and perpendicular to the slope.
On an incline of angle θ, the weight mg splits into: mg sin θ down the slope, mg cos θ into the slope. Always verify the geometry with a small sketch showing the right-angled triangles formed by the weight vector.
在倾角为 θ 的斜面上，重力 mg 分解为：沿斜面向下的 mg sin θ，以及垂直于斜面向下的 mg cos θ。经常用一个显示重力矢量构成的直角三角形的小草图来核对几何关系。
6. Connected Particles and Pulleys | 连接体与滑轮系统
When two or more objects are connected by a light inextensible string over a smooth pulley, the tension is the same throughout the string and the acceleration magnitude is the same for both masses (assuming the string remains taut).
To solve: draw separate free-body diagrams for each mass, write the equation of motion (F = m a) for each using a consistent sign convention, and solve the simultaneous equations. If the pulley is rough or has mass, additional torque equations from rotational dynamics may be needed.
解题步骤：为每个质量单独画出受力图，使用一致的符号规定为每个物体写出运动方程 (F = m a)，然后联立求解。如果滑轮有摩擦或有质量，则可能还需要转动动力学的转矩方程。
A common set-up: mass A hanging vertically pulls mass B along a horizontal surface. The tension accelerates B horizontally, while for A, weight down minus tension up gives mAg − T = mA a. Make sure you assign positive direction of motion correctly and treat the system as a whole to find acceleration, then substitute back to find tension.
一个常见装置：竖直悬挂的物体 A 拉着水平面上的物体 B。拉力沿水平方向加速 B，而对 A 来说，向下的重力减去向上的拉力得出 mAg − T = mA a。务必正确规定运动的正方向，并可将系统视作整体先求出加速度，再代回求出拉力。
7. Motion on Inclined Planes | 斜面上的运动
An object on a smooth inclined plane accelerates down the slope under the component of its weight: mg sin θ. The normal reaction balances the perpendicular component: R = mg cos θ.
在光滑斜面上，物体在重力的分力作用下沿斜面向下加速：mg sin θ。法向反力平衡垂直分量：R = mg cos θ。
a = g sin θ (if no friction)
If friction is present, the net force down the slope becomes mg sin θ − f, where f can be up to μR (where μ is the coefficient of friction). The direction of friction always opposes relative motion (or tendency of motion).
如果有摩擦力，沿斜面向下的合力变为 mg sin θ − f，其中 f 最大可达 μR（μ 为摩擦系数）。摩擦力的方向总是与相对运动（或运动趋势）相反。
A-Level problems often combine inclined planes with pulleys, so an object on a slope is connected to a hanging mass. You must resolve forces parallel to the incline for the block, and vertically for the hanging weight, then link their accelerations.
Friction is a force that opposes the sliding motion of two surfaces in contact. Two types are examined: static friction (when no motion) and kinetic (or dynamic) friction (when surfaces slide).
Static friction varies up to a maximum value: fs ≤ μsR, where μs is the coefficient of static friction. Once motion starts, kinetic friction takes over and is roughly constant: fk = μkR, with μk typically less than μs.
In exam questions, the coefficient of friction may be derived from a limiting equilibrium condition or from a given acceleration. Always check whether the object is about to move (limit of static friction) or already sliding (kinetic).
When an object moves through a fluid (liquid or gas), it experiences a drag force that increases with speed. For a falling object, the net force reduces until drag equals weight; thereafter the object falls at constant terminal velocity.
The drag force for small objects at low speeds is often proportional to velocity (viscous drag), but for larger objects at higher speeds it is often proportional to v². The analysis follows Newton’s second law: mg − kv = m a, leading to terminal velocity vt = mg/k for linear drag.
Questions on terminal velocity often appear in contexts like skydiving or falling spheres in viscous liquids. You must interpret velocity–time graphs and explain the shape in terms of forces and acceleration.
Momentum (p) is defined as the product of mass and velocity: p = m v. It is a vector quantity with units kg m s⁻¹ or N s. Newton’s second law can be stated in terms of momentum: the net force equals the rate of change of momentum.
动量 (p) 定义为质量与速度的乘积：p = m v。它是矢量，单位为 kg m s⁻¹ 或 N s。牛顿第二定律可以用动量的语言表述：合力等于动量的变化率。
F = Δp / Δt
Impulse is the change in momentum of an object when a force acts over a time interval: Impulse = F Δt = Δp. The area under a force–time graph gives the impulse.
冲量是力在一段时间间隔内作用时物体动量的变化量：冲量 = F Δt = Δp。力-时间图下的面积就是冲量。
This formulation is particularly useful when forces vary over time, such as in collisions or when kicking a ball. In A-Level exams, you may need to calculate impulse from a graph or use impulse to find the average force during an impact.
11. Conservation of Momentum in Collisions | 碰撞中的动量守恒
In a closed system with no external forces, total momentum before an event equals total momentum after. This principle is essential for solving collision and explosion problems.
在没有外力的封闭系统中，事件前的总动量等于事件后的总动量。这一原理对于解决碰撞和爆炸问题至关重要。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Collisions can be elastic (kinetic energy conserved) or inelastic (kinetic energy not conserved, objects may stick together). In a perfectly inelastic collision, the two bodies move with a common velocity after impact; momentum is still conserved, though some kinetic energy is converted to other forms.
A-Level problems often ask for final velocities, or the loss of kinetic energy. Always define a positive direction and assign signs to velocities accordingly. Remember: momentum is a vector, so direction matters.
Draw a clear diagram: label all forces, choose a consistent positive direction, and show resolved components. / 绘制清晰的示意图：标注所有力，选择一致的正方向，并标出分解后的分量。
Use the correct mass: when applying F = m a, m is the total mass of the system being accelerated in that direction, not necessarily the entire system. / 使用正确的质量：应用 F = m a 时，m 是沿该方向被加速的系统的总质量，不一定是整个系统的质量。
Avoid confusing equilibrium and acceleration: if an object accelerates, the forces do not balance in the direction of motion. / 避免混淆平衡与加速：如果物体有加速度，那么在运动方向上各力并不平衡。
Distinguish action-reaction from balanced forces: Newton’s third law pairs act on different bodies. / 区分作用力-反作用力与平衡力：牛顿第三定律的力偶作用在不同的物体上。
Double-check sign conventions in connected systems, especially on inclines. / 在连接体问题中，尤其是在斜面上，双重检查符号规定。
Practice interpreting force-time graphs for impulse and momentum change. / 练习解读力-时间图来求冲量和动量变化。
Many marks are lost through algebraic slips rather than misunderstanding physics. Write equations systematically, substitute numbers only at the end, and check units are consistent.
📚 A-Level Physics Unit 5 Insert (Jan19) Concept Analysis | A-Level 物理 Unit 5 插入页 (2019年1月) 概念解析
The Edexcel International A-Level Physics Unit 5 (WPH05) examination includes a data booklet insert that provides essential constants, equations, and reference information. Understanding the concepts behind these equations is critical for solving problems in thermodynamics, nuclear physics, oscillations, and astrophysics. This article breaks down the key concepts linked to the January 2019 insert, helping you master Unit 5 with confidence.
Edexcel 国际 A-Level 物理 Unit 5（WPH05）考试随附的数据插入页提供了关键的常数、公式和参考信息。掌握这些公式背后的物理概念对于解决热力学、核物理、振动和天体物理问题至关重要。本文解析 2019 年 1 月插入页涉及的核心概念，助你自信应对 Unit 5。
1. Radioactive Decay and Half-Life | 放射性衰变与半衰期
The activity A of a radioactive sample is the number of disintegrations per unit time. The fundamental law states A = λN, where λ is the decay constant and N is the number of undecayed nuclei. This relationship appears directly in the Unit 5 insert, emphasising that activity is proportional to the remaining parent nuclei.
放射性样品的活度 A 是指单位时间内发生衰变的次数。基本定律为 A = λN，其中 λ 为衰变常数，N 为尚未衰变的原子核数目。这一关系直接出现在 Unit 5 插入页中，强调了活度与剩余母核数目成正比。
The number of nuclei varies with time as N = N₀ exp(-λt). From this, the half-life T½, the time taken for half the nuclei to decay, is given by T½ = ln 2 / λ. You will often need to convert between half-life and decay constant using the insert’s constants.
On a logarithmic scale, ln A against time t yields a straight line with gradient −λ. This linearisation technique is frequently examined, and knowing how to extract λ from experimental data is essential.
在对数坐标中，ln A 对时间 t 作图得到一条斜率为 −λ 的直线。这种线性化方法经常出现在考题中，掌握如何从实验数据中提取 λ 至关重要。
2. Mass Defect and Binding Energy | 质量亏损与结合能
The mass defect Δm is the difference between the total mass of the individual nucleons and the actual mass of the nucleus: Δm = Zmₚ + Nmₙ − M_nucleus. Here Z is the proton number, N is the neutron number, mₚ is the proton mass and mₙ is the neutron mass. The insert provides these masses in atomic mass units (u) and in kilograms.
The binding energy E = Δm c² represents the energy required to separate a nucleus into its constituent protons and neutrons. A larger binding energy per nucleon indicates a more stable nucleus. The insert often includes a graph of binding energy per nucleon against nucleon number, which you must interpret to explain fission and fusion trends.
结合能 E = Δm c² 表示将原子核拆分成独立的质子和中子所需的能量。每个核子的平均结合能越大，原子核越稳定。插入页中通常包含平均结合能随核子数变化的曲线，你需要解读该曲线来解释裂变与聚变的趋势。
Calculating energy released in a reaction from mass differences is a key skill. Always work in consistent units (kg, J, u and MeV) using the conversion 1 u = 931.5 MeV as given in the insert.
通过质量差计算反应中释放的能量是一项关键技能。务必使用一致的单位(kg、J、u 和 MeV)，并利用插入页给出的换算关系 1 u = 931.5 MeV。
3. Nuclear Fission and Fusion Reactions | 核裂变与核聚变反应
Nuclear fission involves a heavy nucleus, such as uranium‑235, splitting into two lighter nuclei after capturing a neutron. A typical reaction from the Unit 5 syllabus is ²³⁵U + ¹n → ¹⁴¹Ba + ⁹²Kr + 3¹n. The insert provides atomic masses, enabling you to evaluate the energy released, which typically lies around 200 MeV per fission.
Nuclear fusion combines light nuclei, such as deuterium and tritium, to form helium and a neutron: ²H + ³H → ⁴He + ¹n. The fusion gain in binding energy per nucleon is responsible for the large energy release. Fusion requires extremely high temperatures to overcome Coulomb repulsion, as indicated by the insert’s temperature and energy relationships.
4. Kinetic Theory of Gases and Molecular Speed | 气体动理论与分子速率
The kinetic theory model links macroscopic pressure and temperature to microscopic molecular motion. The fundamental equation from the insert is pV = ⅓ N m ‹c²›, where N is the number of molecules, m is the mass of a single molecule, and ‹c²› is the mean square speed. This equation is the bridge to explain why pressure rises with temperature at constant volume.
气体动理论模型将宏观的压强和温度与微观的分子运动联系起来。插入页中的基本方程为 pV = ⅓ N m ‹c²›，其中 N 为分子数，m 为单个分子的质量，‹c²› 为方均速率。这条方程是解释定容条件下压强为何随温度升高而增大的桥梁。
Combining pV = ⅓ N m ‹c²› with the ideal gas equation pV = nRT yields ½ m ‹c²› = (3/2) kT or, for one mole, KE_avg = (3/2) RT. The root‑mean‑square speed c_rms = √(‹c²›) becomes c_rms = √(3RT/M), where M is the molar mass. The insert lists the molar gas constant R and Boltzmann constant k, enabling these calculations.
将 pV = ⅓ N m ‹c²› 与理想气体方程 pV = nRT 结合可得 ½ m ‹c²› = (3/2) kT，对于 1 mol 气体则有 平均动能 = (3/2) RT。方均根速率 c_rms 可表示为 c_rms = √(3RT/M)，其中 M 为摩尔质量。插入页给出了摩尔气体常数 R 和玻尔兹曼常数 k，使这些计算成为可能。
5. The Ideal Gas Equation in Terms of Moles | 以摩尔数表示理想气体状态方程
The insert highlights the ideal gas law in two forms: pV = nRT (molar form) and pV = NkT (molecular form). In the molar form, p is pressure, V is volume, n is the number of moles, and T is the absolute temperature in kelvin. You must be able to rearrange this to find any missing quantity.
The molar gas constant R is given as 8.31 J K⁻¹ mol⁻¹. The number of molecules N is related to n via Avogadro’s constant N_A, which is also in the insert. The Boltzmann constant follows from k = R / N_A. These constants are vital when moving between the microscopic and macroscopic worlds.
摩尔气体常数 R 给出的值为 8.31 J K⁻¹ mol⁻¹。分子数 N 通过阿伏伽德罗常数 N_A 与 n 关联，该常数同样列在插入页中。玻尔兹曼常数由 k = R / N_A 导出。在微观与宏观世界之间切换时，这些常数不可或缺。
6. Internal Energy, Work and the First Law of Thermodynamics | 内能、功与热力学第一定律
The internal energy U of an ideal gas depends only on its temperature. For a monatomic gas, U = (3/2) nRT. The first law of thermodynamics, ΔU = Q + W, is explicitly stated in the Unit 5 insert. Here Q is the heat added to the system and W is the work done on the system. You must apply sign conventions carefully to different processes.
理想气体的内能 U 仅取决于其温度。对于单原子气体，U = (3/2) nRT。热力学第一定律 ΔU = Q + W 明确写在 Unit 5 插入页中。其中 Q 是系统吸收的热量，W 是外界对系统做的功。你需要针对不同过程谨慎使用符号规则。
For a constant‑volume process, W = 0 because W = −pΔV (where work done by the gas is positive). When the volume changes, W = −p ΔV under constant pressure. The insert often provides a table for isothermal, adiabatic, isovolumetric and isobaric processes, and you must be able to analyse energy transfers in each case.
在等体过程中，W = 0，因为 W = −pΔV（气体对外做功则为正）。当体积变化时，等压过程则有 W = −p ΔV。插入页通常给出等温、绝热、等体和等压过程的相关信息，你需要能够分析每一种情况下的能量转移。
The Unit 5 insert provides the defining equation for simple harmonic motion (SHM): a = −ω²x. Here a is acceleration, x is displacement and ω is the angular frequency (ω = 2πf = 2π/T). This equation implies that the restoring force is proportional to displacement and directed towards the equilibrium position.
Unit 5 插入页给出了简谐运动 (SHM) 的定义方程：a = −ω²x。其中 a 为加速度，x 为位移，ω 为角频率 (ω = 2πf = 2π/T)。该方程表明回复力与位移成正比，并指向平衡位置。
The displacement varies as x = A cos(ωt) or x = A sin(ωt). Velocity is maximum at the equilibrium point: v_max = ωA. The speed at any displacement is v = ω √(A² − x²). These formulas, along with the period of a mass‑spring system T = 2π √(m/k) and the simple pendulum T = 2π √(L/g), are commonly assessed.
位移的变化规律为 x = A cos(ωt) 或 x = A sin(ωt)。速度在平衡位置达到最大值：v_max = ωA。任意位移处的速率为 v = ω √(A² − x²)。这些公式以及弹簧振子周期 T = 2π √(m/k) 和单摆周期 T = 2π √(L/g) 均在常见考查范围之内。
8. Gravitational Fields and Circular Orbits | 引力场与圆周轨道
Newton’s law of gravitation, F = GMm / r², and the gravitational field strength g = GM / r² are central to Unit 5. The insert gives the gravitational constant G = 6.67 × 10⁻¹¹ N m² kg⁻². For a satellite in a circular orbit, the gravitational force provides the centripetal force: GMm / r² = m v² / r.
牛顿万有引力定律 F = GMm / r² 和引力场强度 g = GM / r² 是 Unit 5 的核心内容。插入页给出了引力常数 G = 6.67 × 10⁻¹¹ N m² kg⁻²。对于沿圆周轨道运行的人造卫星，引力提供向心力：GMm / r² = m v² / r。
From this, the orbital speed is v = √(GM / r) and the period is T = 2π √(r³ / GM). These relationships show that for a given central mass, outer orbits have slower speeds and longer periods. The insert’s constant for GM of the Earth or Sun often saves time in calculations.
由此可得轨道速率 v = √(GM / r)，轨道周期 T = 2π √(r³ / GM)。这些关系表明，对于给定的中心天体，外层的轨道速率较慢而周期较长。插入页中给出的地球或太阳的 GM 常数往往能简化计算。
9. Hubble’s Law and the Age of the Universe | 哈勃定律与宇宙年龄
Hubble’s law states that the recession speed v of a galaxy is proportional to its distance d: v = H₀ d. The insert provides the Hubble constant H₀ in units of km s⁻¹ Mpc⁻¹. By converting to SI units (s⁻¹), you can estimate the age of the Universe as t ≈ 1 / H₀, assuming a constant expansion rate.
哈勃定律指出，星系的退行速度 v 与其距离 d 成正比：v = H₀ d。插入页以 km s⁻¹ Mpc⁻¹ 为单位给出了哈勃常数 H₀。通过将其转换为 SI 单位（s⁻¹），可以依据 t ≈ 1 / H₀ 估算宇宙的年龄，这是在膨胀速率恒定的假设下得到的。
The Doppler effect for light is used to determine recession velocities from redshift: Δλ / λ ≈ v / c for v << c. Combining redshift measurements with Hubble’s law allows astronomers to map large‑scale cosmic structure. You should be comfortable interchanging wavelength shifts and velocity in this context.
光的多普勒效应被用来通过红移确定退行速度：当 v << c 时有 Δλ / λ ≈ v / c。将红移测量值与哈勃定律结合，能够帮助天文学家绘制宇宙大尺度结构。在此类问题中，你需要熟练地在波长偏移与速度之间进行转换。
10. Important Constants and Units Reference | 重要常数与单位参考
The January 2019 insert includes a comprehensive list of constants: speed of light c = 3.00 × 10⁸ m s⁻¹, Planck constant h = 6.63 × 10⁻³⁴ J s, elementary charge e = 1.60 × 10⁻¹⁹ C, electron mass mₑ = 9.11 × 10⁻³¹ kg, proton mass mₚ = 1.67 × 10⁻²⁷ kg, and the Avogadro constant. Knowing when to use each is essential for efficient exam performance.
2019 年 1 月的插入页包含一份详尽的常数列表：光速 c = 3.00 × 10⁸ m s⁻¹，普朗克常数 h = 6.63 × 10⁻³⁴ J s，元电荷 e = 1.60 × 10⁻¹⁹ C，电子质量 mₑ = 9.11 × 10⁻³¹ kg，质子质量 mₚ = 1.67 × 10⁻²⁷ kg，以及阿伏伽德罗常数。清楚何时使用这些常数对高效答题至关重要。
Additionally, the insert provides conversion factors between electronvolts and joules, atomic mass units and kilograms, and parsec and light‑year. Always ensure that all quantities are expressed in consistent SI units before substituting into equations. Practice with past papers will build fluency in locating these data rapidly.
此外，插入页还提供了电子伏特与焦耳、原子质量单位与千克、以及秒差距与光年之间的换算因子。在将数值代入公式之前，务必确保所有物理量均采用一致的 SI 单位。通过真题练习可以训练快速查找这些数据的能力。
Published by TutorHao | Physics Revision Series | aleveler.com
📚 A-Level Physics: June 2018 Insert 5 – Measuring g by Free Fall | A-Level 物理：2018年6月插入材料5 – 自由落体法测重力加速度
In the June 2018 A-Level Physics examination, Insert 5 provided a diagram of an electromagnet and trapdoor apparatus, along with a table of measured distances and times for a steel ball falling from rest. This investigation is a classic method to determine the acceleration due to gravity, g, and the data must be analysed using graphical techniques and error analysis typical of practical examination questions.
Insert 5 contains a circuit diagram of an electromagnet holding a steel ball, a trapdoor switch connected to a timer, and a metre rule to vary the drop height h. The timer starts when the circuit is broken and stops when the trapdoor opens. A table gives values of h and the corresponding fall time t for several heights.
插入材料 5 包含一个电磁铁吸附钢球的电路图、一个连接计时器的触发门开关，以及一把改变下落高度 h 的米尺。当电路断开时计时器开始计时，触发门打开时停止计时。表格给出了多个高度对应的下落时间 t。
The data in Insert 5 allow students to plot a graph of h against t² and hence determine g from the slope. The experiment assumes negligible air resistance and that the ball is released from rest precisely when the timer starts.
插入材料 5 中的数据使学生能够绘制 h 对 t² 的图像，从而根据斜率求出 g。该实验假设空气阻力可忽略，并且小球在计时器启动的瞬间从静止精确释放。
2. Experimental Apparatus | 实验器材
The key components shown in Insert 5 are an electromagnet powered by a d.c. supply, a steel ball bearing, a trapdoor mechanism that opens on impact, an electronic timer or stopwatch, a metre rule, and clamps and stands.
The electromagnet holds the ball at a measured height h above the trapdoor. When the switch is opened, the magnetic field collapses and the ball begins to fall. The timer starts simultaneously and stops when the ball hits the trapdoor, breaking a contact.
电磁铁将小球保持在触发门上方已测定的高度 h 处。断开开关时，磁场消失，小球开始下落。计时器同步启动，当小球击中触发门并断开触点时计时停止。
3. Procedure and Measurements | 实验步骤与测量
First, set the height h to a small value, using the metre rule to measure from the bottom of the ball to the trapdoor. Ensure the electromagnet is directly above the trapdoor to minimise sideways motion. Record the fall time t three times for each height and calculate an average t to reduce random errors.
首先，将高度 h 设为一个较小值，用米尺从球的底部量到触发门。确保电磁铁位于触发门正上方以减少侧向运动。每个高度记录下落时间 t 三次，计算平均 t 以减小随机误差。
Repeat for at least five different heights, up to about 1.0 m. Insert 5 shows h values ranging from 0.200 m to 1.000 m. For each h, the timer reading t is recorded to 0.001 s. All raw data should be entered into a suitable table.
对至少五个不同的高度重复实验，最高约 1.0 m。插入材料 5 显示 h 值范围从 0.200 m 到 1.000 m。对每个 h，计时器读数 t 记录到 0.001 s。所有原始数据应填入合适的表格。
4. Data Table and Key Calculations | 数据表与关键计算
Insert 5 gives the following typical data. You are expected to calculate t² and then plot h against t². A sample table is shown below:
插入材料 5 给出如下典型数据。你需要计算 t²，然后绘制 h 对 t² 的图像。以下是一个示例表格：
h / m
t₁ / s
t₂ / s
t₃ / s
mean t / s
t² / s²
0.200
0.206
0.199
0.201
0.202
0.0408
0.400
0.288
0.284
0.286
0.286
0.0818
0.600
0.352
0.348
0.350
0.350
0.1225
0.800
0.405
0.403
0.404
0.404
0.1632
1.000
0.454
0.450
0.452
0.452
0.2043
The scatter in repeated readings is small, indicating good precision. The table is used to construct the graph required by Insert 5.
重复读数的离散程度很小，表明精密度良好。这个表格用于构建插入材料 5 所要求的图像。
5. Graph Plotting | 绘图
According to the insert, you must plot a graph of h on the y-axis against t² on the x-axis. Use sensible scales that utilise at least half the graph paper. Label axes as ‘h / m’ and ‘t² / s²’. Plot the points carefully using small crosses and draw a line of best fit.
根据插入材料，你必须绘制 h 在 y 轴、t² 在 x 轴的图像。使用合理的刻度，至少占据坐标纸的一半。坐标轴标注为 ‘h / m’ 和 ‘t² / s²’。用小叉号仔细标出数据点，并画出最佳拟合线。
The expected relationship is a straight line through the origin, because the equation of motion for free fall from rest is:
预期关系是一条过原点的直线，因为从静止自由下落的运动方程为：
h = ½ g t²
Comparing with y = m x, the gradient of the line is ½ g. If the line does not pass through the origin, a small systematic error is present.
与 y = m x 比较，直线的斜率（梯度）为 ½ g。如果直线不通过原点，则存在小的系统误差。
6. Determining g from the Graph | 从图像确定 g
To find g, select two well-separated points on the best-fit line, not the data points. For example, using (t² = 0.040 s², h = 0.196 m) and (t² = 0.200 s², h = 0.980 m):
为了求 g，在最佳拟合线上选取两个间隔较远的点，而非数据点。例如，使用 (t² = 0.040 s², h = 0.196 m) 和 (t² = 0.200 s², h = 0.980 m)：
Insert 5 may ask for an estimate of uncertainty. The main instrumental uncertainty comes from the metre rule (±1 mm) and the timer (±0.001 s). For the smallest h = 0.200 m, the percentage uncertainty in h is (0.001/0.200)×100% = 0.5%.
For time t ≈ 0.202 s, the percentage uncertainty in t is (0.001/0.202)×100% ≈ 0.5%, but t² has double the percentage uncertainty, about 1.0%. The combined uncertainty in the gradient can be found using error bars or by plotting the steepest and shallowest acceptable lines.
If the steepest gradient is 5.00 and the shallowest is 4.80, then g = 2 × gradient would range from 9.6 to 10.0 m/s², giving g = 9.80 ± 0.20 m/s². This range includes the true value, indicating the experiment is valid.
如果最陡斜率为 5.00，最浅斜率为 4.80，那么 g = 2 × 斜率范围为 9.6 至 10.0 m/s²，得出 g = 9.80 ± 0.20 m/s²。该区间包含真值，说明实验是有效的。
8. Significant Sources of Error | 误差的主要来源
The most significant systematic error is the delay in releasing the ball when the circuit is broken. Residual magnetism in the electromagnet can hold the ball for a few milliseconds after the timer starts, effectively increasing the measured fall time and reducing the value of g.
最显著的系统误差是电路断开时小球释放的延迟。电磁铁的剩磁可能在计时器启动后仍吸住小球几毫秒，有效增加了测量的下落时间，使 g 值偏小。
Another source is the reaction time of the trapdoor switch. If the trapdoor does not open instantly, the timer runs slightly longer. Parallax error when measuring height h with a ruler can also affect accuracy, as can any sideways movement causing the ball to miss the trapdoor centre.
另一个来源是触发门开关的反应时间。如果触发门不能瞬间打开，计时器会多运行极短时间。用米尺测量高度 h 时的视差也可能影响准确度，而侧向移动导致小球未击中触发门中心也会引入误差。
Air resistance is usually negligible for steel balls over these heights, but for a very light ball it would reduce the net acceleration, giving a lower apparent g.
对于钢球，在这些高度下空气阻力通常可忽略，但如果球很轻，阻力会减小合加速度，使表观 g 偏小。
9. Improvements to the Experiment | 实验改进
Insert 5 may ask for improvements. Place a thin piece of earthing or use a demagnetised core to reduce residual magnetism. Use a dual-beam oscilloscope or a light gate placed just below the release point to confirm that the ball starts falling exactly when the timer triggers.
Measure the height h with a vernier calliper or digital height gauge for better precision. To minimise the trapdoor reaction time, use an optical sensor that breaks a light beam without any mechanical delay. Repeat measurements more times and use a computer-based data logger to capture the fall time automatically.
Finally, always check that the steel ball is perfectly clean and the electromagnet surface is flat to ensure instantaneous release. Drawing a graph with h vs t² naturally averages out random timing errors.
The experimental investigation in June 2018 Insert 5 successfully demonstrates a free-fall method to determine g using simple apparatus. By plotting h against t², the gradient equals ½ g, and the value obtained is in strong agreement with the accepted 9.81 m/s².
2018 年 6 月插入材料 5 中的实验探究成功地演示了用简单器材通过自由落体测定 g 的方法。通过绘制 h-t² 图像，斜率等于 ½ g，得到的数值与公认值 9.81 m/s² 高度吻合。
Mastering the skills of identifying errors, calculating uncertainties, and suggesting valid improvements is essential for A-Level practical-based questions. Referring to Insert 5 data and applying standard techniques such as using a best-fit line, changing variables, and controlling conditions will help you achieve full marks in such investigations.
📚 Material Physics for AQA A-Level | A-Level AQA 物理：材料物理 考点精讲
Materials physics is a fundamental part of the AQA A-Level Physics course, exploring how solids respond to forces. It covers density, elastic and plastic deformation, stress-strain relationships, and energy storage in materials. Mastery of these concepts is essential for both exam success and practical understanding of material selection in engineering.
Density (ρ) is defined as mass per unit volume: ρ = m / V. The SI unit is kilogram per cubic metre (kg m⁻³). It is a material property that does not depend on the object’s size or shape.
密度（ρ）定义为每单位体积的质量：ρ = m / V。国际单位是千克每立方米（kg m⁻³）。它是一种不依赖于物体尺寸或形状的材料属性。
To find the density of a regular solid, measure its mass using a balance and calculate its volume from geometric dimensions (e.g., length × width × height for a cuboid). The density is then mass divided by calculated volume.
For an irregular solid, use the displacement method: submerge the object in a measuring cylinder partially filled with water and record the rise in water level; this gives the volume. Ensure no air bubbles are trapped. The density is then obtained.
Liquids can be measured using a pycnometer or a measuring cylinder and balance. Always record measurements with appropriate precision and consider uncertainty.
液体可使用比重瓶或量筒和天平测量。始终以适当精度记录测量值并考虑不确定度。
2. Hooke’s Law and Spring Constant | 胡克定律与劲度系数
Hooke’s law states that the extension (ΔL) of a spring or wire is directly proportional to the applied force (F), provided the elastic limit is not exceeded: F = k ΔL. The constant k is the spring constant (stiffness), with units N m⁻¹.
胡克定律指出，只要不超过弹性极限，弹簧或金属丝的伸长量（ΔL）与施加的力（F）成正比：F = k ΔL。常数 k 是劲度系数（刚度），单位为 N m⁻¹。
A graph of force against extension yields a straight line through the origin for an ideal spring obeying Hooke’s law. The gradient equals the spring constant k. The area under the graph represents work done (elastic potential energy).
When springs are combined in series, the effective spring constant k_total is given by 1/k_total = 1/k₁ + 1/k₂. In parallel, k_total = k₁ + k₂. These rules follow from sharing extension or force.
The elastic limit is the point beyond which the material no longer returns to its original length when the force is removed; permanent deformation occurs.
弹性极限是这样一个点，超过该点后，当力撤去时材料不再恢复原长，发生永久形变。
3. Stress and Strain | 应力与应变
To compare materials independently of size and shape, we use stress and strain. Tensile stress (σ) is the force applied per unit cross-sectional area: σ = F / A. Its unit is the pascal (Pa), equivalent to N m⁻².
为了不依赖于尺寸和形状来比较材料，我们使用应力和应变。拉伸应力（σ）是单位横截面积上施加的力：σ = F / A。单位是帕斯卡（Pa），即 N m⁻²。
It is conventional to use the original cross-sectional area A₀ when calculating engineering stress. Tensile strain (ε) is the extension per unit original length: ε = ΔL / L₀. Strain is a dimensionless ratio, often expressed as a decimal or percentage.
For compression, stress and strain are defined similarly but with negative signs indicating reduction in length. The definitions allow us to plot stress-strain graphs that characterise material behaviour.
Young’s modulus (E) is a measure of the stiffness of a material. It is defined as the ratio of tensile stress to tensile strain in the linear elastic region: E = σ / ε. The unit is pascal (Pa).
On a stress-strain graph, the Young’s modulus corresponds to the gradient of the straight-line portion. A steeper gradient indicates a stiffer material. For a given stress, a material with higher E experiences smaller strain.
Typical values: steel has E ≈ 2.0 × 10¹¹ Pa, copper ≈ 1.2 × 10¹¹ Pa, and glass ≈ 7.0 × 10¹⁰ Pa. The Young’s modulus is a fundamental material constant independent of the sample dimensions, as long as measurements stay in the elastic region.
5. Stress-Strain Graphs: Key Features | 应力-应变图：关键特征
A stress-strain curve for a ductile material like copper reveals several important points. The initial straight line represents the linear elastic region, obeying Hooke’s law. The limit of proportionality is where the graph deviates from linearity; beyond it, stress is no longer proportional to strain.
The elastic limit is closely linked: below it, the material returns to its original shape upon unloading. The yield point indicates the onset of significant plastic deformation;
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
📚 AS Physics Formula Derivation Guide (PH02 January 2023 Paper) | AS物理公式推导指南（PH02 2023年1月试卷）
This article delivers a systematic breakdown of essential formula derivations tested in the Edexcel International AS Physics Unit 2 (PH02) paper, with particular reference to the January 2023 sitting. Each section pairs theory with step-by-step reasoning, ensuring you can reproduce every derivation confidently and apply it to novel problem contexts. The derivations span electricity, waves, and the nature of light, which form the core of the PH02 specification.
本文系统梳理了 Edexcel 国际版 AS 物理 Unit 2 (PH02) 考试中必考的重要公式推导，并结合 2023 年 1 月试卷的命题特点进行讲解。每一节都将理论与逐步推导配对呈现，确保你能够独立复现推导过程，并灵活迁移到新的题目情境中。推导内容覆盖电学、波动和光的本质，构成 PH02 考试的核心主干。
1. Resistors in Series | 串联电阻
When resistors are connected end-to-end, the total resistance is the sum of individual resistances because the same current passes through each and the total potential difference divides across them.
当多个电阻首尾相接时，由于经过每个电阻的电流相同而总电压在各电阻上分配，总电阻等于各电阻之和。
By conservation of energy, the total p.d. V = V₁ + V₂ + V₃. Using Ohm’s law, V = IR_total, V₁ = IR₁, V₂ = IR₂, V₃ = IR₃. Substituting gives IR_total = IR₁ + IR₂ + IR₃. Since current I is common, cancelling I yields the series rule.
根据能量守恒，总电压 V = V₁ + V₂ + V₃。应用欧姆定律 V = IR_total，V₁ = IR₁，V₂ = IR₂，V₃ = IR₃。代入得 IR_total = IR₁ + IR₂ + IR₃。因为电流 I 处处相等，约去 I 即得到串联公式。
R_total = R₁ + R₂ + R₃
This derivation is frequently examined in PH02, and the January 2023 paper featured a question requiring you to combine series circuits with internal resistance of a cell.
该推导在 PH02 中常考，2023 年 1 月试卷出现了需要将串联电路与电池内阻结合的题目。
2. Resistors in Parallel | 并联电阻
For a parallel arrangement, the p.d. across each branch is identical, but the total current splits among the branches. Applying charge conservation at a junction yields the reciprocal formula.
在并联连接中，各支路两端电压相同，但总电流在各支路中分配。将电荷守恒应用于节点即可得出倒数关系。
Total current I = I₁ + I₂ + I₃. With the same p.d. V, Ohm’s law gives I = V/R_total, I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃. Substituting, V/R_total = V/R₁ + V/R₂ + V/R₃. Cancelling V (V ≠ 0) produces the familiar parallel-resistance formula.
总电流 I = I₁ + I₂ + I₃。由于各支路电压 V 相同，由欧姆定律 I = V/R_total，I₁ = V/R₁，I₂ = V/R₂，I₃ = V/R₃。代入得 V/R_total = V/R₁ + V/R₂ + V/R₃。约去非零电压 V，即得熟悉的并联电阻公式。
1/R_total = 1/R₁ + 1/R₂ + 1/R₃
The January 2023 paper asked candidates to find the combined resistance of a parallel network and then compute the current drawn from a battery, directly testing this derivation logic.
Resistance of a uniform conductor depends on its length L, cross-sectional area A, and the intrinsic property of the material called resistivity ρ. The formula is derived by assuming the conductor acts as a series of identical resistive segments.
均匀导体的电阻取决于其长度 L、横截面积 A 以及材料的本征特性电阻率 ρ。该公式的推导基于把导体视为一系列相同电阻段串联的模型。
Increasing length L adds more segments in series, so resistance is proportional to L. Increasing cross-sectional area A effectively adds more paths in parallel, so resistance is inversely proportional to A. Thus R ∝ L/A. Introducing a constant of proportionality, resistivity ρ, yields R = ρL/A.
增加长度 L 相当于串联更多电阻段，因此电阻与 L 成正比。增大横截面积 A 等效于增加并联支路，因此电阻与 A 成反比。于是 R ∝ L/A。引入比例常数电阻率 ρ，即得 R = ρL/A。
R = ρL/A
In the PH02 paper, you may be asked to determine ρ from a wire’s dimensions and measured resistance, or to explain how temperature changes affect ρ.
在 PH02 试卷中，你可能需要根据导线的尺寸和测得的电阻求 ρ，或解释温度变化如何影响 ρ。
4. Temperature Dependence of Resistance | 电阻的温度依赖性
For a metallic conductor, resistance increases with temperature because the metal ions vibrate more vigorously, causing more frequent collisions with flowing electrons. A linear approximation gives R ≈ R₀(1 + αθ), where θ is the temperature rise and α is the temperature coefficient of resistance.
对于金属导体，电阻随温度升高而增大，原因是金属离子振动加剧，导致与自由电子的碰撞频率增大。线性近似给出 R ≈ R₀(1 + αθ)，其中 θ 为温升，α 是电阻温度系数。
Starting from the resistivity form R = ρL/A, and noting that ρ changes almost linearly with temperature for many metals over a moderate range: ρ ≈ ρ₀(1 + αθ). Substituting into R gives R = (ρ₀L/A)(1 + αθ) = R₀(1 + αθ), assuming L and A are constant (thermal expansion neglected).
从电阻率形式 R = ρL/A 出发，并注意到在不太宽的温区内多数金属的 ρ 近似随温度线性变化：ρ ≈ ρ₀(1 + αθ)。代入 R 得 R = (ρ₀L/A)(1 + αθ) = R₀(1 + αθ)，假定 L 和 A 不变（忽略热膨胀）。
R ≈ R₀(1 + αθ)
January 2023 candidates needed to interpret a graph of resistance against temperature and extract α, making this derivation crucial for both theoretical and practical questions.
2023 年 1 月的考生需要解读电阻–温度图像并提取 α，因此该推导对理论和实验题都至关重要。
5. Balanced Wheatstone Bridge | 平衡惠斯通电桥
The Wheatstone bridge is balanced when the ratio of resistances in one branch equals the ratio in the other, resulting in zero current through the galvanometer. The condition is derived from equal potential at the mid-points.
Consider four resistors R₁, R₂, R₃, R₄ arranged in a quadrilateral with a galvanometer between the junction of R₁-R₂ and R₃-R₄. When balanced, no current flows through the galvanometer, so the potential at the two junctions is equal. Hence the p.d. across R₁ equals that across R₃, and p.d. across R₂ equals that across R₄. Writing I₁R₁ = I₃R₃ and I₂R₂ = I₄R₄. Since the same current flows through R₁ and R₂ (I₁ = I₂) and through R₃ and R₄ (I₃ = I₄) in the absence of galvanometer current, dividing the two equations yields R₁/R₂ = R₃/R₄.
The January 2023 paper included a practical setup where an unknown resistance was determined using a metre-bridge variant of this principle, making the derivation highly relevant.
2023 年 1 月试卷有一道实验题用到了该原理的滑线电桥变体来测定未知电阻，使得该推导非常切题。
6. The Wave Equation | 波动方程
The relationship between wave speed v, frequency f, and wavelength λ is fundamental to all wave phenomena. It is derived from the definition of speed in terms of distance travelled per unit time.
波速 v、频率 f 和波长 λ 之间的关系是所有波动现象的基础。该关系可从速度的定义（单位时间传播的距离）推出。
In one period T, a wave advances by exactly one wavelength λ. Therefore wave speed v = distance/time = λ/T. Since frequency f = 1/T, substituting gives v = fλ. This applies to both transverse and longitudinal waves, and to mechanical as well as electromagnetic waves.
在一个周期 T 内，波恰好前进一个波长 λ 的距离。因此波速 v = 距离/时间 = λ/T。因为频率 f = 1/T，代入即得 v = fλ。该式既适用于横波与纵波，也适用于机械波与电磁波。
v = fλ
In PH02 January 2023, this equation was used to interpret oscilloscope traces of sound waves and to calculate the speed of light in a double-slit experiment, so clear derivation understanding is expected.
The spacing Δy between adjacent bright (or dark) fringes on a screen, produced by two coherent point sources separated by a distance d and observed at a distance D, is derived using geometry and path-difference arguments.
由间距为 d 的两个相干点光源在相距 D 的屏上产生的相邻亮（或暗）条纹间距 Δy，可通过几何和光程差论证推出。
For a bright fringe at position yₙ from the centre, the path difference between waves from the two slits is nλ. From the geometry of the double-slit arrangement, this path difference ≈ d sinθ, and for small angles sinθ ≈ tanθ = yₙ/D. Hence d (yₙ/D) = nλ, so yₙ = nλD/d. The fringe spacing Δy = yₙ₊₁ − yₙ = (n+1)λD/d − nλD/d = λD/d.
The January 2023 paper required candidates to rearrange this expression to find λ given measured Δy, D, and d, reinforcing the importance of mastering the derivation to handle variations.
2023 年 1 月试卷要求考生重组此式，由测得的 Δy、D 和 d 求 λ，强化了掌握推导以应对变式的重要性。
8. Snell’s Law from Huygens’ Principle | 由惠更斯原理推导斯涅尔定律
Snell’s law n₁ sinθ₁ = n₂ sinθ₂ describes how light refracts at a boundary. It can be derived using Huygens’ principle, which treats each point on a wavefront as a source of secondary wavelets.
Consider a plane wavefront incident at angle θ₁ on a boundary where speed changes from v₁ to v₂. In time Δt, the wavefront at the faster side travels a distance v₁Δt while the lagging side travels v₂Δt. From the geometry of the right triangles formed, the shared hypotenuse of length L gives sinθ₁ = v₁Δt / L and sinθ₂ = v₂Δt / L. Dividing: sinθ₁/sinθ₂ = v₁/v₂. Since refractive index n = c/v, v₁/v₂ = n₂/n₁. Hence sinθ₁/sinθ₂ = n₂/n₁, or n₁ sinθ₁ = n₂ sinθ₂.
Candidates sitting the January 2023 exam needed to apply Snell’s law to fibre optics and to find critical angles, so a solid grasp of the derivation helped with multi-step problems.
9. Critical Angle and Total Internal Reflection | 临界角与全内反射
The critical angle C is the angle of incidence in an optically denser medium for which the angle of refraction is 90°. This is derived directly from Snell’s law and underpins fibre optics.
临界角 C 是光密介质中使折射角为 90° 的入射角。它由斯涅尔定律直接导出，是光纤通信的基础。
Using Snell’s law at the boundary from medium of index n₁ (denser) to n₂ (rarer, often air with n₂ = 1): n₁ sinθ₁ = n₂ sinθ₂. When θ₂ = 90°, sinθ₂ = 1, and the incident angle θ₁ becomes the critical angle C. Thus n₁ sin C = n₂ × 1, so sin C = n₂/n₁. For an interface with air, sin C = 1/n.
在光密介质（折射率 n₁）到光疏介质（折射率 n₂，通常为空气 n₂=1）的边界应用斯涅尔定律：n₁ sinθ₁ = n₂ sinθ₂。当 θ₂ = 90° 时，sinθ₂ = 1，入射角 θ₁ 即为临界角 C。因此 n₁ sin C = n₂ × 1，即 sin C = n₂/n₁。对于与空气的界面，sin C = 1/n。
sin C = 1/n
In the PH02 January 2023 paper, a diagram showed a light ray travelling through a step-index fibre, requiring use of the critical angle condition to ensure total internal reflection.
The photoelectric effect is explained by Einstein’s photon theory: each photon of frequency f carries energy hf, and a single photon interacts with one electron to eject it from a metal surface. The maximum kinetic energy of the ejected electron is derived from energy conservation.
光电效应由爱因斯坦光子理论解释：每个频率为 f 的光子携带能量 hf，一个光子与一个电子相互作用将其从金属表面击出。逸出电子的最大动能从能量守恒中导出。
When a photon hits the surface, its energy hf is used in two ways: part of it overcomes the work function φ (minimum energy to free the electron), and the remainder appears as the electron’s kinetic energy. Hence hf = φ + E_k_max. Since E_k_max = ½ m v_max² (where v_max is maximum speed), we can write hf = φ + ½ m v_max². If the photon energy is exactly equal to φ, the threshold frequency f₀ satisfies hf₀ = φ.
当光子撞击表面时，其能量 hf 被用于两方面：一部分克服逸出功 φ（释放电子的最小能量），剩余部分转化为电子动能。因此 hf = φ + E_k_max。由于 E_k_max = ½ m v_max²（v_max 为最大速率），可写为 hf = φ + ½ m v_max²。若光子能量恰好等于 φ，则极限频率 f₀ 满足 hf₀ = φ。
hf = φ + ½ m v_max²
The January 2023 paper presented a graph of maximum kinetic energy against frequency and required linking the gradient to Planck’s constant, directly testing the significance of this derivation.
11. Standing Waves on a String: Node–Antinode Spacing | 弦上的驻波：波节与波腹间距
A standing wave formed by superposition of two identical progressive waves travelling in opposite directions exhibits nodes (zero displacement) and antinodes (maximum displacement). The distance between adjacent nodes or adjacent antinodes is λ/2, derived from the waveform.
Consider a standing wave described by y = 2A sin(kx) cos(ωt). Nodes occur where sin(kx) = 0, i.e., kx = nπ, n = 0, 1, 2… With k = 2π/λ, we have (2π/λ)x = nπ ⇒ x = nλ/2. Thus the positions of successive nodes differ by Δx = (n+1)λ/2 − nλ/2 = λ/2. Antinodes lie halfway between nodes, so the same spacing applies.
In the January 2023 exam, a question showed a stationary wave on a string fixed at both ends and asked to deduce the wavelength from the number of loops, directly using this spacing rule.
12. Power in Electrical Circuits – P = IV, P = I²R, P = V²/R | 电路中的功率 – P = IV, P = I²R, P = V²/R
Electrical power is the rate at which energy is transferred by a circuit element. The three equivalent forms arise from the definitions of voltage and Ohm’s law.
电功率是电路元件传递能量的速率。三种等价形式源于电压的定义和欧姆定律。
By definition, power P = energy transferred / time = QV / t = (Q/t)V = IV, because current I = Q/t. This yields P = IV. Substituting Ohm’s law V = IR gives P = I × (IR) = I²R. Alternatively, substituting I = V/R gives P = V × (V/R) = V²/R. All three are used in PH02 for analysing circuits and heating effects.
由定义，功率 P = 转移能量/时间 = QV/t = (Q/t)V = IV，因为电流 I = Q/t。由此得 P = IV。代入欧姆定律 V = IR 得 P = I × (IR) = I²R。或者代入 I = V/R 得 P = V × (V/R) = V²/R。这三种形式在 PH02 中都用于分析电路和热效应。
P = IV = I²R = V²/R
The January 2023 paper had a circuit where a heater was connected to a source with internal resistance; candidates needed to calculate power delivered to the load and power wasted internally, making these derivations essential for efficiency questions.
📚 A-Level Physics Unit 3 Mark Scheme Jan19: Key Concepts Explained | A-Level物理 Unit 3 评分方案 2019年1月：核心概念解析
The Edexcel IAL Physics Unit 3 exam (Practical Skills in Physics I) tests your ability to design experiments, analyse data, and evaluate results. The January 2019 mark scheme reveals the precise concepts examiners look for in your answers. This article breaks down those key concepts, providing a detailed explanation to help you understand what is required to score full marks. From uncertainty calculations to graph plotting, each concept is illustrated using typical examples found in this paper.
In Unit 3, every measurement has an associated uncertainty. For a single reading using an analogue instrument, the uncertainty is ± half the smallest scale division. For digital instruments, it is ± the smallest significant digit if not specified. The mark scheme awards marks for correctly estimating and recording uncertainties in raw data. For instance, if a vernier caliper reads to 0.01 mm, the uncertainty is ±0.005 mm. A typical Jan19 question would expect you to state the uncertainty for a micrometer measurement explicitly.
When you add or subtract quantities, absolute uncertainties add: ΔA = ΔB + ΔC. For multiplication or division, percentage uncertainties add: ΔZ/Z = ΔX/X + ΔY/Y. If a quantity is raised to a power n, multiply the percentage uncertainty by n. The mark scheme expects you to show clear steps when combining uncertainties, often awarding marks for calculating percentage uncertainties and then adding them correctly. In the Jan19 paper, you may have had to find the uncertainty in the density of a wire from measurements of mass, length, and diameter.
Accurate graph plotting is essential. The mark scheme checks for correctly labelled axes with units, suitable scales using at least half the grid, and plotting points with small crosses or dots in circles. The line of best fit should be a straight line passing through the error bars if possible, with an even scatter of points on both sides. You may need to draw a worst-acceptable line to determine the uncertainty in the gradient. The gradient uncertainty is given by (gradient of worst-acceptable line – gradient of best line) / 2.
An anomalous point is one that does not follow the trend of the others. The mark scheme rewards circling or identifying such a point, and giving a valid reason, such as an error in measurement or recording. You should not include it when drawing the line of best fit. In the Jan19 mark scheme, simply stating ‘it is wrong’ is insufficient; a scientific justification like ‘likely misread the scale’ is required.
5. Evaluating Methods and Improvements | 评价方法并提出改进
A typical question asks you to criticise the experimental procedure and suggest improvements. The mark scheme looks for specific, physics-based comments. For example, ‘use a video camera to measure time more accurately’ or ‘repeat measurements and take an average’. Avoid vague statements like ‘do it better’. In the Jan19 paper, evaluations might focus on reducing parallax error, using a set square to align the ruler, or using a data logger to record rapid temperature changes.
To assess accuracy, compare your experimental result with a known standard using percentage difference: |measured – accepted| / accepted × 100%. If the percentage difference is less than a given tolerance (often 5–10%), the result is considered accurate. The mark scheme might ask you to comment on accuracy and precision separately. Precision depends on the spread of repeat readings, reflected by the uncertainty. In Jan19, you might have compared your calculated value of g with 9.81 m/s².
Students often lose marks by giving final answers with an inappropriate number of significant figures (s.f.). The rule is: the final result should have the same number of s.f. as the given data with the least s.f. In the mark scheme, answers must be given to 2 or 3 s.f. unless otherwise stated. Intermediate steps should keep extra s.f. to avoid rounding errors. For example, if a mass is 0.050 kg (2 s.f.) and a length is 1.234 m (4 s.f.), the calculated density should be reported to 2 s.f.
The mark scheme often expects you to rearrange a non-linear equation into the form y = mx + c. For instance, T = 2π√(l/g) becomes T² = (4π²/g) l, so plotting T² against l yields a straight line with gradient 4π²/g. You can then determine g from the gradient. Jan19 may have asked to linearise an equation like v² = u² + 2as and hence find the acceleration from a graph of v² against s.
9. Using Micrometer and Vernier Calipers | 使用千分尺和游标卡尺
Precisely measuring small lengths requires a micrometer screw gauge or vernier calipers. The mark scheme insists on correct technique: checking for zero error, reading to 0.01 mm for a micrometer, and combining main scale and vernier scale readings. Understanding how to correct for zero error (subtract positive error, add negative error) is crucial. In the Jan19 experiment, you might have measured the diameter of a wire multiple times and recorded the mean value.
The 6-mark planning question requires a logical method. The mark scheme assesses: independent and dependent variables, controlling other variables, detailed equipment list, step-by-step procedure, data analysis plan (including graphs and equations), and safety considerations. For Jan19, the plan might involve determining the resistivity of a wire or the Young modulus of a material. Always show the derived equation linking the variables, e.g., ρ = RA/l, and explain how the gradient of a R vs. l/A graph gives resistivity.
6分的实验设计题需要逻辑严密的方法。评分方案评估：自变量和因变量、控制其他变量、详细的器材清单、分步程序、数据分析方案（包括图表和方程）、以及安全考虑。2019年1月的设计题可能涉及测量导线电阻率或材料的杨氏模量。务必展示导出变量之间关系的方程，例如ρ = RA / l，并解释如何通过R对l / A图的斜率求出电阻率。
Published by TutorHao | Physics Revision Series | aleveler.com
Mastering practical skills is a vital part of GCSE WJEC Physics. You will be assessed on your ability to plan investigations, use apparatus safely, collect accurate data, and analyse results scientifically. This guide walks you through the essential experimental techniques, common practicals, and how to evaluate your work to achieve top marks in the practical component of your exam.
Before you start any practical, clearly define the aim and identify the independent variable (the one you change), the dependent variable (the one you measure), and the control variables (the ones you keep constant). A well-structured plan ensures you collect valid data that answers the scientific question.
Write a step-by-step method that includes the specific equipment you will use, the range and intervals of the independent variable, and how many repeats you will carry out. For example, if you are investigating how the length of a wire affects its resistance, you might decide to vary the length from 10 cm to 100 cm in steps of 10 cm, taking three readings at each length to calculate a mean.
Always consider safety: identify potential hazards (such as hot components, sharp edges, or electrical risks) and state the precautions you will take. This shows the examiner you can work responsibly.
In any investigation, controlling variables is key to obtaining reliable results. The independent variable is systematically changed, while the dependent variable is recorded. All other variables that could affect the outcome must be kept constant to make the test fair.
Common control variables in physics include temperature, mass of an object, surface area, or the voltage of a power supply. For instance, when measuring the extension of a spring, you must use the same spring throughout and avoid exceeding its elastic limit, otherwise the relationship between force and extension will change.
Use a table to identify and list the variables at the start of your plan. This helps you design a clear method and demonstrates good scientific thinking.
在计划开始时使用表格确定并列出变量。这有助于你设计清晰的方法，并展示良好的科学思维。
3. Measuring Instruments and Precision | 测量仪器与精度
Choosing the correct instrument directly affects the accuracy of your data. A ruler measures length to the nearest millimetre, while a micrometer can measure to 0.01 mm. For electrical quantities, analogue ammeters and voltmeters have different scales, and digital multimeters often offer higher precision.
Always use the instrument with the most appropriate range and sensitivity. For example, to measure the diameter of a thin wire, a micrometer screw gauge is far better than a ruler because it reduces the percentage uncertainty in your reading.
The precision of an instrument is half of the smallest division on its scale (or one division for some digital instruments). Record all measurements with the correct number of significant figures and include units.
No measurement is perfect. Random errors cause readings to be scattered around the true value, while systematic errors shift all readings in one direction. You can reduce random errors by taking multiple readings and calculating a mean, and by using instruments with greater precision.
Systematic errors are harder to spot. A zero error occurs when an instrument does not read zero when it should – for example, a force meter that reads 0.2 N when unloaded. Always check for zero errors before starting and adjust your readings accordingly, or mention them in your evaluation.
Uncertainty is typically quoted as ± half the smallest scale division. When you calculate a quantity from measured values, the uncertainty can be estimated. For instance, a ruler measurement of 2.5 cm has an absolute uncertainty of ±0.05 cm, and a percentage uncertainty of (0.05/2.5) × 100% = 2%.
A well-organised results table is essential. Draw the table with a pencil and ruler, include headings with units separated by a solidus (e.g. Length / cm, Current / A, Voltage / V). List the independent variable in the first column and the dependent variable in subsequent columns, leaving space for repeated readings and the mean.
一个组织良好的结果表格至关重要。用铅笔和直尺绘制表格，表头包括由斜线分隔的单位（例如 Length / cm, Current / A, Voltage / V）。将自变量放在第一列，因变量放在后续列，并为重复读数和平均值留出空间。
Record data directly into the table as you work, not on scraps of paper. Each reading must be taken carefully, and any anomalous result should be identified and repeated if time allows. An anomalous result is one that does not fit the overall pattern and may indicate a mistake.
Use the same number of decimal places for all readings of a given quantity, based on the instrument’s precision. For example, if using a voltmeter reading to 0.01 V, write 1.20 V rather than 1.2 V to show precision.
根据仪器的精度，对同一物理量的所有读数使用相同的小数位数。例如，如果使用读数为 0.01 V 的电压表，写 1.20 V 而不是 1.2 V 以体现精度。
6. Drawing Graphs | 绘制图表
Graphs allow you to see patterns and make predictions. Use graph paper, and label each axis with the quantity and unit. The independent variable goes on the x-axis, and the dependent variable on the y-axis. Choose scales that use at least half the graph paper in both directions and avoid awkward scales like 0.3 per small square.
图表能让你看到变化规律并进行预测。使用坐标纸，在每个轴上标注物理量和单位。自变量在 x 轴，因变量在 y 轴。选择的比例尺应使图形至少占坐标纸每方向的一半，并避免使用不方便的比例，如每小格 0.3。
Plot points with a sharp pencil as small crosses or dots, and draw either a line of best fit or a smooth curve that passes close to most points. Do not ‘join the dots’ with straight segments. The line of best fit can be a straight line if the relationship is linear, or a smooth curve if it is not.
Use the gradient of a straight-line graph to calculate a constant. For example, in a potential difference vs current graph for a fixed resistor, the gradient equals the resistance. Always show a large triangle on the graph when working out the gradient, and include units.
Once you have a graph or processed data, describe the relationship using terms like directly proportional, inversely proportional, or linear. State the pattern in words, and support it with numerical evidence from your graph or table.
If the graph is a straight line through the origin, the two variables are directly proportional. You can write: y ∝ x, or y = kx. Calculate k from the gradient. If the line does not pass through the origin, the relationship is linear but not directly proportional, and you may need to discuss a possible systematic error.
如果图形是一条通过原点的直线，则两个变量成正比。可以写作 y ∝ x 或 y = kx，并由斜率计算 k 值。如果直线不通过原点，则关系是线性但不正比，你可能需要讨论可能的系统误差。
Use mathematical relationships where appropriate. For example, Ohm’s law: R = V/I. Rearrange equations to test proportionality or to calculate unknown quantities. Show all working clearly.
Evaluation is not just about listing what went wrong; it is a critical reflection on the quality of your data and method. Discuss whether your results are repeatable and reproducible, and identify any limitations in your procedure.
Comment on the reliability of your data. If repeat readings are very close together, the data is precise. Compare your experimental value with the accepted true value where possible, and calculate percentage difference. For instance, if the true value of resistivity of a metal is given, you can assess accuracy.
Suggest specific improvements that would reduce errors, such as using a digital sensor instead of a manual stopwatch to measure time for oscillations, or controlling temperature more carefully. Always link each improvement directly to a source of error you have identified.
9. Specific Experiment: Measuring Density | 具体实验：测量密度
To find the density of a regular solid, measure its mass using a top-pan balance and its dimensions using a ruler or vernier callipers. Calculate volume using a formula (e.g. length × width × height for a cuboid). Then use ρ = m / V to compute density.
测量规则固体的密度时，用托盘天平测量质量，用直尺或游标卡尺测量尺寸。用公式计算体积（例如长方体的体积 = 长 × 宽 × 高）。然后使用 ρ = m / V 计算密度。
For an irregular solid, measure its mass, then immerse it in a measuring cylinder partly filled with water. The volume of water displaced equals the volume of the object. Take great care to read the water level at the bottom of the meniscus and to avoid trapping air bubbles.
For a liquid, place an empty measuring cylinder on a balance and zero it, pour in the liquid, and record the mass. Read the volume directly. A common error is forgetting to subtract the mass of the empty cylinder – the zero function avoids this.
10. Specific Experiment: Ohm’s Law and Resistance | 具体实验：欧姆定律与电阻
Set up a simple circuit with a battery, an ammeter in series, and a voltmeter in parallel across the component. Use a variable resistor to change the potential difference across the test component in steps. Record the corresponding current. The ratio V/I should be constant if the temperature does not change significantly.
Plot a graph of potential difference (V) against current (I). For an ohmic conductor at constant temperature, the graph is a straight line through the origin, and the resistance R equals the gradient. Include the unit ohm (Ω).
绘制电位差 (V) 对电流 (I) 的图形。对于恒温下的欧姆导体，图形是一条通过原点的直线，电阻 R 等于斜率。单位使用欧姆 (Ω)。
To investigate factors affecting resistance, you can change the length of a wire while keeping the ammeter reading constant, or measure resistance with a multimeter. Use a metre rule for length and a micrometer for diameter to calculate resistivity using R = ρL / A.
为了研究影响电阻的因素，你可以改变导线长度并保持电流表读数不变，或者使用万用表测量电阻。使用米尺测量长度，用千分尺测量直径，然后利用 R = ρL / A 计算电阻率。
11. Specific Experiment: Hooke’s Law | 具体实验：胡克定律
Suspend a spring from a clamp stand and add known masses to its hanger. Measure the extension (the increase in length from the original position) for each mass. Use a metre rule and a pointer to read the position accurately, and take care not to overload the spring beyond its elastic limit.
Plot a graph of force (F = mg) against extension (x). Within the elastic limit, the graph is a straight line through the origin, satisfying F = kx, where k is the spring constant in N/m. The gradient gives k.
绘制力 (F = mg) 对伸长量 (x) 的图形。在弹性极限内，图形是一条通过原点的直线，满足 F = kx，其中 k 是弹簧常数，单位为 N/m。斜率即为 k 值。
Anomalous points may indicate where the spring was slightly distorted or the masses were swinging. Wait for the spring to settle before taking readings, and use a fiducial marker (a small piece of tape) to help align your eye level perpendicular to the rule, reducing parallax error.
12. Specific Experiment: Waves in a Ripple Tank | 具体实验：波纹槽中的波
Using a ripple tank, you can measure the frequency and wavelength of water waves. Place a vibrating dipper in the water and illuminate the tank from above so that wave crests appear as bright lines on a screen below. Count the number of waves passing a point in a known time to find frequency.
To measure wavelength, quickly freeze the pattern with a strobe light or take a photograph. Measure the distance covered by, say, 10 wavelengths using a ruler, and divide by 10 to improve accuracy. Then apply the wave equation v = fλ.
测量波长时，用频闪灯或拍照将波形快速定格。用直尺测量例如 10 个波长所覆盖的距离，再除以 10 以提高精度。然后应用波动方程 v = fλ。
Controlling the depth of the water is crucial, as wave speed changes in shallower water. Keep the water depth uniform by placing the tank on a level surface. Also, avoid reflections from the sides interfering with the pattern by using absorbing materials or damping the edges.
In A2 Physics, many concepts appear similar but have distinct meanings and applications. Understanding these subtle differences is essential for exam success and deeper comprehension. This article clarifies some of the most commonly confused pairs in electricity, magnetism, mechanics, waves, thermal physics, and modern physics.
1. Electric Field Strength vs. Electric Potential | 电场强度与电势
Electric field strength (E) is a vector quantity that measures the force per unit positive charge at a point: E = F/q. It points in the direction of the force on a positive test charge and has units N C⁻¹ or V m⁻¹. Electric potential (V) is a scalar quantity that represents the work done per unit charge to bring a positive test charge from infinity to that point: V = W/q, measured in volts (J C⁻¹).
电场强度（E）是矢量，衡量单位正电荷在某点所受的力：E = F/q，方向与正电荷受力方向相同，单位是 N C⁻¹ 或 V m⁻¹。电势（V）是标量，表示将单位正电荷从无穷远处移到该点所做的功：V = W/q，单位是伏特（J C⁻¹）。
While E is related to the spatial rate of change of potential (E = -dV/dr in a radial field), they describe different aspects: E quantifies the force environment, whereas V quantifies the energy landscape. In a uniform field, E = V/d, linking them directly. In a radial field, E = kQ/r² and V = kQ/r, showing E decreases more rapidly.
2. Magnetic Flux Density vs. Magnetic Flux | 磁通量密度与磁通量
Magnetic flux density (B) is a vector field that describes the strength and direction of a magnetic field at a point. It is defined by the force on a moving charge: F = qvB sin θ, with units tesla (T). Magnetic flux (Φ) is a scalar quantity representing the total magnetic field passing through a given area: Φ = BA cos θ, measured in webers (Wb).
磁通量密度（B）是矢量场，描述某点磁场的强弱和方向。由运动电荷受力定义：F = qvB sin θ，单位特斯拉（T）。磁通量（Φ）是标量，表示穿过某一给定面积的总磁场：Φ = BA cos θ，单位韦伯（Wb）。
B determines the local magnetic influence and is independent of area, while Φ depends on both B and the orientation of the area. In electromagnetic induction, the induced emf is proportional to the rate of change of Φ (Faraday’s law), not directly to B. A uniform B can produce zero net flux if oriented parallel to the surface (θ = 90°, cos 90° = 0).
Capacitance (C) is a property of a system that quantifies its ability to store electric charge per unit potential difference: C = Q/V. The SI unit is the farad (F). A capacitor is a physical device designed to have a specific capacitance, consisting of two conductors separated by an insulator (dielectric). The capacitance depends on geometry and dielectric material: for a parallel-plate capacitor, C = ε₀εᵣ A/d.
While a capacitor is a component, capacitance is its rated value. The energy stored in a capacitor is given by E = ½ CV² = ½ QV = ½ Q²/C. In circuits, capacitors can be combined in series or parallel, affecting the total capacitance (series: 1/Cₜₒₜ = Σ 1/Cᵢ; parallel: Cₜₒₜ = Σ Cᵢ).
电容器是元件，而电容是它的额定值。电容器储存的能量由E = ½ CV² = ½ QV = ½ Q²/C给出。在电路中，电容器可串联或并联，影响总电容（串联：1/Cₜₒₜ = Σ 1/Cᵢ；并联：Cₜₒₜ = Σ Cᵢ）。
Aspect
Capacitance
Capacitor
What it is
Physical quantity
Electronic component
Symbol
C
—
Unit
Farad (F)
Rated in farads
Depends on
Geometry & dielectric
Design & materials
4. Work Function vs. Threshold Frequency | 逸出功与阈频率（光电效应）
In the photoelectric effect, the work function (φ) is the minimum energy required to remove an electron from the surface of a metal. It is a material property measured in joules (J) or electronvolts (eV). The threshold frequency (f₀) is the minimum frequency of incident light needed to eject an electron: hf₀ = φ. Photons with frequency below f₀ cannot cause emission, regardless of intensity.
Work function is an energy threshold; threshold frequency is a frequency threshold directly proportional to φ. They are linked by the Planck constant: φ = h f₀. The stopping potential (Vₛ) relates to the maximum kinetic energy: eVₛ = hf – φ. Understanding the distinction helps explain why ultraviolet light can emit electrons from zinc while intense red light cannot.
逸出功是能量阈值；阈频率是频率阈值，与φ成正比。由普朗克常数联系：φ = h f₀。遏止电势（Vₛ）与最大动能相关：eVₛ = hf – φ。理解这一区别有助于解释为什么紫外光能使锌发射电子，而强红光却不能。
Feature
Work Function (φ)
Threshold Frequency (f₀)
Nature
Energy
Frequency
Units
J or eV
Hz
Relation
φ = h f₀
Depends on
Metal type
Metal type (via φ)
5. Temperature vs. Heat | 温度与热量
Temperature (T) is a measure of the average random kinetic energy of particles in a substance. It is a scalar quantity measured in kelvin (K) or degrees Celsius (°C). Heat (Q) is the transfer of energy from a region of higher temperature to a region of lower temperature due to a temperature difference. Heat is measured in joules (J) and is not a property stored in an object.
An object does not ‘contain’ heat; it has internal energy. When heat flows, the temperature may change according to Q = mcΔθ, where m is mass and c is specific heat capacity. During a phase change, heat transfer occurs without a temperature change (latent heat: Q = mL). Confusing temperature and heat leads to misconceptions about thermal equilibrium and energy transfer.
Force (F) is an interaction that changes the motion of an object, given by Newton’s second law: F = dp/dt. It is a vector with unit newton (N). Impulse (J) is the change in momentum resulting from a force acting over a time interval: J = F Δt = Δp. Impulse equals the area under a force–time graph.
力（F）是改变物体运动的相互作用，由牛顿第二定律给出：F = dp/dt。它是矢量，单位牛顿（N）。冲量（J）是力在一段时间内作用导致的动量变化：J = F Δt = Δp。冲量等于力-时间图下的面积。
While force describes the instantaneous interaction, impulse describes the cumulative effect over time. In collisions, a large force acting for a short time can produce the same impulse as a smaller force over a longer time. This is the principle behind airbags and crumple zones – increasing Δt reduces the average force for the same Δp.
Momentum (p) is a vector quantity defined as p = m v, measured in kg m s⁻¹. Kinetic energy (Eₖ) is a scalar quantity representing the energy of motion: Eₖ = ½ m v², with unit joule (J). Both depend on mass and velocity, but momentum is linear in v while kinetic energy is quadratic, leading to different conservation properties.
动量（p）是矢量，定义为 p = m v，单位 kg m s⁻¹。动能（Eₖ）是标量，代表运动能量：Eₖ = ½ m v²，单位焦耳（J）。两者都取决于质量和速度，但动量与v成正比，动能与v²成正比，导致不同的守恒性质。
In an isolated system, momentum is always conserved (vector sum constant). Kinetic energy is conserved only in perfectly elastic collisions; in inelastic collisions, some Eₖ is converted to other forms. This distinction is critical when analysing collisions, explosions, and particle interactions.
Relationship: p = √(2m Eₖ) and Eₖ = p²/(2m). Thus, a small percentage change in speed produces a larger percentage change in kinetic energy than in momentum.
Nuclear physics is a cornerstone of the CCEA A-Level Physics specification, bridging the microscopic world of the nucleus with macroscopic phenomena like radioactivity and nuclear energy. This article distills the essential concepts that every student must master, from the structure of the atom and the nature of radioactive decay to the principles of nuclear fission and fusion. We will walk through key definitions, decay laws, mass–energy equivalence, binding energy, and practical applications, ensuring clarity and depth for exam success.
The atom consists of a small, dense, positively charged nucleus surrounded by negatively charged electrons in discrete energy levels. The nucleus contains protons and neutrons (collectively called nucleons), held together by the strong nuclear force. Rutherford’s alpha-particle scattering experiment provided evidence for this nuclear model, as most alpha particles passed straight through a thin gold foil, but a small fraction were deflected through large angles, indicating a concentrated positive charge at the centre.
The atomic number Z equals the number of protons, defining the element. The mass number A is the total number of nucleons (protons + neutrons). Isotopes are atoms of the same element (same Z) with different numbers of neutrons (different A). The notation for a nuclide is ᴬZX, e.g. ¹⁴₆C for carbon-14.
原子序数 Z 等于质子数，决定了元素的种类。质量数 A 是核子总数（质子 + 中子）。同位素是指质子数相同而中子数不同的原子，即同一元素但质量数不同。核素符号记作 ᴬZX，例如碳-14 表示为 ¹⁴₆C。
2. Radioactive Decay and Types of Radiation | 放射性衰变与辐射类型
Unstable nuclei spontaneously emit radiation to become more stable. The three main types are alpha (α), beta (β), and gamma (γ) radiation. An alpha particle is a helium nucleus (⁴₂He), consisting of two protons and two neutrons; it is highly ionising but has low penetrating power, stopped by a sheet of paper. Beta-minus (β⁻) decay occurs when a neutron turns into a proton, emitting an electron (⁰₋₁e) and an antineutrino; beta particles are moderately ionising and can be stopped by a few millimetres of aluminium. Gamma radiation is electromagnetic radiation of very short wavelength, emitted from an excited nucleus; it is weakly ionising but highly penetrating, requiring thick lead or concrete to reduce intensity.
The activity A of a radioactive source is the number of decays per unit time, measured in becquerels (Bq), where 1 Bq = 1 decay per second. The decay constant λ is the probability per unit time that a given nucleus will decay. The equation A = λN links activity, decay constant, and the number N of undecayed nuclei.
放射源的活度 A 定义为单位时间内发生的衰变次数，单位是贝克勒尔（Bq），1 Bq = 每秒 1 次衰变。衰变常量 λ 表示单个原子核在单位时间内发生衰变的概率。方程 A = λN 将活度、衰变常量与尚未衰变的核数目 N 联系起来。
3. Exponential Decay Law and Half-Life | 指数衰变规律与半衰期
Radioactive decay is a random process, and the number of undecayed nuclei N decreases exponentially with time: N = N₀ e^(−λt). Here N₀ is the initial number of nuclei. This exponential law can also be expressed in terms of activity A = A₀ e^(−λt) or count rate, provided background radiation is subtracted and the same geometry is maintained. The half-life T₁/₂ is the time taken for the number of undecayed nuclei (or the activity) to halve. It is related to the decay constant by T₁/₂ = ln 2 / λ ≈ 0.693 / λ.
Students often need to determine half-life from a graph of activity vs time, or use the decay equation to calculate remaining activity after a certain number of half-lives. For example, after n half-lives, the fraction remaining is (½)ⁿ.
学生经常需要根据活度-时间图读取半衰期，或者利用衰变方程计算经过若干个半衰期后的剩余活度。例如，经过 n 个半衰期，剩余比例为 (½)ⁿ。
4. Nuclear Stability and the N–Z Curve | 原子核稳定性与 N–Z 曲线
Stable nuclei tend to lie along a narrow band on a plot of neutron number N against proton number Z. For light nuclei, N ≈ Z; for heavier nuclei, N > Z, because extra neutrons are needed to counteract the increasing electrostatic repulsion between protons. The N–Z curve shows that nuclides lying above the stability band (excess neutrons) typically undergo β⁻ decay to convert a neutron into a proton, moving diagonally down and right on the chart. Nuclides below the stability band (excess protons) may undergo β⁺ (positron) emission or electron capture, moving diagonally up and left. Very heavy nuclei (Z > 83) are all unstable and tend to undergo α decay.
在以中子数 N 对质子数 Z 作图得到的 N–Z 曲线上，稳定核素分布在一条狭窄的带内。轻核的 N ≈ Z；较重核的 N > Z，因为需要额外的中子来抵消质子间不断增大的静电斥力。位于稳定带上方（中子过多）的核素通常发生 β⁻ 衰变，将一个中子转变为质子，图上表现为向右下方移动。位于稳定带下方（质子过多）的核素可能发生 β⁺（正电子）发射或电子俘获，向左上方移动。极重的核（Z > 83）全都不稳定，倾向于发生 α 衰变。
The strong nuclear force is responsible for binding nucleons together. It is a short-range attractive force that acts between all nucleons (protons and protons, neutrons and neutrons, protons and neutrons). At very short ranges it becomes repulsive, preventing the nucleus from collapsing. This force is much stronger than the electrostatic repulsion between protons at distances around 1 fm (10⁻¹⁵ m), which explains the stability of nuclei.
5. Mass–Energy Equivalence and Atomic Mass Unit | 质能等价与原子质量单位
Einstein’s famous equation E = mc² expresses the equivalence of mass and energy. In nuclear physics, mass is often measured in atomic mass units (u). 1 u is defined as 1/12 of the mass of a carbon-12 atom, approximately 1.66 × 10⁻²⁷ kg. The energy equivalent of 1 u is 931.5 MeV (megaelectronvolts), a conversion factor that is used extensively to calculate energy released in nuclear reactions.
爱因斯坦著名的方程 E = mc² 表达了质量和能量的等价关系。在核物理中，质量常用原子质量单位 (u) 来计量。1 u 定义为碳-12 原子质量的 1/12，约等于 1.66 × 10⁻²⁷ kg。1 u 的能量当量为 931.5 MeV（兆电子伏特），这是一个广泛用于计算核反应释放能量的转换因子。
When a nucleus is formed from its constituent protons and neutrons, the total mass of the nucleus is found to be slightly less than the sum of the masses of the individual nucleons. This difference is called the mass defect Δm. Therefore, Δm = Z mₚ + (A − Z) mₙ − M_nucleus, where mₚ is the proton mass, mₙ is the neutron mass, and M_nucleus is the actual nuclear mass. In calculations, it is often convenient to use atomic masses (including electrons) rather than bare nuclear masses.
当一个原子核由其组成的质子和中子形成时，原子核的总质量略小于各个核子的质量之和。这一差值称为质量亏损 Δm 。因此，Δm = Z mₚ + (A − Z) mₙ − M_nucleus ，其中 mₚ 为质子质量，mₙ 为中子质量，M_nucleus 为原子核的真实质量。在计算中，通常使用原子质量（包含电子）而不用裸核质量更为方便。
6. Binding Energy and Binding Energy per Nucleon | 结合能与平均结合能
The binding energy E_b of a nucleus is the energy equivalent of the mass defect: E_b = Δm c². It represents the energy required to separate a nucleus completely into its individual protons and neutrons. A higher binding energy indicates a more stable nucleus. To compare stability across nuclei, we use the binding energy per nucleon: E_b / A. A graph of binding energy per nucleon against mass number A is a fundamental curve in nuclear physics. It rises steeply for light nuclei, peaks around iron-56 (Fe) with about 8.8 MeV per nucleon, and then gradually decreases for heavier nuclei.
原子核的结合能 E_b 是质量亏损对应的能量：E_b = Δm c²。它表示将原子核完全分解为质子和中子所需的能量。结合能越大，原子核越稳定。为了比较不同原子核的稳定性，我们使用平均结合能：E_b / A 。平均结合能对质量数 A 的曲线是核物理中一条基础曲线。对轻核，平均结合能迅速上升，在铁-56（Fe）附近达到峰值，约每核子 8.8 MeV，之后随着质量数增大而逐渐下降。
This curve explains why energy can be released in both fission (splitting heavy nuclei into medium-mass nuclei) and fusion (combining light nuclei into heavier ones up to iron). In both processes, the products have a higher binding energy per nucleon than the reactants, meaning that mass is converted into energy according to ΔE = Δm c².
Nuclear fission is the splitting of a heavy nucleus into two smaller nuclei of roughly equal mass, accompanied by the release of a large amount of energy and typically two or three neutrons. A common example is the fission of uranium-235 when it captures a slow thermal neutron: ²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n + energy. The energy released (about 200 MeV per fission) comes from the difference in binding energy per nucleon between the heavy parent nucleus and the daughter products.
The neutrons released can induce further fission events, leading to a chain reaction. In a controlled chain reaction, such as in a nuclear reactor, control rods (often boron or cadmium) absorb excess neutrons to keep the multiplication factor k exactly 1, providing steady heat output. If k > 1, the reaction is supercritical and can lead to a rapid release of energy, as in a nuclear weapon. A moderator (e.g., water or graphite) slows down the fast fission neutrons to thermal speeds, greatly increasing the probability of inducing further fission in ²³⁵U.
释放出的中子可以引发更多的裂变事件，形成链式反应。在受控链式反应中，例如核反应堆里，控制棒（常由硼或镉制成）吸收多余的中子，使倍增系数 k 精确保持在 1，从而提供稳定的热量输出。如果 k > 1，反应处于超临界状态，可能导致能量的急剧释放，如同核武器中那样。慢化剂（如水或石墨）将快裂变中子减速至热速度，从而大幅提高它们引发 ²³⁵U 裂变的概率。
8. Nuclear Fusion and Stellar Energy | 核聚变与恒星能源
Nuclear fusion is the process in which two light atomic nuclei combine to form a heavier nucleus, releasing a substantial amount of energy because the product has a higher binding energy per nucleon than the reactants. For example, in the Sun’s core, the proton-proton chain fuses hydrogen into helium: 4 ¹₁H → ⁴₂He + 2 e⁺ + 2 νₑ + energy. Approximately 26.7 MeV is released per helium nucleus formed. Fusion requires extremely high temperatures (on the order of 10⁸ K) to overcome the Coulomb repulsion between the positively charged nuclei, allowing them to approach close enough for the strong nuclear force to act.
Fusion is the energy source of stars. On Earth, controlled fusion for power generation has not yet been achieved commercially due to the difficulty of confining a plasma at temperatures of hundreds of millions of degrees. Current research focuses on magnetic confinement (tokamaks) and inertial confinement. Fusion offers the advantages of abundant fuel (deuterium from seawater) and reduced long-lived radioactive waste compared with fission.
9. Radiation Detection, Background Radiation, and Radiation Hazards | 辐射探测、本底辐射与辐射危害
Ionising radiation can be detected using devices such as the Geiger-Müller (GM) tube, which produces an electrical pulse for each ionising particle that enters the tube, allowing the count rate to be measured. Background radiation is always present from natural sources (cosmic rays, radon gas, rocks) and artificial sources (medical uses, fallout). When conducting experiments, the background count rate must be subtracted to obtain the corrected count rate for a source.
Radiation can damage living tissue. Alpha particles are highly damaging if ingested or inhaled because of their strong ionisation within a short range, but external exposure is less hazardous since they are stopped by dead skin cells. Beta and gamma radiation can penetrate the body and damage cells, increasing the risk of cancer. Safety precautions include minimising exposure time, maximising distance from a source, and using appropriate shielding. The dose equivalent is measured in sieverts (Sv), quantifying the biological effect of radiation.
10. Radioactive Tracers and Applications | 放射性示踪剂及其应用
Radioisotopes are widely used in medicine, industry, and research. In medical imaging, short-lived gamma-emitting isotopes such as technetium-99m (half-life 6 hours) are injected into the body as tracers to observe organ function. The gamma camera detects the emitted radiation to form an image. The short half-life and the fact that it emits gamma rays (which can escape the body) make Tc-99m particularly suitable. In radiotherapy, targeted doses of radiation are used to destroy cancerous cells.
Industrial applications include thickness gauging, where beta particles are used to monitor the thickness of paper or plastic films during production by measuring the attenuation of radiation. Radioactive tracers can also map leaks in pipelines or track the flow of fluids. Carbon-14 dating uses the decay of ¹⁴C to determine the age of organic materials up to about 50 000 years, based on the known half-life of 5730 years.
11. Nuclear Equations and Conservation Laws | 核方程与守恒定律
In all nuclear reactions and decays, several quantities must be conserved: total mass–energy, electric charge, and nucleon number (mass number). These conservation laws allow us to balance and predict the products of nuclear reactions. For example, in alpha decay: ᴬZX → ᴬ⁻⁴Z₋₂Y + ⁴₂He. In beta-minus decay: ᴬZX → ᴬZ₊₁Y + ⁰₋₁e + ν̄ₑ (antineutrino). Notice how charge and mass number balance on both sides.
When writing nuclear equations, it is vital to clearly indicate the mass number (superscript) and atomic number (subscript) for each species. Electron (beta particle) is ⁰₋₁e; positron is ⁰₊₁e; neutron is ¹₀n; proton is ¹₁p (or ¹₁H). The neutrino (ν) and antineutrino (ν̄) carry no charge and negligible mass, but they carry energy and momentum, ensuring conservation of lepton number in weak interactions.
12. Radioactive Decay Calculations and Data Handling | 放射性衰变的计算与数据处理
Typical exam questions ask students to calculate the number of nuclei remaining after a given time, determine the half-life from data, or find the decay constant. For example, given initial activity 800 Bq and half-life 2 hours, after 6 hours (three half-lives) the activity is 800 × (½)³ = 100 Bq. Alternatively, using N = N₀ e^(−λt), first find λ = 0.693 / T₁/₂. If T₁/₂ = 2 hours, convert to seconds if necessary. These calculations require careful unit handling and often involve logarithms for more complex situations.
Students should also be comfortable interpreting exponential decay graphs and using the concept of half-life to estimate ages in radiometric dating. The key relationship linking mass defect, binding energy, and energy released in a nuclear reaction is ΔE = Δm c², where Δm is the mass difference between reactants and products, usually expressed in u and converted to MeV using 1 u = 931.5 MeV.
学生还应能够解读指数衰变曲线，并运用半衰期的概念来估算放射性年代。联系质量亏损、结合能和核反应释放能量的关键关系是 ΔE = Δm c²，其中 Δm 是反应物与产物之间的质量差，通常以 u 表示，并利用 1 u = 931.5 MeV 转换为 MeV。
Published by TutorHao | Physics Revision Series | aleveler.com
📚 Common Misconceptions in IGCSE Physics | IGCSE 物理常见误区
Many IGCSE Physics students lose marks not because they lack knowledge, but because they hold onto common misconceptions. These misunderstandings often arise from everyday language or simplified early models and can persist into the exam. This article highlights the most frequent errors and provides clear, examiner-friendly explanations to help you avoid them. Mastering the correct concepts will strengthen your understanding and boost your confidence in Paper 2, Paper 4, and the practical paper.
许多 IGCSE 物理学生丢分并不是因为缺乏知识，而是因为他们固守着一些常见的误区。这些误解往往源于日常语言或早期简化模型，并可能一直延续到考试中。本文重点剖析最常见的错误，并提供清晰、符合考官视角的解释，帮助你避开这些坑。掌握正确的概念将加深你的理解，并增强你在 Paper 2、Paper 4 和实验考试中的信心。
1. Mass and Weight Are Not the Same | 质量和重量不是一回事
Students often use ‘mass’ and ‘weight’ as if they mean the same thing. In physics, mass is a scalar quantity that measures the amount of matter in an object; it is measured in kilograms (kg) and does not change with location. Weight is a force, specifically the gravitational pull on an object, and is measured in newtons (N). Confusing the two can lead to errors in calculations involving gravitational field strength g.
学生经常把“质量”和“重量”混为一谈。在物理学中，质量是标量，衡量物体所含物质的多少，单位是千克（kg），且不随位置改变。重量是一种力，确切地说是作用在物体上的引力，单位是牛顿（N）。混淆二者会导致在涉及引力场强度 g 的计算中出现错误。
The relationship is given by the equation:
W = m × g
On Earth, g ≈ 10 N/kg, so an object with a mass of 60 kg has a weight of 600 N. If you take the same object to the Moon, where g ≈ 1.6 N/kg, its weight becomes 96 N, but its mass stays exactly 60 kg. A common exam mistake is to say the mass ‘got lighter’ on the Moon—never do that.
关系式如下：
W = m × g
在地球上，g ≈ 10 N/kg，所以一个质量为 60 kg 的物体重量为 600 N。如果你把同一个物体带到月球上，那里 g ≈ 1.6 N/kg，它的重量变为 96 N，但质量仍然是 60 kg。一个常见的考试错误是声称物体在月球上“质量变轻了”——千万不要这么说。
Remember: weight is a vector and always points towards the centre of the planet; mass is just a number representing how much ‘stuff’ there is. When answering questions about astronauts floating in space, clarify that they are weightless due to free fall, not because mass disappears.
Many learners treat speed and velocity as interchangeable. Speed is a scalar—it tells you how fast something moves but not in which direction. Velocity is a vector—it includes both magnitude and direction. In IGCSE questions, a car going round a roundabout at constant speed is accelerating because its direction, and hence its velocity, changes continuously.
Acceleration is defined as the rate of change of velocity, not speed. If velocity is constant, acceleration is zero. But if speed is constant while direction changes, there is acceleration towards the centre of the circle—centripetal acceleration. Students often wrongly think that ‘constant speed means no acceleration’, which loses marks on circular motion and satellite orbit topics.
For straight-line motion, speed and the magnitude of velocity are the same. The formula v = s / t works for both if direction is ignored. But always check whether the question asks for speed or velocity—if it asks for velocity, you must state the direction as well (e.g. 5 m/s north).
对于直线运动，速率和速度的大小是一样的。如果忽略方向，公式 v = s / t 对两者都适用。但一定要看清楚题目问的是速率还是速度——如果问的是速度，你必须同时给出方向（例如 5 m/s 向北）。
3. A Constant Force Does Not Give Constant Velocity | 恒定的力不会产生恒定的速度
This is perhaps the most deeply rooted misconception, dating back to Aristotle. Everyday experience with friction tricks us into believing that a constant push is needed to keep something moving at constant speed. Newton’s first law states that if resultant force is zero, a body remains at rest or moves with constant velocity. Constant velocity requires zero net force, not a constant force.
When a constant resultant force is applied, Newton’s second law applies:
F = m × a
A constant net force produces constant acceleration, meaning the velocity changes by the same amount every second. For example, if you push a shopping trolley on a frictionless surface with a steady force, it will speed up continuously, not settle at a fixed speed.
In real life, as speed increases, air resistance builds up until it balances the driving force, resulting in zero net force and thus constant (terminal) velocity. Students should distinguish between the driving force and the resultant force: a constant driving force can eventually lead to constant velocity only if opposing forces increase with speed to cancel it out.
4. Current Is Not Used Up in a Circuit | 电流在电路中不会被消耗
A widespread mistake is believing that electric current is ‘consumed’ by components like bulbs or resistors. Students often draw diagrams where the current leaving a bulb is smaller than the current entering it. In reality, charge is conserved. In a series circuit, the current is the same at every point; an ammeter placed anywhere in the loop will read the same value.
What gets used up or transferred is energy, not current. The bulb converts electrical energy into light and heat, making the potential difference drop across it. The current carries the energy, but the number of charges per second (the current) stays constant. Think of water in a pipe: the flow rate (current) does not decrease after passing a water wheel, even though energy is extracted.
In parallel circuits, the current splits at junctions, but the total current entering a junction equals the total current leaving it (Kirchhoff’s first law). A common exam error is to say ‘current is shared equally’—this is only true if all parallel branches have identical resistance.
5. Heat and Temperature Are Not the Same | 热量和温度不是一回事
In everyday language we say ‘a hot object contains a lot of heat’, but this is physically incorrect. Temperature is a measure of the average kinetic energy of particles; it is not energy itself. Heat, in physics, refers to the transfer of thermal energy from a hotter body to a colder one. An object does not ‘contain’ heat—it contains internal energy.
Two objects at the same temperature can have very different amounts of internal energy depending on their mass and material. For example, a swimming pool at 25 °C has far more internal energy than a cup of coffee at 80 °C, but the coffee is at a higher temperature. Saying the pool has ‘more heat’ is misleading.
两个温度相同的物体，其内能的多少可能因质量和材料的不同而差异巨大。例如，一个 25 °C 的游泳池所含的内能远比一杯 80 °C 的咖啡要多，但咖啡的温度更高。笼统地说游泳池有“更多的热量”会带来误导。
During phase changes (e.g., melting or boiling), the temperature remains constant even though energy is being supplied. The supplied energy breaks bonds rather than raising kinetic energy. Students often forget this and incorrectly state that temperature must rise while ice melts.
6. Energy Cannot Be Destroyed, Only Transferred or Transformed | 能量不能被消灭，只能被转移或转化
Phrases like ‘energy is used up’ are common in conversation, but they contradict the principle of conservation of energy. Energy is never created or destroyed; it changes from one form to another. When a battery runs flat, its chemical energy has not vanished—it has been transferred into light, heat and kinetic energy in the circuit and surroundings.
The total energy in a closed system remains constant. In efficiency calculations, the ‘lost’ energy is often dissipated as thermal energy, raising the temperature of the surroundings. The phrase ‘energy is lost’ should be avoided in exam answers unless you specify it means transferred to the thermal store of the environment.
When a ball bounces, it does not reach its original height because some kinetic energy is converted to thermal energy during the impact. Students may think energy ‘disappears’, but it merely spreads out and becomes less useful. This relates to the idea of energy dissipation and resource efficiency.
7. Action and Reaction Forces Act on Different Bodies | 作用力与反作用力作用在不同物体上
Newton’s third law states that if object A exerts a force on object B, then object B exerts an equal and opposite force on object A. The critical detail that many students miss is that these two forces act on different objects. A common exam trap is to ask whether the weight of a book and the normal force from the table are an action-reaction pair.
牛顿第三定律说：如果物体 A 对物体 B 施加一个力，那么物体 B 会同时对物体 A 施加一个大小相等、方向相反的力。许多学生遗漏的关键细节是：这两个力作用在不同的物体上。一个常见的考试陷阱就是问：书的重力和桌子的支持力是不是一对作用力与反作用力。
The weight of a book is the gravitational pull by Earth on the book. The reaction to this weight is the gravitational pull by the book on Earth—not the normal force from the table. The normal force and the weight that act on the same book are a pair of balanced forces if the book is stationary, but they are not an action-reaction pair.
To identify action-reaction pairs, use the phrase: ‘A exerts a force on B, B exerts a force on A’. Check that the forces are of the same type (e.g. both gravitational, both contact) and act on different bodies. A rocket moving forward is a classic example: the rocket pushes exhaust gases backwards, and the gases push the rocket forwards.
识别作用力与反作用力时，可以用这样的表述：“A 对 B 施加一个力，B 对 A 施加一个力”。检查这两个力是否属于同一类型（例如都是引力，或者都是接触力），并且作用在不同物体上。向前飞行的火箭就是一个经典例子：火箭向后推出燃气，燃气则向前推动火箭。
8. Refraction Is Caused by a Change in Speed | 折射是由速度变化引起的
When light enters a denser medium, it slows down. If it hits the boundary at an angle, this change in speed causes the light to change direction—this is refraction. Many students mistakenly believe that light bends towards the normal simply because the medium is ‘thicker’ or that the ray ‘prefers’ the normal. The correct reasoning connects bending directly to the change in wave speed.
A useful memory rule: if the speed decreases (entering denser medium), the ray bends towards the normal; if speed increases (entering less dense medium), it bends away from the normal. When light enters glass from air, its speed drops from 3.0 × 10⁸ m/s to about 2.0 × 10⁸ m/s, and the ray bends towards the normal.
一个有用的记忆规则：如果速度减小（进入光密介质），光线向法线偏折；如果速度增大（进入光疏介质），光线远离法线偏折。当光从空气进入玻璃时，速度从 3.0 × 10⁸ m/s 降低到约 2.0 × 10⁸ m/s，光线便向法线偏折。
Refraction explains apparent depth illusions: a swimming pool looks shallower than it really is because light from the bottom speeds up as it leaves the water, bending away from the normal. Students sometimes incorrectly attribute this to reflection or scattering. Always draw a clear ray diagram with arrows and label the normal.
9. Floating and Sinking Depend on Density, Not Just Weight | 浮沉取决于密度，而不只是重量
A heavy ship floats while a small pebble sinks—this contradicts the intuition that ‘heavier things sink’. Whether an object floats or sinks depends on its average density compared to the fluid. If the object’s density is less than that of the liquid, it floats; if greater, it sinks. The large steel ship floats because its overall volume includes a lot of air, making the average density less than water.
Archimedes’ principle states that the upthrust on an object equals the weight of fluid displaced. When an object is fully submerged, the upthrust depends only on the volume of displaced fluid and the fluid’s density. Whether the object then sinks or rises depends on whether this upthrust is less than or greater than the object’s weight.
A common exam answer to ‘what makes an object float?’ is simply ‘upthrust equals weight’. That is true only when the object is floating (equilibrium). For a submerged object that is rising, upthrust is greater than weight. Avoid mixing up the conditions for floating versus rising.
10. Series and Parallel Circuit Misconceptions | 串联和并联电路误区
When a cell is connected to bulbs in series, many students expect the bulb nearest the positive terminal to be brighter because ‘it gets the current first’. In a series circuit, the rate of charge flow is established almost instantaneously everywhere, and the current is identical through all components. Brightness differences in series are due to resistance differences, not position.
Adding more resistors in series increases the total resistance, which reduces the current from the cell. In parallel, adding an extra branch provides an additional path for current, and the total resistance decreases, so the current drawn from the cell increases. Students often reverse these effects, thinking that parallel circuits always have larger resistance.
Voltage behaves differently: in a series circuit, the supply voltage is shared between components (potential divider); in a parallel circuit, each branch receives the full supply voltage. A classic error is to say that ‘voltage is the same in all series components’—that is only true for current. Always use V, I, R relationships carefully.
📚 AS Physics: Essay Writing Template | AS 物理：论文写作模板
AS Physics exams often require extended essay responses that test not only your knowledge but also your ability to structure arguments, apply concepts, and communicate clearly. A reliable template can help you organize your thoughts and meet the marking criteria. This guide provides a step‑by‑step template for tackling any AS Physics essay, from describing experiments to evaluating conclusions.
AS 物理考试经常要求撰写扩展论文式答案，不仅考查知识掌握，还考查论证结构、概念应用和清晰表达的能力。一个可靠的模板能帮助你组织思路并满足评分标准。本指南提供逐步模板，适用于从描述实验到评价结论的各种 AS 物理论文题。
1. Understanding Keywords in Questions | 理解问题中的关键词
Before you write, highlight the command words: ‘describe’ means state what happens without explanation; ‘explain’ requires you to link cause and effect using physics principles; ‘compare’ demands similarities and differences; ‘evaluate’ asks you to weigh evidence and state a judgment. Misinterpreting these can cost marks even if your physics is correct.
Always check if the question specifies a practical context, such as ‘In an experiment to determine the Young modulus…’. Your essay must then include relevant apparatus, measurement techniques, and safety considerations. Many marks are allocated to structuring the answer around that context.
Spend 3–5 minutes creating a skeleton: list the key physics concepts, the logical sequence from observation to conclusion, and any equations or diagrams you will include. A clear plan prevents waffle and keeps you focused on the markscheme expectations.
For example, if the essay asks you to explain terminal velocity, your plan might be: (1) forces acting – weight and drag, (2) net force and acceleration, (3) drag increasing with speed, (4) condition for terminal velocity, (5) graph of velocity vs time. This five‑point structure directly maps to the mark allocation.
Your introduction should define the topic and set the scope. Avoid repeating the question; instead, rephrase it to show understanding. State the key variables and the overarching physics relationship you will explore.
For instance, ‘This essay will examine how the resistance of a metallic conductor varies with temperature, linking the macroscopic observation to the microscopic behaviour of free electrons and lattice ions.’ This instantly signals to the examiner that you are addressing the core concept.
4. Using the PEEL Structure for Body Paragraphs | 在主体段落中使用 PEEL 结构
Each body paragraph should follow PEEL: Point, Evidence, Explanation, Link. State your point in the first sentence, support it with data or an observation, explain using physics laws, and link back to the question or the next point.
For example, when discussing the effect of temperature on the resistance of a thermistor: Point – ‘As temperature rises, the resistance of an NTC thermistor decreases.’ Evidence – ‘At 20 °C the resistance is 3 kΩ, dropping to 200 Ω at 80 °C.’ Explanation – ‘Thermal energy excites more electrons across the band gap, increasing the number of charge carriers, so current increases for the same p.d., meaning R = V/I falls.’ Link – ‘This non‑linear behaviour makes thermistors ideal for temperature sensing circuits.’
例如，讨论温度对热敏电阻的影响时：观点 – “温度升高时，NTC 热敏电阻的阻值下降。”证据 – “20 °C 时电阻为 3 kΩ，80 °C 时降至 200 Ω。”解释 – “热能激发更多电子穿过带隙，增加载流子数量，因此相同电压下电流增大，由 R = V/I 可知电阻下降。”链接 – “这种非线性特性使热敏电阻非常适合温度传感电路。”
5. Describing Experimental Setups | 描述实验装置
When describing an experiment, name each piece of apparatus and its purpose. Use a logical order: power supply, measuring instruments, the specimen, and any environmental controls. A well‑described setup can be visualised by the examiner.
For a resistivity experiment on a wire: ‘A nichrome wire of length 1.00 m is clamped on a metre rule. A voltmeter is connected across two crocodile clips separated by 0.80 m, and an ammeter is in series with a d.c. power supply. A rheostat adjusts the current to avoid heating effects. The diameter is measured with a micrometer screw gauge at three positions.’ Include a labelled diagram if possible.
对于导线电阻率实验：“一根 1.00 m 长的镍铬丝固定在米尺上。电压表接在与导线相距 0.80 m 的两个鳄鱼夹之间，电流表与直流电源串联。变阻器调节电流以避免加热效应。用于分尺在三个位置测量直径。”可能的话附上带标注的示意图。
6. Explaining Physical Principles | 解释物理原理
Explanations must move from the simple to the complex. Start with a law or principle (e.g. Ohm’s law, conservation of momentum), then apply it to the specific scenario. Use cause‑and‑effect language: ‘because’, ‘therefore’, ‘as a result’.
解释必须从简单到复杂。先给出定律或原理（如欧姆定律、动量守恒），再将其应用于具体情境。使用因果关系的连接词：’because’、’therefore’、’as a result’。
When explaining why the current in a wire increases with p.d., do not just state Ohm’s law. Say: ‘A higher p.d. produces a stronger electric field along the wire. This exerts a greater force on the free electrons, increasing their drift velocity v. Since I = nAve, a larger v leads directly to a larger current, assuming the number density n remains constant.’ This shows deep understanding.
解释导线中电流随电压升高而增大时，不要只给出欧姆定律。要说明：“更高的电压在导线中产生更强的电场。这使施加在自由电子上的力增大，提高了漂移速度 v。由 I = nAve 可知，在载流子密度 n 不变时，v 增大直接导致电流增大。”这体现了深层理解。
7. Incorporating Equations and Diagrams | 融入公式和图示
Equations are essential but must be explained. Centre them on a new line and define every symbol immediately. For example, the resistivity equation can be presented as:
公式必不可少，但必须加以解释。将其居中单独成行，并立即定义每个符号。例如，电阻率公式可以这样呈现：
ρ = RA / L
where ρ is resistivity (Ω m), R is resistance (Ω), A is cross‑sectional area (m²), and L is length (m). Then explain how you used the formula to calculate results. Diagrams of circuits, ray paths, or force vectors should be clear, labelled, and referred to in the text.
Never just drop an equation without linking it to the physics. Show how it follows from first principles or how the experimental data validate it.
切勿只放公式而不与物理原理联系。应展示公式如何从基本原理推导出来，或实验数据如何验证它。
8. Handling Data and Error Analysis | 处理数据和误差分析
If the essay involves calculations or experimental data, present them in a well‑structured table. Use appropriate headings with units and correct significant figures. For example:
如果论文涉及计算或实验数据，要用结构清晰的表格呈现。表头带单位，有效数字正确。例如：
Length L / m
Resistance R / Ω
R/L (Ω m⁻¹)
0.100
1.25
12.5
0.200
2.47
12.4
Discuss sources of uncertainty: instrumental (e.g. ammeter ±0.01 A), random (fluctuations in readings), and systematic (zero error on the voltmeter). Explain how you minimised or accounted for these. If you calculated a percentage difference from an accepted value, comment on whether the result is accurate.
Evaluation is a high‑order skill. Identify two or three limitations of the procedure and suggest realistic improvements. For instance, ‘The wire was not perfectly uniform in cross‑section, leading to scatter in the R/L values. This could be improved by measuring the diameter at more points or using a travelling microscope.’
Do not simply list ‘human error’. Be specific: ‘Parallax error when reading the voltmeter introduced a random uncertainty of about ±0.05 V. This could be reduced by using a digital voltmeter.’ Linking limitations to the calculated uncertainties shows analytical thinking.
不要只是列出“人为误差”。要具体：“读取电压表时的视差引入了约 ±0.05 V 的随机不确定度。使用数字电压表可以减小这一误差。”将局限性与你计算的不确定度联系起来，展示分析性思维。
10. Writing an Effective Conclusion | 撰写有效的结论
A conclusion must directly answer the question. Summarise the main findings without introducing new information. State whether the results support the initial hypothesis and, if applicable, quote the final numerical result with its uncertainty.
For example: ‘The experiment confirmed that the resistance of a constantan wire is directly proportional to its length, with a constant R/L ratio of 12.45 Ω m⁻¹. The small percentage difference from the accepted value (within 2%) indicates systematic errors were minimal and the method was valid.’
One major pitfall is writing a ‘story’ instead of a structured argument. Avoid chronological descriptions like ‘First we did this, then we did that’. Focus on the physics reasoning. Also, do not confuse common laws: Ohm’s law only applies to ohmic conductors at constant temperature.
Another mistake is ignoring units and significant figures. Writing ‘R = 12.5 Ω’ is fine, but ‘R = 12.512’ without units is meaningless. Always convert to SI units before substituting into formulae, and round answers to match the least precise measurement.
12. Worked Example: Temperature and Resistance | 范例：温度与电阻
Let us apply the template to a typical AS essay: ‘Describe and explain how the resistance of a metal wire and a thermistor vary with temperature, and suggest how each could be used in a sensing circuit.’
让我们将这个模板应用于一道典型的 AS 论文题：“描述并解释金属导线和热敏电阻的电阻如何随温度变化，并说明它们各自如何用于传感电路。”
Plan: (1) Metal wire – positive temperature coefficient, R ∝ T due to increased lattice vibrations; (2) Thermistor (NTC) – negative temperature coefficient, more charge carriers; (3) Graphs of R vs T for both; (4) Circuit uses – wire for temperature measurement in a platinum resistance thermometer, thermistor in a potential divider to trigger a switch.
Introduction: ‘This essay compares the temperature dependence of resistance in a metal conductor and an NTC thermistor. While the metal shows a linear increase due to enhanced electron scattering, the semiconductor thermistor exhibits an exponential decrease because thermal energy liberates more charge carriers.’
Body paragraph on metal: ‘For a metal such as copper, resistance increases approximately linearly with temperature. As temperature rises from θ₁ to θ₂, the lattice ions vibrate with greater amplitude. This increases the frequency of collisions between free electrons and ions, reducing the mean free path and thus the drift velocity. Using R = ρL/A and ρ = ρ₀ (1 + αΔθ), where α ≈ 3.9×10⁻³ K⁻¹ for copper, the resistance follows R = R₀ (1 + αΔθ). Experimental data confirm this linearity within modest temperature ranges.’
Body paragraph on thermistor: ‘An NTC thermistor, typically made of a semiconductor, shows a rapid decrease in resistance with temperature. The relationship is not linear but exponential, represented by R = A e^{B/T}, where A and B are constants. The physics differs: as thermal energy is supplied, more valence electrons jump the band gap into the conduction band, drastically increasing the number density n of charge carriers. Since I = nAve, a larger n for a given p.d. means a lower resistance. At 20 °C the resistance may be several kilo‑ohms, dropping to tens of ohms at 100 °C.’
关于热敏电阻的主体段落：“NTC 热敏电阻通常由半导体材料制成，其电阻随温度迅速下降。该关系不是线性而是指数式的，可用 R = A e^{B/T} 表示，其中 A 和 B 为常数。物理机制不同：随着热能供给，更多价电子越过带隙进入导带，大大增加了载流子数密度 n。由 I = nAve，给定电压下 n 更大意味着电阻更低。20 °C 时电阻可能为数千欧，100 °C 时降至几十欧。”
Conclusion: ‘In summary, a metal wire’s positive temperature coefficient makes it suitable for a linear resistance thermometer, while the thermistor’s high sensitivity is exploited in potential‑divider circuits for thermostat control. The contrasting microscopic mechanisms—phonon scattering versus band‑gap excitation—elegantly explain the macroscopic behaviour.’
📚 AS Physics Unit 1 Mark Scheme Jun19: Experimental Investigation of Young’s Modulus | AS物理第一单元2019年6月评分方案：杨氏模量实验探究解析
In AS Physics Unit 1, the experimental investigation question is a reliable differentiator, and the June 2019 mark scheme provides a blueprint for exactly what examiners expect. By unpacking this mark scheme, students can see how marks are allocated for measurement techniques, data handling, graph work and uncertainty analysis. This article breaks down the Young’s modulus experiment featured in that exam, translating examiner shorthand into actionable revision.
1. Experiment Overview and the Mark Scheme’s Mindset | 实验概览与评分方案的思维
The June 2019 Unit 1 required candidates to determine the Young modulus of a metal wire using a simple setup: a wire fixed at one end, a marker attached, a metre rule behind the marker, and a hanger for adding masses. The mark scheme rewards a logical sequence of practical steps, not just the final number. Every detail from zeroing instruments to plotting a best-fit line is assessed.
Examiners look for evidence that you understand why a particular tool is chosen, how to minimise random and systematic errors, and how uncertainties propagate. The mark scheme explicitly links each mark to a specific skill: using a micrometer, reading a metre rule without parallax, tabulating repeated readings, calculating a gradient, and combining percentage uncertainties.
2. Measuring the Diameter with a Micrometer | 用千分尺测量直径
The mark scheme insists that the diameter must be measured with a micrometer screw gauge, not a ruler or vernier callipers. This is because the wire is typically less than 0.5 mm thick, so a micrometer with a resolution of 0.01 mm is required. A candidate who chooses a vernier calliper (resolution 0.1 mm) will lose the method mark immediately.
You must also check for a zero error by closing the jaws before taking readings. The mark scheme often gives a mark for ‘check zero error and subtract/add correction’. Take at least three diameter readings at different points along the wire and in perpendicular directions to average out any non-uniformity. In June 2019, the raw data showed readings of 0.28 mm and 0.27 mm; the mean diameter was 0.275 mm. The cross-sectional area A is then calculated using A = πd²/4.
你还必须在测量前将测砧闭合以检查零误差。评分方案常为“检查零误差并减去/加上修正值”留出1分。沿着金属丝的不同位置、在相互垂直的方向上至少取三次直径读数，以消除粗细不均的影响。2019年6月的原始数据显示两个读数为0.28 mm和0.27 mm，平均直径为0.275 mm。接着用公式 A = πd²/4 计算截面积。
Use a micrometer with 0.01 mm resolution; check zero error before use and record the correction. — 使用分辨率为0.01 mm的千分尺；使用前检查零误差并记录修正值。
Take readings at several positions and in two perpendicular directions; record the mean diameter. — 在几个位置和两个正交方向上测量直径；记录平均直径。
Calculate cross-sectional area using A = πd²/4 and keep the value in m². — 用 A = πd²/4 计算截面积，并保留平方米单位。
3. Determining the Original Length and Avoiding Parallax | 测定原长与避免视察
The original length L of the wire should be measured with a metre rule, since the length is typically around 1–2 m and does not need micrometre precision. However, an examiner will award a mark for stating that you must clamp the rule vertically, align the zero mark carefully, and read the scale at eye level to avoid parallax error. The mark scheme in June 2019 rewarded ‘use a set square to align the marker with the rule’ or ‘use a travelling microscope’ as an alternative method for measuring extension, but for original length the rule is sufficient.
金属丝的原长 L 通常用米尺测量，因为长度一般在1到2米左右，无需微米级精度。不过，考官会为“将米尺竖直固定、小心对齐零刻线并在眼睛水平位置读数以避免视差”给出得分。2019年6月的评分方案对“使用三角板将标记物与直尺对齐”或“改用移动显微镜测量伸长量”都给予认可，但对于原长来说，米尺即可满足要求。
Always record the original length with its absolute uncertainty, usually ±1 mm for a metre rule, unless a more precise instrument is used. The mark scheme demands that you state this uncertainty explicitly in a table or in the data section. If you fail to provide an uncertainty, the mark for ‘appreciation of uncertainty’ may be lost.
In the June 2019 setup, extension is measured by recording the new position of the marker on the metre rule after each mass is added. The mark scheme highlights two essential precautions: first, wait for the wire to stop stretching before taking a reading (to avoid dynamic effects); second, always subtract the initial marker reading from each subsequent reading to obtain extension. Do not simply record the new scale reading as extension.
Where the extension is very small, candidates can suggest using a travelling microscope or a digital indicator to improve precision. The mark scheme treats these as valid improvements for an extra mark in planning questions, but in this particular assessed task the metre rule method was fixed. Still, showing awareness of parallax correction by placing the eye level with the marker and using an outstretched ruler could earn a ‘detail’ mark.
The mark scheme insists that all raw data be presented in a clear table with headings and units. In the June 2019 paper, the table required columns for mass, weight (force), marker reading, extension, and possibly the calculated extension. The unit for extension must be mm or m, but be consistent. The force is calculated as mass × 9.81 N kg⁻¹, and the mark scheme expects ‘weight = mg’ to be used, with m in kg.
评分方案坚持所有原始数据都要在清晰的表格中呈现，包含表头与单位。2019年6月的试题中，表格需要的列包括质量、重量（力）、标记物读数、伸长量，以及可能再列一列计算出的伸长量。伸长量的单位必须使用mm或m，并保持一致。力由质量乘以9.81 N kg⁻¹算出，评分方案要求使用“重量=mg”，且质量以kg为单位。
Significant figures matter. The mark scheme penalises over-quoting or under-quoting. Normally, raw data should reflect the precision of the instrument: if the metre rule reads to the nearest mm, then the reading should be given as 235 mm, not 235.0 mm. Calculated extensions may be given to one extra decimal place to avoid rounding errors.
A force-extension graph must be plotted with force (weight) on the y-axis and extension on the x-axis. The June 2019 mark scheme allocated marks for appropriate scales, accurately plotted points, and a best-fit straight line. The axes must be labelled with quantities and units, e.g. ‘Weight / N’ and ‘Extension / mm’.
The line of best fit does not have to pass through the origin unless the data strongly suggest it. The mark scheme accepts a line that passes through the error bars of all points. If a point is clearly anomalous, it should be circled and ignored in the trend line. A mark is often available for identifying an anomalous point.
Choose a scale that uses at least half the graph paper in each direction. — 选取能使图线在各方向至少占据格子一半范围的坐标尺度。
Label axes with ‘Weight / N’ and ‘Extension / mm’ (or m). — 用“Weight / N”和“Extension / mm”（或m）标注坐标轴。
Plot points as small crosses or dots with circles; draw error bars if uncertainties are known. — 用细叉或加点套圈的方式描点；如果已知不确定度则画误差棒。
Draw one smooth best-fit line, not joining point to point. — 画一条平滑的最佳拟合线，而不要逐点连接。
7. Calculating the Gradient and Its Uncertainty | 计算斜率及其不确定度
The gradient k of the force-extension graph is needed because Young modulus E = (L/A) × k, provided L and A are in consistent SI units. The mark scheme expects the gradient to be calculated from a large triangle covering at least half the line, with the coordinates clearly read from the best-fit line, not data points. Show the values used, e.g., ΔF/Δx = (8.0 N) / (6.5 mm).
需要力-伸长图的斜率 k，因为杨氏模量 E = (L/A) × k，前提是 L 和 A 采用一致的国际单位。评分方案期望你用覆盖至少半条线的大三角形求斜率，坐标应从最佳拟合线上读取，而非数据点。列出所用数值，例如 ΔF/Δx = (8.0 N) / (6.5 mm)。
To find the uncertainty in the gradient, draw a ‘worst acceptable line’ – either the steepest or shallowest line that still fits the error bars. The mark scheme awards a separate mark for calculating the difference between best and worst gradients, then the half-range gives the absolute uncertainty in k. This mirrors the typical ‘uncertainty from graph’ mark.
要得到斜率的不确定度，可画一条“最差可接受线”——即仍通过误差棒的最陡或最平线。评分方案单独给1分用于计算最佳与最差斜率之差，再用半区间作为 k 的绝对不确定度。这对应于常见的“从图中求不确定度”得分点。
8. Computing Young’s Modulus Step by Step | 逐步计算杨氏模量
E = FL / (A ΔL)
Since the gradient k = F/ΔL, the expression becomes E = (L/A) × k. The mark scheme gives clear step marks: first calculate k from graph, then multiply by L/A. For example, if L = 2.500 m (±0.001 m) and A = 5.94 × 10⁻⁸ m² (from d = 0.275 mm), then L/A = 4.21 × 10⁷ m⁻¹. Multiplying by k (e.g., 1.23 × 10³ N m⁻¹) gives E ≈ 5.2 × 10¹⁰ Pa.
因为斜率 k = F/ΔL，表达式变为 E = (L/A) × k。评分方案给出了清晰的步骤分：先由图求出 k，再乘以 L/A。例如，若 L = 2.500 m (±0.001 m)，A = 5.94 × 10⁻⁸ m²（d = 0.275 mm 得到），则 L/A = 4.21 × 10⁷ m⁻¹。乘以 k（如 1.23 × 10³ N m⁻¹）得到 E ≈ 5.2 × 10¹⁰ Pa。
Always convert all units to SI before substituting: length in m, area in m², force in N. A mark is frequently reserved for the correct unit of Young modulus (Pa or N m⁻²) and a sensible answer, typically in the range of 10¹¹ Pa for steel. The mark scheme also credits comparing the calculated E with known values and calculating percentage difference when required.
9. Uncertainty Analysis for Young’s Modulus | 杨氏模量的不确定度分析
The total percentage uncertainty in E is the sum of the percentage uncertainties of L/A and k. Since A depends on d², the percentage uncertainty in A is twice that of d. The June 2019 mark scheme expected candidates to handle this propagation: %U(E) = %U(k) + %U(L) + 2 × %U(d). L is measured with a metre rule, so %U(L) is usually very small (e.g., 0.04% for 2.5 m ± 1 mm). d is measured with a micrometer, so %U(d) = (0.01 mm / 0.275 mm) × 100% ≈ 3.6%; thus 2 × %U(d) ≈ 7.2%. The gradient uncertainty %U(k) comes from the worst acceptable line method.
E 的总百分不确定度是 L/A 和 k 的百分不确定度之和。由于 A 取决于 d²，A 的百分不确定度是 d 的两倍。2019年6月的评分方案期望考生能处理这种传递关系：%U(E) = %U(k) + %U(L) + 2 × %U(d)。L 用米尺测量，因此 %U(L) 通常非常小（例如，对 2.5 m ± 1 mm 约为 0.04%）。d 用千分尺测量，因此 %U(d) = (0.01 mm / 0.275 mm) × 100% ≈ 3.6%；2 × %U(d) ≈ 7.2%。斜率不确定度 %U(k) 来自最差可接受线法。
The mark scheme explicitly penalises missing the factor of 2 for diameter uncertainty. It also expects a statement of the final E value together with its absolute uncertainty or percentage uncertainty, written as E = (5.2 ± 0.6) × 10¹⁰ Pa or E = 5.2 × 10¹⁰ Pa ± 12%. Showing the combination of uncertainties in clear steps is the key to securing full marks.
评分方案明确会对漏掉直径不确定度的因子2进行扣分。它还期望学生写出最终的 E 值及其绝对不确定度或百分不确定度，表达为 E = (5.2 ± 0.6) × 10¹⁰ Pa 或 E = 5.2 × 10¹⁰ Pa ± 12%。将不确定度合成过程以清晰步骤展示，是拿到满分的关键。
%U(d) = (resolution / mean d) × 100%; then multiply by 2 for area. — %U(d) = (分辨率 / 平均直径) × 100%；然后将其乘以2得到面积的不确定度。
Add all %U contributions; never subtract or cancel them. — 将所有的百分不确定度相加，切勿相减或抵消。
10. Common Mistakes and Examiner Tips from the Mark Scheme | 评分方案中的常见错误与考官提示
One recurring error is forgetting to subtract the initial marker reading. Candidates sometimes plot the scale reading directly, resulting in a curved or offset graph that does not earn the best-fit line mark. Another is using the raw diameter instead of dividing by 2 to get radius; the area formula A = πr² must be used, or equivalently A = πd²/4.
一个常见错误是忘记减去初始标记读数。考生有时直接把刻度读数来画图，结果得到一条曲线或平移的直线，拿不到最佳拟合线的分数。另一个错误是直接用直径代替半径，面积公式必须使用 A = πr² 或等价的 A = πd²/4。
Unit confusion is heavily penalised. The mark scheme shows that candidates who kept extension in mm but force in N often forgot to convert gradient to m when calculating E. The easiest route is to convert extension to metres as soon as possible. Additionally, some students draw a ‘line of worst fit’ that is parallel to the best fit; this is mathematically incorrect as the worst acceptable line should pivot around the centroid of the data to capture the maximum plausible gradient variation.
Finally, when asked to comment on accuracy, the mark scheme expects candidates to compare their answer with a reference value (e.g., 2.0 × 10¹¹ Pa for steel). A simple ‘my value is close’ is insufficient; you must calculate a percentage difference and relate it to the experimental uncertainty. If the percentage difference lies within the total percentage uncertainty, the result is considered accurate within experimental limits.
Particle physics reveals the fundamental building blocks of matter and the forces that govern their interactions. In the IB Physics syllabus, this topic bridges quantum ideas and conservation laws, introducing quarks, leptons, exchange particles and Feynman diagrams. Mastering these concepts will not only help you solve classification problems but also deepen your understanding of how the universe operates at the smallest scales.
The Standard Model classifies all known elementary particles into two families: fermions (matter particles) and bosons (force carriers). Fermions are further divided into quarks and leptons, each with six flavours organised in three generations. Everyday matter is built from the first generation only: up and down quarks, the electron and the electron neutrino.
The gauge bosons mediate the fundamental forces: the photon (electromagnetic), W⁺, W⁻ and Z⁰ (weak interaction), and eight gluons (strong interaction). Gravity is not yet incorporated into the Standard Model. In IB, you need to be able to state the categories of particles and give examples of their properties.
Quarks are elementary fermions that carry fractional electric charge and participate in the strong interaction. The six flavours are up (u, charge +⅔), down (d, –⅓), charm (c, +⅔), strange (s, –⅓), top (t, +⅔) and bottom (b, –⅓). Each quark has a corresponding antiquark with opposite charge.
Quarks are never observed in isolation due to colour confinement; they combine to form colourless hadrons. Baryons consist of three quarks (qqq), while mesons consist of a quark and an antiquark (qq̄). In IB questions, you will often deduce the quark composition of a particle from its charge, baryon number and strangeness.
Leptons are elementary fermions that do not experience the strong interaction. The six leptons are the electron (e⁻), muon (μ⁻), tau (τ⁻) and their associated neutrinos (νₑ, ν_μ, ν_τ). Each lepton carries a lepton number L = +1, while antileptons have L = –1.
轻子是不参与强相互作用的基本费米子。六种轻子分别是电子 (e⁻)、μ子 (μ⁻)、τ子 (τ⁻) 以及它们对应的中微子 (νₑ, ν_μ, ν_τ)。每个轻子携带轻子数 L = +1，反轻子的轻子数则为 L = –1。
Lepton number is strictly conserved in all interactions. This means the total lepton number for each generation (electron number, muon number, tau number) remains constant, apart from the tiny effects of neutrino oscillations which are not required in IB. For example, in beta-minus decay a neutron changes into a proton, emitting an electron and an antineutrino: n → p + e⁻ + ν̄ₑ. The electron lepton number before is 0, and after is (+1 for e⁻) + (–1 for ν̄ₑ) = 0.
在所有相互作用中，轻子数严格守恒。这意味着每一代轻子数（电子数、μ子数、τ子数）保持恒定，中微子振荡的微小效应不包括在IB要求内。例如，在β⁻衰变中，中子变为质子，放出一个电子和反中微子：n → p + e⁻ + ν̄ₑ。反应前电子轻子数为0，反应后为（e⁻的+1） + （ν̄ₑ的–1）= 0。
4. Antimatter | 反物质
Every particle has a corresponding antiparticle with the same mass and spin but opposite values of all charges (electric charge, baryon number, lepton number and strangeness). For instance, the positron (e⁺) is the antiparticle of the electron; it has charge +1e and lepton number –1.
When a particle and its antiparticle meet, they annihilate, converting their mass into energy according to E = mc², typically producing a pair of photons. Conversely, pair production creates a particle-antiparticle pair from a high-energy photon in the presence of a nucleus to conserve momentum. In IB, you may be asked to apply these concepts in energy calculations.
当粒子和它的反粒子相遇时，它们会发生湮灭，按照 E = mc² 将质量转化为能量，通常产生一对光子。相反地，电子对产生是指高能光子在有原子核存在的条件下转化为粒子-反粒子对，以保持动量守恒。在IB中，你可能需要将这些概念应用于能量计算。
5. Hadrons: Baryons and Mesons | 强子：重子和介子
Hadrons are composite particles made of quarks, held together by the strong interaction. They are categorised into baryons (three quarks) and mesons (quark-antiquark). Baryons include protons (uud) and neutrons (udd), with baryon number B = +1. Antibaryons have B = –1.
强子是由夸克通过强相互作用组成的复合粒子，分为重子（三个夸克）和介子（夸克-反夸克）。重子包括质子 (uud) 和中子 (udd)，重子数 B = +1；反重子的重子数为 –1。
Mesons are bosons with baryon number zero. Pions (π⁺, π⁻, π⁰) are the lightest mesons and play a key role in the strong nuclear force between nucleons. Kaons (K⁺, K⁻, K⁰) contain a strange quark or antiquark. All mesons are inherently unstable and decay via the weak or electromagnetic interaction.
6. Conservation Laws in Particle Physics | 粒子物理中的守恒定律
When analysing particle reactions, you must apply conservation of electric charge, baryon number and lepton number. These quantities are absolute conservation laws in the Standard Model. Strangeness is conserved in strong and electromagnetic interactions but can change by ±1 in weak decays.
The strangeness quantum number S is assigned to particles containing strange quarks: s quark has S = –1, s̄ has S = +1. For example, a K⁺ meson (us̄) has strangeness +1. When a strange particle decays via the weak interaction, a strange quark transforms, and strangeness is not conserved. This explains why strange particles are always produced in pairs (associated production).
奇异量子数 S 赋予含有奇异夸克的粒子：s夸克 S = –1，s̄ 反夸克 S = +1。例如，K⁺ 介子 (us̄) 的奇异数为 +1。当奇异粒子通过弱相互作用衰变时，奇异夸克发生转变，奇异数不再守恒。这解释了为什么奇异粒子总是成对产生（协同产生）。
7. The Four Fundamental Forces | 四种基本相互作用
The Standard Model describes three fundamental forces: electromagnetic, weak nuclear and strong nuclear, plus gravity which is negligible at the particle scale. Each force is mediated by specific gauge bosons. The electromagnetic force acts on charged particles via photon (γ) exchange, has infinite range, and is responsible for binding electrons to nuclei.
The strong interaction binds quarks together inside hadrons and is mediated by gluons (g). It acts only on particles with colour charge. Its residual effect holds protons and neutrons together in the nucleus, mediated by mesons. The weak interaction is responsible for beta decay and quark flavour changes, mediated by W⁺, W⁻, Z⁰ bosons. It is the only force that can change the flavour of a quark.
8. Exchange Particles and Feynman Diagrams | 交换粒子和费曼图
Forces are understood as the exchange of virtual gauge bosons between particles. Feynman diagrams are a visual tool to represent these interactions. In an IB Feynman diagram, time usually runs from left to right, fermions are shown as solid lines with arrows, and bosons as wavy (photon, W, Z) or curly lines (gluon).
When drawing a diagram, ensure conservation of charge at each vertex. For instance, beta-minus decay at quark level is d → u + W⁻, where the W⁻ then decays into e⁻ + ν̄ₑ. The diagram shows a down quark emitting a W⁻ boson, turning into an up quark, and the W⁻ boson splitting into an electron and an antineutrino.
画图时，确保每个顶点处电荷守恒。例如，夸克层面的β⁻衰变是 d → u + W⁻，随后 W⁻ 衰变成 e⁻ + ν̄ₑ。费曼图展示一个下夸克放出一个 W⁻ 玻色子，转变为上夸克，然后 W⁻ 玻色子分裂成电子和反中微子。
9. Weak Interaction and Quark Flavour Change | 弱相互作用与夸克味变
The weak interaction is unique in its ability to change quark flavour, thereby causing hadrons to decay. This occurs through the emission or absorption of a charged W boson. The Cabibbo–Kobayashi–Maskawa (CKM) matrix describes the probability of flavour changes, but for IB you only need to know that transitions between generations are suppressed compared to those within a generation.
弱相互作用的独特之处在于它能改变夸克味，从而导致强子衰变。这通过发射或吸收带电 W 玻色子实现。Cabibbo–Kobayashi–Maskawa (CKM) 矩阵描述了味变的概率，但IB只要求你知道跨代跃迁相对于同代跃迁受到抑制。
Common weak decays include neutron decay (n → p + e⁻ + ν̄ₑ), muon decay (μ⁻ → e⁻ + ν̄ₑ + ν_μ), and kaon decays. In all these, lepton number and baryon number are strictly conserved, while strangeness can change. A useful rule is that the total charge of the W boson involved equals the change in charge of the quark: for d (–⅓) to u (+⅔), a W⁻ is required.
常见的弱衰变包括中子衰变 (n → p + e⁻ + ν̄ₑ)、μ子衰变 (μ⁻ → e⁻ + ν̄ₑ + ν_μ) 和K介子衰变。在这些衰变中，轻子数和重子数严格守恒，而奇异数可能改变。一条有用的规则是，涉及的 W 玻色子的电荷等于夸克电荷的变化量：从 d (–⅓) 到 u (+⅔) 需要 W⁻。
10. Hadron Decays and Strangeness | 强子衰变与奇异性
Strange particles contain at least one strange quark or antiquark. They are produced via the strong interaction (always in pairs, e.g., π⁻ + p → K⁰ + Λ⁰), conserving strangeness. However, they decay through the weak interaction, where strangeness is not conserved, giving them relatively long lifetimes of order 10⁻¹⁰ to 10⁻⁸ s.
An example is the neutral kaon K⁰ (ds̄). It decays mainly into pions via weak interaction. The Λ⁰ baryon (uds) decays weakly into p + π⁻ or n + π⁰. In IB questions, you may need to construct quark-level equations to show the weak decay, ensuring charge and baryon number are conserved while allowing strangeness to change by ±1.
例如中性 K 介子 K⁰ (ds̄)，主要通过弱相互作用衰变为π介子。Λ⁰ 重子 (uds) 弱衰变为 p + π⁻ 或 n + π⁰。在IB考题中，你可能需要构建夸克层面的方程来表示弱衰变，确保电荷、重子数守恒，同时允许奇异数改变 ±1。
11. Particle Interactions and Feynman Diagram Examples | 粒子相互作用与费曼图实例
Beyond decay, Feynman diagrams can illustrate scattering and annihilation events. Electron-electron repulsion is shown as two electron lines exchanging a virtual photon. Electron-positron annihilation into two photons is shown by the incoming e⁻ and e⁺ meeting, annihilating and producing photon lines emerging from the vertex.
Deep inelastic scattering of electrons off protons gave evidence for quarks. The Feynman diagram shows an electron emitting a virtual photon, which is absorbed by a single quark within the proton, causing the proton to break apart. In IB, you only need to recognise that such experiments confirmed the existence of point-like constituents inside nucleons.
12. The Higgs Boson and Mass Generation (HL Extension) | 希格斯玻色子与质量生成（HL拓展）
The Higgs boson is a scalar boson associated with the Higgs field, which permeates all space. Particles acquire mass through their interaction with this field; the stronger the coupling, the greater the mass. The discovery of the Higgs boson at CERN in 2012 confirmed the mechanism of electroweak symmetry breaking.
In IB HL, you may be asked to discuss the role of the Higgs boson in the Standard Model and explain why the W and Z bosons are massive while the photon remains massless. The Higgs mechanism gives mass to the weak gauge bosons but leaves the photon unaffected, preserving U(1) electromagnetic gauge symmetry.
📚 A-Level Physics: Experimental Investigation of Energy Sources | A-Level物理：能量来源实验探究
This article provides a comprehensive guide to the experimental investigations of energy sources covered in the OxfordAQA International A-Level Physics course. Students are expected to design, carry out, and analyse experiments related to renewable and non‑renewable energy sources, such as solar cells and wind turbines. These practical investigations develop essential skills in data collection, graphical analysis, uncertainty estimation, and evaluation of energy conversion efficiencies. Understanding both the physical principles and the experimental limitations helps learners appreciate the challenges involved in real‑world energy systems and prepares them for topic tests and the practical endorsement.
1. Overview of Energy Source Experiments | 能量来源实验概览
In the International A-Level Physics specification, the topic of energy sources requires students to investigate the performance of devices that convert natural energy flows into usable electrical power. Common experiments focus on the photovoltaic effect in solar cells, the kinetic energy conversion in wind turbines, and occasionally chemical cells or electromagnetic induction generators. These experiments are designed to reinforce concepts such as energy transfer, efficiency, power, and the relationship between physical variables like irradiance, wind speed, load resistance, and electrical output.
The two most representative investigations are the determination of a solar cell’s efficiency under varying illumination and the measurement of a model wind turbine’s power output as a function of wind speed. Both involve circuit assembly, systematic variation of an independent variable, and the application of physics equations to calculate efficiency and power, making them excellent vehicles for developing practical skills.
2. Safety Precautions in Energy Experiments | 能量实验中的安全措施
Before beginning any investigation involving electrical circuits, light sources, or rotating machinery, a thorough risk assessment must be carried out. Standard laboratory safety rules apply, but additional attention is required for specific hazards in energy source experiments.
Electrical safety: Keep voltages low (typically below 12 V for solar cells and small turbines); check for damaged insulation; avoid short circuits. 电气安全：使用低压（太阳能电池和小型风机通常低于12 V）；检查绝缘层是否损坏；避免短路。
Light sources: High‑power lamps used to simulate sunlight can become extremely hot. Allow bulbs to cool before handling and keep flammable materials away. 光源：用于模拟阳光的高功率灯会变得极烫。待灯泡冷却后再操作，并远离易燃物。
Rotating parts: Wind turbines have fan blades that can cause injury. Use a guard mesh or stand behind a safety screen, and secure the turbine firmly. 旋转部件：风力发电机的扇叶可能造成伤害。使用防护网或站在安全屏后面，并牢固固定风机。
Always wear eye protection and tie back long hair. For solar cell experiments, avoid staring directly at intense lamps to prevent eye strain.
始终佩戴护目镜并将长发束起。对于太阳能电池实验，避免直接注视强光灯具以防止眼疲劳。
3. Experiment 1: Solar Cell Efficiency Investigation | 实验一：太阳能电池效率探究
This investigation aims to determine how the efficiency of a photovoltaic cell varies with the intensity of incident light. By measuring the current–voltage (I–V) characteristic of the solar cell at different lamp distances, students can calculate the maximum power output and the overall efficiency. The circuit used includes a solar cell, a variable resistor (rheostat), an ammeter connected in series, and a voltmeter connected in parallel across the variable resistor.
The irradiance (G) incident on the cell is varied by moving a lamp along a metre ruler. The distance from the lamp filament to the cell surface is measured with an uncertainty of ±1 mm. A blackened tube or aperture can be used to ensure that only direct light reaches the active area of the cell, and a correction for background light should be made by taking a dark reading.
4. Procedure for Solar Cell I–V Measurements | 太阳能电池I–V测量步骤
Set the lamp at a chosen distance (e.g. 30 cm) and switch it on, allowing the filament to stabilise for a few minutes. During this time, cover the solar cell to avoid pre‑heating. The variable resistor is initially set to its maximum resistance. Remove the cover and record the voltmeter and ammeter readings. Then reduce the resistance in small steps, taking at least ten pairs of V and I values across the range from open‑circuit voltage to short‑circuit current.
It is crucial to record readings rapidly to minimise temperature effects on the solar cell’s performance. The cell temperature should be monitored with an infrared thermometer if available, and the ambient temperature noted. Repeat the entire I–V sweep for at least four different lamp distances, ensuring the lamp position and alignment are identical for each distance.
5. Varying Irradiance and Data Collection | 改变辐照度和数据收集
As the lamp distance increases, the irradiance at the cell decreases approximately as 1/d², assuming a point source. Students may confirm this inverse‑square relationship using a light sensor, or they may simply treat distance as a proxy for irradiance. A more rigorous approach is to measure the actual irradiance with a calibrated photometer and then plot results directly against irradiance. The area (A) of the solar cell must be measured with a ruler, giving an uncertainty of about ±1 mm².
For each data set, calculate the electrical power P = V × I at every point. This allows the construction of a power–voltage (P–V) curve. The peak of this curve gives the maximum power point Pmax. The corresponding voltage Vmp and current Imp should be recorded.
对每组数据，计算每个数据点的电功率P = V × I。这样可以构建功率–电压（P–V）曲线。该曲线的峰值给出最大功率点Pmax。应记录对应的电压Vmp和电流Imp。
6. Analysis: Plotting I–V and P–V Curves | 分析：绘制I–V和P–V曲线
Plot the I–V characteristics on the same axes for each distance. Typical curves show a region of nearly constant current at low voltages, bending downwards as the voltage approaches the open‑circuit value. The P–V curves are derived by multiplying ordinate values. On the same graph or separate axes, plot P against V and identify Pmax by drawing a smooth curve through the points and finding the maximum.
The fill factor (FF) may also be calculated as FF = Pmax / (Voc × Isc), where Voc is the open‑circuit voltage and Isc the short‑circuit current. Students often compare fill factors at different irradiances to see if cell performance remains consistent.
7. Calculating Efficiency and Uncertainty | 计算效率和不确定度
The instantaneous incident power is Pin = G × A, where G is the measured irradiance in W m⁻² and A is the cell area in m². The conversion efficiency is given by:
瞬时入射功率为Pin = G × A，其中G为测得的辐照度（W m⁻²），A为电池面积（m²）。转换效率由下式给出：
η = (Pmax / (G × A)) × 100%
Uncertainties in efficiency arise from the resolution of the voltmeter and ammeter, the measurement of A, and the accuracy of the photometer. Percentage uncertainties should be combined using standard rules: for multiplication/division, add the percentage uncertainties of the components. For G, the uncertainty in the inverse‑square law or photometer reading must be propagated.
A typical student result might give an efficiency between 5% and 15%, heavily dependent on cell quality and spectral match between the lamp and sunlight. The final quoted efficiency should be expressed as η ± Δη.
8. Experiment 2: Wind Turbine Power Output | 实验二：风力涡轮机功率输出
A model wind turbine can be tested in the laboratory using a controllable electric fan to provide an air flow. The turbine is connected to a load resistor, and the voltage across the load is measured as the fan speed is varied. The wind speed can be estimated using a handheld anemometer placed at the turbine’s rotor plane, or by calculating from the fan’s rating and geometry.
The theoretical power available in the wind is given by Pwind = ½ ρ A v³, where ρ is the air density (≈1.2 kg m⁻³ at room temperature), A is the swept area of the rotor blades, and v is the wind speed. The electrical power output is Pelec = V² / R, assuming a purely resistive load. Students plot Pelec against v and expect a cubic relationship, although frictional losses and generator efficiency mean the measured power is much lower.
风中可获取的理论功率为Pwind = ½ ρ A v³，其中ρ为空气密度（室温下≈1.2 kg m⁻³），A为桨叶扫风面积，v为风速。假设纯电阻负载，电输出功率为Pelec = V² / R。学生绘制Pelec对v的关系图，预期得到三次方关系，但摩擦损失和发电机效率意味着实测功率会低得多。
9. Data Analysis for Wind Energy | 风能数据分析
To verify the v³ dependence, take logs of both power and wind speed: if P ∝ v³, a graph of ln(Pelec) against ln(v) should give a straight line with gradient 3. The y‑intercept then contains information about the system’s overall efficiency ηturbine = Pelec / Pwind. Students should comment on whether the gradient is significantly different from the theoretical value and propose reasons for any discrepancy.
Additional measurements might include the effect of blade pitch angle or the number of blades on the power output. Such investigations are excellent for extended projects and can link to the physics of fluid dynamics and Betz’s limit, which states that no turbine can capture more than 59.3% of the kinetic energy in the wind.
10. Comparison of Energy Sources and Limitations | 能源比较与局限性
When comparing experimental results for solar and wind energy, several key differences emerge. Solar cell efficiency is highly sensitive to the spectral distribution of the light source and the angle of incidence, neither of which is perfectly controlled in a classroom setting. Wind turbine models suffer from significant aerodynamic losses and are affected by turbulence from the fan, which is not characteristic of natural laminar wind.
Both experiments illustrate that real energy converters operate far below their theoretical maxima. For instance, silicon solar cells have an upper theoretical efficiency of about 29% (the Shockley–Queisser limit), yet student cells rarely exceed 10% under lab lamps. Similarly, the power coefficient of small educational turbines seldom exceeds 20%. Highlighting these limitations is crucial for evaluating the reliability of the conclusions.
A thorough evaluation should identify the main sources of systematic and random error. In the solar cell experiment, the lamp’s output may drift due to voltage fluctuations, and the cell’s temperature rise changes its I–V characteristic. Using a regulated DC power supply for the lamp and cooling the cell with a small fan can reduce these effects. For the wind turbine, the anemometer’s accuracy and the fan’s non‑uniform airspeed are dominant uncertainties.
Other improvements include the use of a data logger to record V and I automatically, which allows more data points and reduces timing errors. For the wind experiment, a wind tunnel would provide a much steadier flow, but such equipment is often unavailable; a collimator made from drinking straws can help straighten the airflow from a fan.
In topic tests and written examinations, OxfordAQA International A-Level Physics questions on energy sources often ask candidates to describe an experimental method, identify variables, suggest improvements, or analyse given data to calculate efficiency and power. Students should be familiar with the key equations, be able to sketch I–V and P–V curves, and understand how to propagate uncertainties for efficiency calculations.
When writing up an experimental investigation, always state the independent, dependent, and controlled variables clearly. Use precise scientific vocabulary: “irradiance” rather than “light brightness”, “power coefficient” rather than “how good the turbine is”. Finally, include a discussion of the environmental and economic context of the energy source, as the specification links physics to real‑world applications.
Understanding resistance is fundamental to mastering electricity in IGCSE Edexcel Physics. This article breaks down every key concept, formula, graph and experiment you need to score full marks on resistance-related questions.
Resistance is the opposition a component offers to the flow of electric current. When electrons move through a conductor, they collide with vibrating ions, converting electrical energy into thermal energy. The greater the resistance, the smaller the current for a given potential difference.
The resistance of a component is defined by the ratio of the potential difference (V) across it to the current (I) flowing through it. This relationship is the core of Ohm’s law, but it is important to remember that not all components follow Ohm’s law.
2. Ohm’s Law and the Formula R = V / I | 欧姆定律与公式 R = V / I
Ohm’s law states that the current through a metallic conductor is directly proportional to the potential difference across it, provided its temperature remains constant. This gives the equation: R = V / I, where R is resistance in ohms (Ω), V is voltage in volts (V) and I is current in amperes (A).
欧姆定律指出，对于金属导体，在温度保持不变时，通过它的电流与两端的电势差成正比。由此得出公式：R = V / I，其中 R 是电阻（Ω），V 是电压（V），I 是电流（A）。
A component that follows Ohm’s law is called an ohmic conductor. For an ohmic conductor, the I-V graph is a straight line passing through the origin, and the resistance is constant at constant temperature.
遵循欧姆定律的元件称为欧姆导体。欧姆导体的 I-V 图像是一条过原点的直线，温度不变时电阻恒定。
It is common to rearrange the formula to find V or I. V = I × R and I = V / R are equally important when analysing circuits. Always convert mA to A by dividing by 1000 before substituting into the formula.
公式变形也很常用：V = I × R 和 I = V / R。分析电路时，务必先将 mA 除以 1000 转换为 A 再代入计算。
3. Ohmic and Non-Ohmic Conductors | 欧姆导体与非欧姆导体
Ohmic conductors have a constant resistance at constant temperature. Their I-V characteristic is a straight line, e.g., a fixed resistor or a metal wire at a steady temperature.
欧姆导体在温度恒定时电阻不变，其 I-V 特性曲线为一条直线，例如定值电阻或温度稳定的金属导线。
Non-ohmic conductors do not have a constant resistance. Their I-V graphs are not straight lines. Examples include filament lamps, diodes and thermistors. The resistance changes due to heating or the direction of current.
For a filament lamp, as the voltage increases, the current increases, causing the temperature to rise. The ions vibrate more, increasing resistance. The I-V graph curves, showing a decreasing gradient as V increases.
对于白炽灯，电压升高时电流增大，温度上升，离子振动加剧，电阻随之增大。其 I-V 图像弯曲，随着 V 增大，斜率减小。
4. Factors Affecting Resistance | 影响电阻的因素
The resistance of a wire depends on four main factors: length (l), cross-sectional area (A), temperature and the material of the wire. These are summarised in the resistivity equation which is not required at IGCSE but helps explain the trends.
Resistance is directly proportional to length. Doubling the length doubles the resistance. Resistance is inversely proportional to cross-sectional area: a thicker wire has lower resistance. Temperature increase generally raises the resistance of metals because ions vibrate more, causing more collisions.
In the practical measuring resistance of a wire, students vary the length of a resistance wire and measure V and I using a voltmeter and ammeter. Plotting V against I gives a straight line for constant temperature. The slope is the resistance. Average multiple readings to reduce random error.
在“测量导线电阻”实验中，学生改变电阻丝的长度，用电压表和电流表测 V 和 I。绘制 V-I 图可得一条过原点的直线，斜率即电阻。多次读数取平均值可减少随机误差。
5. I-V Characteristics of Key Components | 关键元件的 I-V 特性
IGCSE Edexcel requires you to sketch and interpret I-V graphs for a fixed resistor, filament lamp, and diode. For a fixed resistor, the graph is a straight line through the origin in both directions if the current direction is reversed.
For a filament lamp, the I-V graph is a curve that is symmetrical about the origin. At low voltages, the resistance is lower; as voltage increases, the curve bends towards the voltage axis, indicating higher resistance.
A diode only allows current to flow in one direction (forward bias). In the forward direction, a large current flows once the threshold voltage (about 0.6 V for silicon) is exceeded. In reverse bias, almost no current flows, giving infinite resistance. The I-V graph shows zero current for negative voltages and a steep rise after the threshold.
6. Resistance and Resistivity (Quantitative Approach) | 电阻与电阻率（定量分析）
Although the full resistivity equation (R = ρl / A) is not always explicitly examined, understanding it deepens your grasp of conductor properties. Resistivity (ρ) is a property of the material, measured in ohm-metres (Ω m).
Low resistivity materials like copper (ρ ≈ 1.7 × 10⁻⁸ Ω m) are excellent conductors. Nichrome, used for heating elements, has a much higher resistivity (ρ ≈ 1.1 × 10⁻⁶ Ω m). Insulators like glass have extremely high resistivity.
The experiment to determine resistivity involves measuring resistance for several lengths of a wire of known diameter, then plotting R against l. The gradient is ρ / A, so resistivity = gradient × A.
7. Thermistors and LDRs (Variable Resistors) | 热敏电阻和光敏电阻（可变电阻）
A thermistor is a temperature-dependent resistor. For most NTC (negative temperature coefficient) thermistors used in IGCSE, resistance decreases as temperature rises. This makes them useful in temperature-sensing circuits, e.g., fire alarms and thermostats.
An LDR (light-dependent resistor) has a resistance that decreases when light intensity increases. In bright light, resistance can fall to a few hundred ohms; in darkness, it can be millions of ohms. LDRs are used in automatic street lighting and light meters.
In potential divider circuits, a thermistor or LDR can be used to produce a voltage that varies with the environment. For example, an LDR in series with a fixed resistor can switch on a transistor when it gets dark.
8. Series and Parallel Circuits – Resistance Rules | 串联和并联电路 – 电阻规律
In a series circuit, the total resistance (Rtotal) is the sum of individual resistances: Rtotal = R₁ + R₂ + R₃ … . Adding more resistors in series increases the total resistance because the same current must pass through each, and the total p.d. is shared.
In a parallel circuit, the total resistance is given by 1/Rtotal = 1/R₁ + 1/R₂ + … . The total resistance of a parallel combination is always smaller than the smallest individual resistance. Adding more resistors in parallel provides more paths, reducing overall resistance.
Remember: for two identical resistors R in parallel, the total resistance is R/2. For two resistors R₁ and R₂ in parallel, a quick formula is Rtotal = (R₁ × R₂) / (R₁ + R₂).
9. Power Dissipation and Heating Effect | 功率消耗与热效应
When current flows through a resistor, electrical energy is converted to heat. The power (P) dissipated is given by P = I × V, which combined with Ohm’s law gives P = I² R and P = V² / R. This heating effect is used in kettles, heaters, and filament lamps, but is wasted in power lines.
电流通过电阻时，电能转化为热量。功率 P = I × V，结合欧姆定律可得 P = I² R 和 P = V² / R。这种热效应用于电水壶、取暖器和白炽灯，但在输电线中却是能量损耗。
In the syllabus, you may be asked why resistors get hot or to calculate energy transferred using E = I V t. High resistance is not the only factor; for a given current, a larger resistance gives more heat, but in parallel circuits, a lower resistance path draws more current and dissipates more power.
考纲可能问电阻为何发热，或要求用 E = I V t 计算能量转移。并非电阻越大产热越多：在给定电流下，电阻越大产热越多；但在并联电路中，低电阻支路电流更大，功率更高。
10. Understanding Potential Dividers | 理解分压器
A potential divider is a circuit with two resistors in series, dividing the input voltage into two parts. The output voltage Vout across R₂ is given by Vout = Vin × [R₂ / (R₁ + R₂)]. It allows you to obtain a voltage smaller than the source.
If a variable resistor or sensor replaces one resistor, the output voltage changes with resistance. For example, with a thermistor replacing R₂, as temperature rises, R₂ falls, so Vout decreases. This can be used to trigger a warning light.
The potential divider is a favourite exam topic. You must be able to calculate Vout and explain how output changes when resistance changes, especially with sensors like LDRs and thermistors.
11. Variable Resistors and Practical Circuit Control | 可变电阻与实际电路控制
A variable resistor (rheostat) allows you to change the resistance in a circuit smoothly. Moving the slider changes the length of resistive wire in the circuit, thereby changing the resistance. It is used to control current, e.g., in dimmer switches or volume knobs.
Potentiometer wiring is a three-terminal variable resistor used as a potential divider. All three terminals are connected: the total voltage is applied across the whole resistor, and the slider provides a variable output voltage from zero to full voltage.
In circuits, always protect components by including a fixed resistor in series to limit current, especially with LEDs which require a current-limiting resistor to prevent damage.
When describing the relationship in an I-V graph, always mention the shape (straight line or curve) and what it implies about resistance. Use ‘directly proportional’ only for straight lines through the origin.
Never forget units: resistance in ohms (Ω), voltage in volts (V), current in amps (A). Convert milliamperes to amperes (÷1000) and kilohms to ohms (×1000). Show all steps in calculations.
A common error is confusing series and parallel resistance rules. In series, Rtotal always increases; in parallel, Rtotal decreases. For parallel, the reciprocal formula must be used – simply adding resistances is wrong for parallel circuits.
For circuits with a fuse, remember that a fuse is a thin wire that melts when current exceeds its rating due to a fault. A low-resistance path (short circuit) draws a huge current, blowing the fuse and protecting the appliance.
Light, as a wave, bends around obstacles and spreads into regions that would otherwise be shadowed. In the Edexcel A Level Physics specification, diffraction of light is a core component of the ‘Waves and Particle Nature of Light’ topic. Understanding how a single slit, a double slit, and a diffraction grating produce characteristic patterns is essential for both theoretical explanations and practical examinations. This revision guide systematically covers the key concepts, formulas, experimental setups, and common pitfalls students encounter when studying light diffraction.
光作为一种波，在遇到障碍物时会发生弯曲并传播到几何阴影区。在 Edexcel A Level 物理大纲中，光的衍射是“波与光的粒子性”主题的核心内容。理解单缝、双缝和衍射光栅如何产生特征图样，对于理论解释和实验考试都至关重要。本复习指南系统地梳理了学生在学习光的衍射时需要掌握的关键概念、公式、实验装置以及常见易错点。
1. What Is Diffraction? | 什么是衍射？
Diffraction is the spreading of a wave as it passes through an aperture or moves past an obstacle. The amount of spreading depends on the size of the aperture relative to the wavelength. Noticeable diffraction occurs when the aperture width is comparable to, or smaller than, the wavelength of the incident wave.
In geometrical optics, light travels in straight lines and would produce a sharp shadow. However, diffraction explains why we see fringes of light and dark regions near the edge of an object, even when the source is coherent. This confirms the wave nature of light.
Huygens’ principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at a later time is the envelope of all these wavelets. When a wavefront encounters a slit, the points across the slit width emit wavelets, which then interfere with each other, producing a diffraction pattern.
If the slit is much wider than the wavelength, the wavelets from the centre and the edges largely cancel in the shadow region, resulting in a narrow beam. If the slit is narrow, the wavelets spread out significantly into the shadow region, demonstrating pronounced diffraction.
3. Single-Slit Diffraction – The Experiment | 单缝衍射实验
A typical single-slit diffraction experiment uses a monochromatic laser directed at a narrow vertical slit. A screen is placed several metres away. The resulting pattern consists of a broad, bright central maximum flanked by alternating dark and bright fringes of decreasing intensity.
The central maximum is twice as wide as the secondary maxima and is the brightest. Dark fringes (minima) occur where destructive interference is complete. The intensity distribution can be predicted using phasor addition of wavelets from points across the slit.
4. Dark Fringe Condition for Single Slit a sinθ = nλ | 单缝暗纹条件 a sinθ = nλ
For a single slit of width a, illuminated by light of wavelength λ, destructive interference occurs at angles θ satisfying the equation:
a sinθ = nλ
where n = ±1, ±2, ±3, … (n is the order of the dark fringe). The central maximum lies between n = -1 and n = +1.
对于宽度为 a 的单缝，以波长为 λ 的光照射时，在满足以下方程的角度 θ 处发生相消干涉：a sinθ = nλ，其中 n = ±1, ±2, ±3, …（n 是暗纹的级次）。中央明纹位于 n = -1 和 n = +1 之间。
This formula is derived by dividing the slit into two halves and requiring the path difference between a wavelet from the top of the slit and one from the midpoint to be λ/2. The same principle extends to higher orders by dividing the slit into an even number of equal segments.
The angular half-width of the central maximum is given by the first minimum: sinθ ≈ θ = λ / a for small angles. The full angular width is therefore 2λ / a. On a screen at distance D from the slit, the linear width of the central bright region is approximately:
This relationship shows that narrowing the slit (smaller a) increases the width of the central maximum, making diffraction more pronounced. It also indicates that longer wavelengths produce wider fringes, which is crucial when observing white light diffraction.
6. Diffraction Gratings – Construction and Principle | 衍射光栅——结构与原理
A diffraction grating consists of a large number of equally spaced parallel slits (or grooves), typically several hundred per millimetre. The distance between adjacent slits is called the grating spacing d. It is the reciprocal of the number of lines per unit length: d = 1 / N (e.g., if N = 300 lines per mm, d = 1×10⁻³ m / 300 = 3.33×10⁻⁶ m).
衍射光栅由大量等间距的平行狭缝（或刻槽）组成，通常每毫米有数百条。相邻狭缝之间的距离称为光栅常数 d，它是单位长度内线条数的倒数：d = 1 / N（例如，若 N = 300 条/毫米，则 d = 1×10⁻³ m / 300 = 3.33×10⁻⁶ m）。
Because there are many slits, the constructive interference peaks are very sharp and bright. The multiple-slit interference pattern is superimposed on the single-slit diffraction envelope, but for a grating with narrow slits, the single-slit effect mainly modulates the intensity of the orders.
7. The Grating Equation d sinθ = nλ | 光栅方程 d sinθ = nλ
For a transmission diffraction grating, the principal maxima occur when the path difference between waves from adjacent slits is an integer multiple of the wavelength. This gives the grating equation:
d sinθ = nλ
where n = 0, ±1, ±2, ±3, … (the order number). The zeroth order (n = 0) is the direct beam; first order (n = 1) appears on either side, and so on.
This equation is the key tool for measuring the wavelength of light. By finding the angle θ for a given order and using the known grating spacing d, the wavelength λ can be calculated. The maximum possible order is limited by n ≤ d / λ, because sinθ cannot exceed 1.
该方程是测量光波长的关键工具。通过测得某一级次的衍射角 θ 并使用已知的光栅常数 d，即可计算出波长 λ。最高可能级次受限于 n ≤ d / λ，因为 sinθ 不能大于 1。
8. Determining the Wavelength of Light Using a Grating | 利用光栅测定光波长
In a practical, students set up a laser perpendicular to a grating and record the positions of the bright spots on a screen. The angle θ is found by measuring the distance from the central maximum to the nth order bright spot, x, and the grating-to-screen distance D, using tanθ = x / D. If the angle is small, θ ≈ x / D, but for larger angles the full trig calculation is needed.
在实验中，学生将激光垂直照射光栅，并记录屏幕上亮斑的位置。通过测量从中央亮纹到第 n 级亮斑的距离 x，以及光栅到屏幕的距离 D，利用 tanθ = x / D 求得 θ。若角度较小，θ ≈ x / D，但角度较大时需要进行完整的三角函数计算。
Substituting d, θ, and n into d sinθ = nλ gives λ. Common safety precautions include avoiding direct eye exposure to the laser and handling the grating by its edges. The experiment is a standard way to demonstrate the wave nature of light and to verify the relationship between wavelength, grating spacing, and fringe spacing.
将 d、θ 和 n 代入 d sinθ = nλ 即可求得 λ。常见的安全预防措施包括避免激光直射眼睛以及手持光栅边缘。该实验是证明光的波动性并验证波长、光栅常数及条纹间距之间关系的标准方法。
9. White Light Diffraction and Spectra | 白光衍射与光谱
When a diffraction grating is illuminated by white light, each wavelength component is diffracted at a slightly different angle. This produces a continuous spectrum for each order (except n = 0). The violet end of the spectrum is closest to the central maximum, while red is deviated the most, because sinθ is proportional to λ.
At higher orders, the spectra may overlap: the blue end of the second order can coincide with the red end of the first order. This overlap can be predicted by comparing the angular positions for different n and λ. The grating acts as a dispersive element, similar to a prism, but with a linear relationship between sinθ and λ rather than a material-dependent dispersion.
在更高级次，光谱可能会重叠：第二级的紫端可能与第一级的红端重合。通过比较不同 n 和 λ 的角度位置可以预测这种重叠。光栅充当了类似于棱镜的色散元件，但其 sinθ 与 λ 呈线性关系，而非依赖于材料的色散。
10. Comparison: Single Slit vs Diffraction Grating | 对比：单缝与衍射光栅
It is important to distinguish the patterns and formulas. A single wide slit produces a broad central maximum and weaker secondary fringes, with minima given by a sinθ = nλ. A grating produces a series of very sharp, equally spaced (in sinθ) principal maxima, with the equation d sinθ = nλ.
区分图样和公式非常重要。单条宽缝产生宽阔的中央明纹和较弱的次级条纹，暗纹条件为 a sinθ = nλ。光栅产生一系列非常锐利、在 sinθ 上等间距分布的主极大，其方程为 d sinθ = nλ。
Key differences:
Single slit: minima condition a sinθ = nλ; central maximum width is 2λD / a.
单缝：暗纹条件 a sinθ = nλ；中央明纹宽度为 2λD / a。
Diffraction grating: maxima condition d sinθ = nλ; angular separation between orders is determined by d.
衍射光栅：明纹条件 d sinθ = nλ；级次之间的角间隔由 d 决定。
Grating fringes are much narrower and brighter, enabling precise wavelength measurements.
光栅条纹更窄更亮，从而能够实现精确的波长测量。
Double slit (Young’s fringes) is an interference pattern with a diffraction envelope; fringe width w = λD / s where s is slit separation.
双缝（杨氏条纹）是带有衍射包络的干涉图样；条纹宽度 w = λD / s，其中 s 为缝间距。
11. Exam Tips and Common Misconceptions | 考试技巧与常见误区
Edexcel exam questions often ask students to calculate the number of lines per mm from a given d, or to find the highest order visible. Always convert distances to metres and recall that d = 1 / N. When calculating d from ‘lines per mm’, remember that 300 lines per mm means 300 000 lines per metre, so d = 1 / 300 000 m.
Edexcel 考题经常要求学生根据给定的 d 计算每毫米的刻线数，或找出可见的最高级次。始终将距离单位转换为米，并牢记 d = 1 / N。当根据“每毫米线数”计算 d 时，记住 300 线/毫米即每米 300 000 线，因此 d = 1 / 300 000 m。
Common pitfalls include confusing a sinθ = nλ with d sinθ = nλ, using the wrong angle (e.g., confusing the angle from the central maximum with the angle between two first-order maxima), and forgetting that the central maximum is twice as wide for single-slit diffraction. Also, in white light questions, be prepared to describe the appearance of overlapping orders.
常见错误包括混淆 a sinθ = nλ 与 d sinθ = nλ，使用了错误的角度（例如将偏离中央明纹的角度误认为两个一级明纹之间的夹角），以及忘记单缝衍射中中央明纹宽度是其他明纹的两倍。此外，在白光问题中，要准备好描述光谱重叠现象。
Always check the unit of wavelength (often nanometres) and ensure your calculator is in degree mode when working with sinθ. For graph questions, plotting sinθ against n gives a straight line with gradient λ / d, from which λ can be found if d is known.
务必检查波长的单位（通常为纳米），并确保在使用 sinθ 时计算器处于角度模式。在图像分析题中，绘制 sinθ 对 n 的图像会得到一条斜率为 λ / d 的直线，若已知 d 则可求得 λ。
Published by TutorHao | Edexcel Physics Revision Series | aleveler.com
Welcome to the Dynamics revision guide for A-Level Physics. This module explores the relationship between motion and the forces that cause it, bridging kinematics (the description of motion) with Newton’s laws and momentum. Understanding these core principles is essential for solving problems ranging from projectile motion to collisions, and it lays the groundwork for more advanced topics like circular motion and oscillations.
Scalars are physical quantities that have only magnitude, such as distance, speed, mass, and time. Vectors, however, possess both magnitude and direction, including displacement, velocity, acceleration, and force. In dynamics, distinguishing between these two is crucial because vector addition follows specific rules, and direction often determines the sign of a quantity.
When adding vectors, you can use the tip-to-tail method or resolve them into perpendicular components. Resolving a vector into horizontal and vertical components using trigonometry (e.g. Fx = F cos θ, Fy = F sin θ) simplifies two-dimensional motion analysis. Always define a positive direction before beginning calculations.
进行矢量相加时，可以使用三角形法则（头尾相接法）或将矢量分解为相互垂直的分量。利用三角函数将矢量分解为水平与竖直分量（例如 Fx = F cos θ，Fy = F sin θ）可以简化二维运动分析。在开始计算之前，请务必规定一个正方向。
2. Displacement, Velocity and Acceleration | 位移、速度和加速度
Displacement is the vector quantity that measures the straight-line distance from the initial position to the final position in a specified direction. It differs from distance, which is a scalar path length. Velocity is the rate of change of displacement, while speed is the rate of change of distance.
Acceleration is the rate of change of velocity. An object accelerates if its speed changes or if its direction changes. The definition is a = Δv / Δt, and the unit is m s⁻². In the context of constant acceleration, the five kinematic quantities — initial velocity u, final velocity v, acceleration a, displacement s, and time t — are linked by a set of equations.
加速度是速度的变化率。如果一个物体的速度大小改变或方向改变，它就具有加速度。其定义为 a = Δv / Δt，单位是 m s⁻²。在匀加速直线运动中，五个运动学量——初速度 u、末速度 v、加速度 a、位移 s 和时间 t——通过一组方程联系起来。
3. Kinematic Equations for Constant Acceleration | 匀加速运动学方程
For motion in a straight line with uniform acceleration, the following SUVAT equations apply. These are derived from the definitions of velocity and acceleration assuming a is constant. You must be able to select the correct equation based on which quantities are known and which are required.
对于匀加速直线运动，以下 SUVAT 方程成立。它们是根据速度和加速度的定义在 a 恒定的假设下推导出来的。你必须能够根据已知量和待求量选择正确的方程。
v = u + at
v = u + at
s = ut + ½at²
s = ut + ½at²
v² = u² + 2as
v² = u² + 2as
s = ½(u + v)t
s = ½(u + v)t
Remember that s is the displacement, not the total distance travelled. If an object returns to its starting point, s = 0. For problems involving vertical motion under gravity, simply replace a with g (9.81 m s⁻²) and define upward or downward as positive consistently.
请记住，s 是位移，而不是通过的总路程。如果物体回到出发点，则 s = 0。对于涉及重力作用下的竖直运动问题，只需用 g（9.81 m s⁻²）代替 a，并始终如一地规定向上或向下为正方向。
4. Free Fall and Projectile Motion | 自由落体与抛体运动
Free fall is a special case of uniformly accelerated motion where the only force acting is gravity. The acceleration a becomes g, directed downward. When air resistance is negligible, all objects fall with the same acceleration regardless of mass.
自由落体是一种特殊的匀加速运动，其中唯一作用的力是重力。加速度 a 变为 g，方向向下。当空气阻力可以忽略时，所有物体无论质量大小都以相同的加速度下落。
Projectile motion is analysed by separating the horizontal and vertical components. Horizontally, velocity remains constant (if air resistance is ignored). Vertically, the motion is governed by constant acceleration g. The time of flight is determined by the vertical motion, while the range is found from the horizontal velocity and time.
抛体运动通过分解水平与竖直分量来分析。水平方向的速度保持不变（若忽略空气阻力）。竖直方向的运动由恒定加速度 g 控制。飞行时间由竖直运动决定，而射程则由水平速度和时间求出。
Use these equations for projectile motion: horizontal displacement x = ux t, vertical displacement y = uy t − ½gt², where ux = u cos θ and uy = u sin θ. Maximum height occurs when vertical velocity becomes zero, and the trajectory is parabolic in the absence of air resistance.
抛体运动使用以下方程：水平位移 x = ux t，竖直位移 y = uy t − ½gt²，其中 ux = u cos θ，uy = u sin θ。最大高度出现在竖直速度为零时，在没有空气阻力的情况下轨迹为抛物线。
5. Motion Graphs | 运动图像
Graphs of displacement-time, velocity-time, and acceleration-time provide powerful visual tools for interpreting motion. The gradient of a displacement-time graph gives the velocity; the gradient of a velocity-time graph gives the acceleration; the area under a velocity-time graph gives the displacement.
In a displacement-time graph, a straight line indicates constant velocity. A curved line indicates changing velocity (acceleration). On a velocity-time graph, a horizontal line shows constant velocity, a sloping straight line shows constant acceleration, and the area can be calculated using geometry or integration for uniform acceleration.
Newton’s First Law states that an object will remain at rest or move with constant velocity unless acted upon by a resultant external force. This is the principle of inertia, and it explains why observation of uniform motion implies zero net force.
Newton’s Second Law quantifies the relationship: resultant force F = m a, where m is the mass and a is the acceleration in the direction of the net force. This vector equation is fundamental in dynamics problems, linking kinematics with forces.
牛顿第二定律给出了定量关系：合外力 F = m a，其中 m 是质量，a 是沿合外力方向的加速度。这个矢量方程是动力学问题的核心，将运动学与力联系起来。
Newton’s Third Law states that if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These action-reaction pairs act on different objects and never cancel each other out in a free-body diagram of a single object.
牛顿第三定律指出，若物体 A 对物体 B 施加一个力，则物体 B 会同时对物体 A 施加一个大小相等、方向相反的力。这些作用力与反作用力分别作用在不同物体上，绝不能在单个物体的受力图中相互抵消。
7. Force, Mass and Acceleration | 力、质量和加速度
Mass is the measure of an object’s inertia — its resistance to change in motion. It is a scalar quantity measured in kilograms. Weight, on the other hand, is the gravitational force on an object, W = m g, and is a vector directed toward the centre of the Earth. Do not confuse mass and weight in force equations.
质量是物体惯性——即其抵抗运动状态变化的能力——的量度。它是一个标量，单位是千克。而重量是作用在物体上的重力，W = m g，是一个方向指向地心的矢量。在力方程中切勿混淆质量与重量。
When solving problems, draw a clear free-body diagram showing all forces acting on the object. Resolve forces into components parallel and perpendicular to the direction of motion. Apply F = m a in the net force direction. Typical exam questions involve objects on slopes, pulleys, and connected particles.
解题时，应画出清晰的受力图，标明作用在物体上的所有力。将力沿运动方向和垂直于运动方向分解，在合外力方向上应用 F = m a。常见考题涉及斜面上的物体、滑轮系统和连接体问题。
Normal reaction is perpendicular to the surface. / 法向支持力垂直于接触面。
Friction opposes relative motion and is μR where R is the normal reaction. / 摩擦力阻碍相对运动，大小为 μR，R 为法向支持力。
8. Momentum and Impulse | 动量和冲量
Linear momentum p is defined as the product of mass and velocity: p = m v. It is a vector quantity with the same direction as velocity. The rate of change of momentum is equal to the resultant force — in fact, Newton’s second law can be expressed as F = Δp / Δt.
线动量 p 定义为质量与速度的乘积：p = m v。它是矢量，方向与速度相同。动量的变化率等于合外力——事实上，牛顿第二定律可以表示为 F = Δp / Δt。
Impulse is the change in momentum caused by a force acting over time. It is given by Impulse = F Δt = Δp. This concept is particularly useful when a force varies over time, such as during collisions. The area under a force-time graph represents the impulse.
冲量是由力在一段时间内作用而引起的动量变化。冲量 = F Δt = Δp。当力随时间变化（例如碰撞过程中）时，这一概念特别有用。力-时间图像下方的面积表示冲量。
9. Conservation of Momentum | 动量守恒
In a closed system with no external resultant force, total linear momentum is conserved. This means the vector sum of momenta before an interaction (collision or explosion) equals the vector sum after. This principle is essential for solving problems involving collisions and recoil.
For a two-body collision: m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂, where u denotes velocities before and v after. Remember to assign positive and negative signs according to direction. Momentum conservation can be applied even if kinetic energy is not conserved, as in inelastic collisions.
Distinguish between elastic collisions (KE conserved) and inelastic collisions (KE not conserved). In perfectly inelastic collisions, the bodies stick together and move with a common velocity after the impact.
10. Problem-Solving Strategies for Dynamics | 动力学解题策略
Begin by identifying known quantities and converting units to SI (metres, seconds, kilograms, Newtons). Sketch a diagram showing all forces and velocities, including their directions. For constant-acceleration kinematics, write down the five SUVAT variables and circle those you know and the one you seek; then pick the equation that links them.
For dynamics problems involving forces, always draw a free-body diagram and apply Newton’s second law in two perpendicular directions if necessary. Check whether the system is in equilibrium (a = 0) or accelerating. In connected-particle systems, treat the whole system to find acceleration, then analyse individual parts to find internal forces such as tension.
Conservation of momentum is most effective when dealing with collisions or explosions with no external horizontal force. Always define a clear positive direction before writing momentum equations, and treat velocities as signed quantities.
Revise by practising past-paper questions under timed conditions, and pay attention to command words such as ‘state’, ‘explain’, and ‘calculate’. Clear working and unit handling are essential to secure full marks.
📚 AS Level Physics Paper 2: Examiner Reports on Experimental Investigations | AS物理Paper 2：实验探究考官报告解析
In AS Level Physics, Paper 2 is often a structured question paper that includes a significant experimental investigation component. Every year, examiner reports highlight the skills candidates need to demonstrate: from designing a practical to analysing data, evaluating errors, and drawing valid conclusions. This article distills key lessons from recent examiner reports, helping you understand exactly what examiners look for and how to avoid common pitfalls.
1. Overview of Paper 2 Experimental Questions | Paper 2 实验题概览
Paper 2 experimental questions typically carry a high weight of marks and are designed to test practical skills without actually being in a laboratory. You may be asked to plan an investigation, describe how to collect data, process results, draw a graph, or comment on sources of uncertainty. Examiners report that many candidates treat these questions as merely ‘common sense’, failing to use precise scientific language and rigour.
Paper 2 实验题通常占分很重，旨在脱离实验室环境考查实践技能。你可能会被要求设计探究方案、描述如何收集数据、处理结果、绘制图表或评论不确定度来源。考官报告指出，许多考生将这些题目仅视为“常识题”，未能使用精确的科学语言和严谨的逻辑。
2. Understanding Command Words | 理解指令词
Examiner reports stress that misinterpreting command words like ‘describe’, ‘explain’, ‘determine’, or ‘suggest’ is a major cause of lost marks. ‘Describe’ requires a step-by-step account of what you would do, without giving reasons; ‘explain’ demands scientific reasoning. In experimental design, you must state not only what equipment you use, but how and why you use it, if the question asks you to explain.
A frequent examiner note is that candidates write vague statements such as ‘measure the length’. You should always indicate the instrument (e.g. metre rule, vernier calipers), relevant precision, and a repeat-and-average approach to improve reliability.
一个常见的考官点评是，考生常写模糊的陈述，如“measure the length”。你应当始终指明所用的仪器（如米尺、游标卡尺）、相关精度，以及通过重复测量取平均值来提高可靠性的做法。
3. Designing a Fair Test | 设计公平实验
Many candidates lose marks because they do not identify the independent, dependent, and control variables. Examiner reports reveal that a good plan must explicitly state: ‘Keep the mass constant, change the length, and measure the period.’ You must mention how you will control each control variable and confirm that only one independent variable is changed at a time.
Examiners also expect a clear step-by-step method, not a bullet list without details. Include safety precautions only if the experiment involves potential hazards (e.g. heating, electricity). Do not add generic ‘wear goggles’ unless genuinely justified.
Choosing the right measuring instrument directly affects the quality of data. Examiner reports frequently note that candidates select equipment with insufficient resolution. For measuring the diameter of a wire, a micrometre screw gauge is appropriate; a ruler is not. You should justify your choice by comparing the resolution to the expected smallest reading.
A helpful table summarising typical instruments and their resolutions:
下面是一个总结典型仪器及其分辨率的表格：
Physical quantity 物理量
Instrument 仪器
Typical resolution 典型分辨率
Length (few cm)
Vernier calipers 游标卡尺
0.1 mm or 0.05 mm
Small thickness/diameter
Micrometre screw gauge 千分尺
0.01 mm
Time
Digital stopwatch 数字秒表
0.01 s (human reaction ~0.2 s)
Current
Ammeter 电流表
0.01 A or 0.001 A
5. Recording Data and Building Tables | 记录数据与构建表格
Examiner reports repeatedly emphasise that tables must have correct headings with units separated by a slash or brackets, e.g. ‘Length l / cm’ or ‘Time t (s)’. Raw data should be recorded to the resolution of the instrument, and the number of decimal places should be consistent in each column. Marks are lost for inconsistent significant figures.
考官报告反复强调，表格必须有正确的标头，单位用斜线或括号分隔，例如“Length l / cm”或“Time t (s)”。原始数据应记录到仪器的分辨率，且每一列的小数位数应保持一致。因有效数字不一致而失分的情况十分常见。
You should also design the table to include space for calculated quantities, such as average values or t², even if they are not yet computed. This shows you are planning the full data analysis pathway.
Graph work is a major area flagged in examiner reports. Common errors include: choosing awkward scales (e.g. multiples of 3 or 7), not using more than half the graph paper, forgetting axis labels with units, and drawing small, thick plots. The line of best fit must be thin, continuous, and balanced with equal numbers of points above and below.
Examiners look for accurate plotting to within half a small square. Use a sharp pencil and plot points as small crosses or encircled dots. For non-linear trends, you may be asked to calculate and plot transformed quantities like T² against l to obtain a straight line.
When calculating a gradient, you must use a large triangle that covers more than half of the drawn best-fit line. Show the coordinates used, clearly labelled on the graph. Examiner reports criticise candidates who read off small intervals or their own plotted points instead of points on the line. The gradient formula is:
Intercepts should be read directly from the graph if extended, and you must state whether the intercept is on the y-axis or x-axis. Significant figures in the final gradient must be consistent with the precision of the raw data.
8. Dealing with Uncertainty and Errors | 处理不确定度与误差
Examiner reports often mention that candidates confuse ‘error’ with ‘mistake’. A random error can be reduced by taking repeats and averaging; a systematic error (e.g. zero error, wrongly calibrated instrument) cannot be reduced by repetition and must be addressed by adjusting the apparatus or subtracting the zero offset. You may be asked to estimate absolute uncertainty from half the range of repeats.
When combining uncertainties, for addition or subtraction, add absolute uncertainties. For multiplication or division, add percentage uncertainties. You can show the logic:
Examiners expect you to comment on the reliability of your results by comparing repeated values or discussing the spread of points around the best-fit line. If a graph does not pass through the origin, you must state that this suggests a systematic error or a non-zero intercept of physical significance.
Suggesting improvements is another key skill. Avoid generic answers like ‘use more precise instruments’. Instead, suggest specific ways to reduce a named source of error: e.g. ‘use an electronic timer triggered by a light gate to eliminate reaction time error in measuring swing period’. This ties the improvement directly to a limitation you identified.
10. Common Pitfalls from Examiner Reports | 考官报告指出的常见陷阱
Reviewing examiner reports reveals recurring mistakes that you can avoid. The following table summarises the most frequent errors and how to tackle them.
回顾考官报告可以揭示那些能够避免的常见错误。下表总结了最常见的问题及应对方法。
Common mistake 常见错误
Why it costs marks 为何失分
How to avoid 如何避免
Not stating how to control variables
Plan is incomplete; marks for control variables are lost.
Explicitly write ‘Keep…constant by…’ for each variable.
Forgetting units in table headings
The data column becomes meaningless.
Use ‘Quantity / unit’ format.
Choosing an awkward scale for axes
Readings become difficult and plotting errors increase.
Use 1, 2, 5, 10… multiples; never 3, 7, 11…
Drawing a ‘dot-to-dot’ line instead of best-fit
Does not show trend; can’t be used for gradient.
Draw a single smooth straight line or curve with balanced scatter.
Using vague language in evaluation
Improvements not credited because they are not linked to specific errors.
Name the error, then state a precise improvement that reduces that error.
11. Practice and Revision Tips | 练习与复习技巧
To master experimental investigation questions, you should practise past paper questions under timed conditions, then carefully study the corresponding examiner reports. Pay attention to the mark scheme phrases that attract marks, such as ‘repeat and average’ or ‘measure perpendicular height with a set square’. Build a personal checklist of common apparatus, resolution values, and appropriate graph scales.
In revision, do not just read about experiments — try to sketch apparatus diagrams, write stepwise methods, and practise plotting data from given tables. Examiner reports show that active recall of practical procedures significantly improves performance.
Examiners consistently remind candidates that experimental questions are not about having carried out the exact experiment before. They assess transferable skills of scientific inquiry. Therefore, think critically, read the question carefully, and always respond with precise terminology and logical reasoning. Use the number of marks as a guide to the depth of answer required.
Finally, manage your time well. Leave enough room for graph plotting and analysis, a section that often carries the highest marks. A well-drawn graph with a clear gradient calculation can make the difference between grade boundaries.
📚 Experimental Investigation Skills for A-Level Physics | A-Level 物理实验探究技能
Experimental investigations form the heart of A-Level Physics, testing your ability to plan, execute, analyse and evaluate practical work. Whether you are determining the acceleration of free fall, measuring the resistivity of a wire, or investigating simple harmonic motion, a structured approach to experimental design and error analysis is essential. This guide breaks down the complete investigation cycle, from formulating a clear aim to writing an effective evaluation, and provides detailed strategies for tackling the practical questions found in Oxford AQA International A-Level Physics and other exam boards.
Before picking up any apparatus, you must clearly identify the independent variable (the quantity you deliberately change), the dependent variable (the quantity you measure as a result), and all control variables (quantities that must be kept constant to ensure a fair test). A well-written aim states precisely what relationship you are investigating, for example: “To investigate how the time period of a simple pendulum depends on its length, keeping mass and amplitude constant.” Ambiguous aims lead to confused data collection.
For A-Level practicals, variables are often linked by a known equation, such as T = 2π√(L/g) for a pendulum. You should rearrange the equation to show how a straight-line graph can be plotted. In this case, squaring both sides gives T² = (4π²/g) L, so plotting T² against L yields a straight line through the origin with gradient 4π²/g. Identifying this linearised form at the planning stage guides your choice of data range and intervals.
对于 A-Level 实验，变量之间通常由一个已知方程联系，比如单摆的 T = 2π√(L/g)。你应当重新排列方程，以表明如何绘制一条直线图像。在这里，两边平方得到 T² = (4π²/g) L，因此以 T² 对 L 作图，会得到一条过原点的直线，其斜率为 4π²/g。在规划阶段就识别出这种线性化形式，能指导你选择数据范围和间隔。
Control variables are often overlooked but are frequently the source of systematic error. For instance, when investigating the extension of a spring, the temperature of the room affects the spring constant, so performing the experiment in a draft-free environment and allowing the spring to settle is vital. List every control variable and state exactly how you will monitor or fix it.
A valid procedure is one that produces reliable data with the available apparatus. Your step-by-step method should be written in impersonal, imperative language (e.g. “Measure the mass of the trolley using a digital balance.”). Include details such as how many readings will be taken, the range and intervals of the independent variable, and how you will ensure repeats to identify anomalies.
Preliminary trials are a vital part of the design. By testing extremes of your intended range, you can check whether the apparatus responds linearly, whether the dependent variable is measurable with sufficient precision, and whether any safety issues arise. For example, when determining the resistivity of a metal wire, a preliminary run helps you choose a current that does not cause excessive heating, which would change resistance systematically.
Your plan must also include a clear diagram of the apparatus arrangement. Labels should show how measuring instruments are connected and indicate any reference points used for parallax-free readings, such as a set square for ruler measurements or a fiducial marker for timing oscillations.
Choosing the right instrument for a measurement is a crucial skill. Accuracy is often specified by the manufacturer as a percentage of full-scale deflection or as a number of least-count units. A digital multimeter may be needed instead of an analogue voltmeter to achieve small uncertainties in potential difference. The rule of thumb is to select an instrument whose resolution is at least ten times smaller than the smallest change you wish to detect.
When using a metre rule, always state that you will avoid parallax error by placing your eye directly above the scale or by using a digital sensor such as a motion encoder. For timing, electronic timers activated by light gates eliminate human reaction time, which can dominate errors in free-fall experiments. Detail how you will zero or calibrate instruments before use.
In experiments with multiple instruments, such as measuring the Young modulus of a wire, you need a micrometer screw gauge for the diameter (resolution typically 0.01 mm) and a travelling microscope or Vernier callipers for extension. Always justify your choice of instrument with reference to the magnitudes being measured and the required precision.
Random errors cause readings to scatter about a mean value and can be reduced by taking multiple repeat readings and calculating a mean. Systematic errors shift all readings in one direction; they persist despite repeats and can be identified by comparing with an accepted value or by employing a different method. Record how you will minimise both types of error: for example, measuring the diameter of a wire at several points along its length and in perpendicular directions reduces the systematic effect of non-uniformity.
Uncertainty is a quantitative expression of the doubt in a measurement. The absolute uncertainty in a single analogue reading is normally half the smallest scale division; for a digital instrument, it is the last significant digit unless stated otherwise. Percentage uncertainty = (absolute uncertainty / measured value) × 100%. Reducing percentage uncertainty often involves measuring a larger quantity, e.g. timing 20 oscillations and dividing by 20, rather than timing just one.
Data should be recorded in neat tables with clear headings that include the quantity and its unit, for example “Time for 20 oscillations, t/s”. The independent variable is best placed in the leftmost column, with repeats and the mean in subsequent columns. Never include units in the body of the table; place them in the heading. Show the raw data, the mean, and possibly the range of the repeats to allow later scatter analysis.
Consistent significant figures are vital. If your instrument reads to 0.01 V, then every voltage measurement should be recorded to two decimal places, even if the last digit is zero. Similarly, calculated means should be expressed to the same number of decimal places or perhaps one additional place for intermediate calculations. This discipline prevents rounding errors and maintains the integrity of uncertainty estimates.
6. Processing Data and Plotting Graphs | 处理数据与绘制图形
Graphs are the most powerful tool for revealing relationships. Use graph paper or software that allows precise plotting. Choose scales that occupy at least half of the grid in each direction and use simple multiples such as 1, 2, 5 or 10 units per large square. Axes must be labelled with the quantity and unit, e.g. “T² / s²”. Data points should be marked with fine crosses or dots surrounded by a small circle; error bars may be added to show the absolute uncertainty in each point.
A line of best fit balances the scatter above and below it, and should be drawn with a transparent ruler so that anomalous points are easily identified. Outliers that lie far from the line should be excluded from the analysis but circled and commented on. The gradient and intercept should be determined using a large triangle on a printed graph, selecting points on the line (not data points) as far apart as possible to minimise relative uncertainty.
If the relationship is y = mx + c, the gradient m equals Δy/Δx. Calculate the gradient using intervals: m = (y₂ − y₁) / (x₂ − x₁). The intercept c is read directly from the axis or calculated after finding m. If the line passes through the origin, do not force it — let the data decide. Instead, comment whether the intercept is close to zero compared with its uncertainty.
如果关系式为 y = mx + c，斜率 m 等于 Δy/Δx。用间隔计算斜率：m = (y₂ − y₁) / (x₂ − x₁)。截距 c 可从轴上直接读取，或在求出 m 后计算得出。如果直线过原点，不要强行让它过——让数据说话。相反，应评论截距与零的接近程度是否在其不确定度之内。
7. Calculating Uncertainties in Measured and Derived Quantities | 计算测量量和导出量的不确定度
When measurements are combined to calculate a final quantity, uncertainties propagate. The rules are summarised below. In each case, we assume the individual absolute uncertainties Δx, Δy are independent and of similar order.
For example, if a length is (12.0 ± 0.1) cm and a width is (5.0 ± 0.1) cm, the area uncertainty ΔA/A = 0.1/12.0 + 0.1/5.0 ≈ 0.0083 + 0.02 = 0.0283 ≈ 2.8%. Thus A = 60.0 cm² with absolute uncertainty roughly 60.0 × 0.0283 ≈ 1.7 cm², giving (60 ± 2) cm². Always round the uncertainty to one significant figure (or two if the first digit is 1) and round the value to the same decimal place.
When the same measurement appears in a multiplication, as in kinetic energy ½mv², the uncertainty in v² is Δ(v²)/v² = 2 Δv/v. The mass uncertainty adds at the same percentage level. Practice building an uncertainty budget table that lists each source, its percentage uncertainty, and justifies why you have neglected any source smaller than one-third of the largest.
8. Analysing Results and Drawing Conclusions | 分析结果与得出结论
Your conclusion must refer back to the original aim and use the processed data to state whether the data supports a particular relationship. If you obtained a straight line through the origin within experimental uncertainty, you can claim direct proportionality. If the intercept is significantly non-zero, suggest a systematic cause. Quote the gradient with its absolute uncertainty: e.g. gradient = 4.1 ± 0.2 m s⁻². Use this to determine the target physical constant, such as g from the pendulum gradient, and compare it with the accepted value.
你的结论必须回溯到原始目的，并利用处理过的数据说明数据是否支持某种特定关系。如果通过实验不确定度得到一条过原点的直线，你就可以声称是正比关系。如果截距明显不为零，则提出一个系统性原因。引用斜率及其绝对不确定度：例如，斜率 = 4.1 ± 0.2 m s⁻²。用斜率确定目标物理常数，比如从单摆的斜率求出 g，并与公认值相比较。
A percentage difference calculation quantifies agreement: % difference = |experimental value − accepted value| / accepted value × 100%. If this percentage difference is comparable to or smaller than your largest percentage uncertainty, the result is considered accurate. If not, there is an unaccounted systematic error. Never claim your experiment “proves” a law; say it is consistent with the expected relationship.
An excellent evaluation identifies the most significant sources of uncertainty and suggests realistic, concrete improvements. Do not just list generic problems like “human error”; instead specify that “the stopwatch was started and stopped manually, introducing a reaction time of about 0.2 s, which could be eliminated by using a photogate and data logger.” Link the improvement to the category of error it reduces.
Discuss the reliability of your data. Are the repeated readings consistent? If the range of repeats for a given independent variable is large relative to the instrument’s precision, it indicates random fluctuations due to environmental factors such as vibrations or air currents. Mention how you might dampen these effects or take more repeats to improve the reliability of the mean.
Finally, comment on any procedural limitations. For example, when using a simple pendulum, the small-angle approximation (θ < 10°) must be satisfied; if your amplitude exceeded this, a recommendation would be to use a protractor to monitor release angles. Also suggest extending the investigation, such as measuring g by a different method (free fall) to corroborate the result.
10. Common Pitfalls and How to Avoid Them | 常见陷阱及避免方法
Many students lose marks by confusing precision with accuracy. Precision relates to the spread of repeats (low random error), while accuracy describes closeness to the true value. A set of readings clustered around an incorrect mean is precise but inaccurate due to systematic error. Always discuss both aspects separately.
Another common mistake is using a “big triangle” on a computer-generated graph but selecting data points instead of points on the line of best fit. The large triangle method only reduces gradient uncertainty if the points taken are on the best-fit line. Also, do not forget to state that the gradient is calculated using a large interval on the line, not on the axes.
Omitting units from table headings or axis labels leads to ambiguity. An axis labelled “T” without units is meaningless. Similarly, always ensure that calculated quantities include compound units, e.g. “g / m s⁻²”. Use superscript notation (m s⁻²) consistently in all text, tables and graphs.
在表格标题或坐标轴标签中遗漏单位会导致歧义。一条标注为“T”而没有单位的轴是毫无意义的。同样，始终确保计算出的物理量包含复合单位，例如“g / m s⁻²”。在所有的正文、表格和图形中一贯地使用上标记法（m s⁻²）。
Lastly, do not just state “systematic error” without identifying the specific culprit. Examples include: a zero error in a micrometer, a meter rule with a worn end, a thermometer not immersed fully, or parallax when reading a meniscus. By naming the specific error and suggesting a calibration check or revised technique, your evaluation becomes highly creditworthy.
📚 A-Level AQA Physics: Past Paper Analysis | A-Level AQA 物理：历年真题解析
Going through years of AQA Physics past papers reveals consistent patterns, question types, and examiner expectations. This analysis breaks down the core themes, common pitfalls, and effective strategies to help you maximise your marks. Whether you are revising AS topics or the full A-level, understanding how the exam board frames questions is just as important as knowing the content.
回顾多年的 AQA 物理历年真题，会发现清晰的出题规律、题型框架与阅卷人期望。这篇解析将拆解核心主题、常见失分点以及高效的应对策略，助你最大化得分。无论你正在复习 AS 内容还是完整 A-level，理解考试局的命题思路与掌握知识点同等重要。
1. Overview of AQA Physics Exam Structure | AQA 物理考试结构概览
The AQA Physics specification (7407/7408) is assessed through Paper 1, Paper 2, and the practical-oriented Paper 3. Paper 1 covers topics 1–5 and 6.1 (measurements and their errors, particles, waves, mechanics, and materials), while Paper 2 explores topics 6.2–8 and assumed knowledge from earlier sections. Paper 3 includes a practical data analysis section and a written exam on the optional topic.
Past papers consistently allocate roughly 40% of marks to mathematical application and problem-solving, with the remainder split between conceptual explanation and practical-based questions. Recognising this balance helps you prioritise your revision; never neglect the written explanations, as a correct numerical answer alone rarely earns full marks without clear working and justification.
Thermal, fields, nuclear, plus assumed earlier topics
Paper 3
32%
Practical skills, data analysis, optional topic
2. Mechanics and Materials: Recurring Themes | 力学与材料：常见主题
Questions on kinematics, Newton’s laws, and conservation of energy appear in virtually every Paper 1. A classic past-paper item asks you to calculate the velocity of a projectile at a given time, requiring resolution of vectors and the suvat equations. The examiner expects you to state the direction as well as the magnitude, often awarding marks for a clear vector diagram.
Materials questions regularly explore stress, strain, and the Young modulus. You must be able to interpret force–extension graphs for ductile, brittle, and polymeric materials. A frequent exam task is to determine the Young modulus from a given gradient, converting units to pascals:
E = stress / strain = (F/A) / (ΔL/L₀)
Missing the conversion from mm² to m² is one of the most common unit errors.
3. Waves and Optics: Essential Graph Questions | 波与光学：核心图像题
Wave topics contribute around 15–20% of Paper 1 marks. Past papers show a strong emphasis on stationary waves, superposition, and interference. A typical question provides a graph of displacement against distance for a progressive wave and asks you to determine amplitude, wavelength, and phase difference. You must remember that phase difference is measured in radians:
Double-slit interference appears regularly, often combined with the diffraction grating equation nλ = d sin θ. Examiner reports note that candidates frequently forget to measure the angle from the centre or confuse orders with maxima. A labelled diagram of the apparatus can safeguard marks and clarifies your reasoning.
双缝干涉规律出现，常与光栅方程 nλ = d sin θ 结合考查。阅卷报告指出考生经常忘记从中心线测量角度，或混淆级数与亮纹。画出带标注的实验装置图能守住分数，并让你的推理更清晰。
4. Electricity and Circuits: Calculations and Practicals | 电学与电路：计算与实验
Circuit analysis routinely features internal resistance, potential dividers, and the use of logarithmic scales for components like thermistors. A favourite exam task is to determine the internal resistance of a cell from a V–I graph. The gradient equals –r and the y-intercept gives the emf ε. This graph-based approach tests both data analysis and formula rearrangement:
Resistivity questions demand careful unit handling. Candidates often forget to convert the diameter to radius, or leave cross-sectional area in mm² instead of m². The relationship
ρ = RA / L
is straightforward, yet past-paper data reveals that more than half of errors arise from sloppy unit conversions.
Particle physics constitutes a significant chunk of the new specification and appears in both years of the A-level. Conservation laws – charge, baryon number, lepton number, and strangeness – are the examiners’ favourites. A typical question presents an interaction and asks you to state whether it is possible, requiring a systematic check of each conservation rule.
The photoelectric effect appears almost annually. You must be able to explain why the maximum kinetic energy depends on frequency and not intensity, using the Einstein equation:
Eₖ(max) = hf – φ
Graph skills are tested when you sketch how photocurrent varies with applied potential for different frequencies or intensities.
光电效应几乎每年必考。你必须能解释最大动能取决于频率而非光强，并使用爱因斯坦方程：
Eₖ(max) = hf – φ
当题目要求画出不同频率或光强下光电流随外加电压的变化时，考查的正是图像绘制能力。
6. Further Mechanics and Thermal Physics | 进阶力学与热物理
Circular motion and simple harmonic motion (SHM) provide a rich source of challenging multi-step problems. Centripetal acceleration a = v²/r = ω²r often needs to be combined with Newton’s second law. A common exam request is to derive the time period of a simple pendulum from first principles; such derivations are frequently omitted in revision but are high-scoring opportunities.
圆周运动和简谐运动 (SHM) 催生了许多有难度的多步问题。向心加速度 a = v²/r = ω²r 通常要与牛顿第二定律结合使用。考试中常见要求是从基本原理推导单摆周期；这类推导常被复习忽略，却正是高分机会。
Thermal physics questions blend conceptual understanding with calculations of specific heat capacity and latent heat. Past papers show that the distinction between thermal energy and temperature is often misunderstood. The interrogation of a kinetic theory graph, such as the Maxwell–Boltzmann distribution, is a staple: you must explain why the curve flattens and shifts at higher temperatures, linking r.m.s. speed to
Gravitational and electric fields appear in almost every Paper 2, with a heavy weighting on comparisons between field strengths, potentials, and inverse-square laws. The definition of electric field strength
E = F/q
is deceptively simple, yet examiners often embed it in contexts requiring you to calculate the force on a charge moving in a uniform field between parallel plates.
引力场和电场几乎出现在每份试卷二中，且高度侧重场强、电势与平方反比律的比较。电场强度的定义式
E = F/q
看似简单，但阅卷人常常将其植入需要计算均匀场中运动电荷受力的情境中。
Capacitor charging and discharging is a perennial topic. The time constant τ = RC governs the exponential decay equations:
Q = Q₀ e^(–t/RC)
and
V = V₀ e^(–t/RC)
Multiple-choice and structured questions both test the interpretation of exponential graphs, so practice using logarithmic plots to deduce τ from a straight-line graph.
电容的充放电是永恒主题。时间常数 τ = RC 主导指数衰减方程：
Q = Q₀ e^(–t/RC)
及
V = V₀ e^(–t/RC)
选择与结构题都会考查指数图像的解读，因此需练习使用对数图像从直线中推求 τ。
8. Nuclear Physics and Option Topics | 核物理与选修主题
Nuclear radius estimation via electron diffraction or closest-approach methods is a frequent high-mark question. The relationship
R = R₀ A^(1/3)
requires you to calculate the radius or density of a nucleus and discuss its nearly constant density. A graph of R against A^(1/3) often appears, and the gradient yields R₀.
通过电子衍射或最近距离法估算原子核半径是常见的高分题。关系式
R = R₀ A^(1/3)
要求计算核半径或密度，并讨论核密度近乎恒定这一特点。常出现 R 对 A^(1/3) 的图像，其斜率即 R₀。
The optional topic (Astrophysics, Medical Physics, Engineering Physics, Turning Points in Physics, or Electronics) appears in Section B of Paper 3. Whichever option your centre has chosen, past papers consistently test detailed definitions and derivations. For instance, Astrophysics candidates often sketch the Hertzsprung–Russell diagram and explain stellar evolution; a solid understanding of Wien’s displacement law and the Stefan–Boltzmann law is essential.
选修主题（天体物理、医学物理、工程物理、物理学的转折点或电子学）出现于试卷三的 B 部分。无论学校选择哪一门，真题总是考查细致的定义与推导。例如，选修天体物理的学生经常需要绘制赫罗图并解释恒星演化；透彻理解维恩位移定律和斯特藩-玻尔兹曼定律至关重要。
9. Practical Skills and Data Analysis | 实验技能与数据分析
Paper 3 Section A tests your ability to analyse experimental data that you have not necessarily seen before. Questions typically provide a table of measurements and ask you to calculate the percentage uncertainty in a derived quantity, identify anomalous results, and plot a graph of best fit. You must know that the uncertainty in a gradient or intercept can be found from the worst acceptable line.
试卷三 A 部分考查你对未必接触过的实验数据的分析能力。题目通常给出一张测量数据表，要求计算导出量的百分不确定度，识别异常结果，并描出最佳拟合图像。你必须知道斜率或截距的不确定度可从最差可接受线求得。
Common practicals such as determining g by free-fall, the Young modulus of a wire, and the internal resistance of a cell are heavily featured. For instance, the determination of g using a simple pendulum requires a plot of T² against L. The gradient gives
4π²/g
, and examiners deduct marks if you fail to describe how to measure the pendulum length precisely from point of suspension to the centre of mass.
Examiner reports consistently highlight a few recurring errors. One is confusing scalar and vector quantities: claiming velocity is speed or that momentum is just mass times speed without considering direction. Another is failing to quote final answers to an appropriate number of significant figures, typically matching the least precise data given in the question.
In extended response questions, many candidates write vague statements such as ‘the resistance changes with temperature’ without specifying how or why. The mark scheme requires precise language: ‘As current increases, the temperature of the filament rises, causing increased lattice vibrations which scatter electrons more, thereby increasing resistance.’ Practising such chains of causal reasoning elevates your answers.
11. Effective Revision Using Past Papers | 利用真题高效复习
Treat past papers as a diagnostic tool rather than just a mock exam. Start by attempting a full paper under timed conditions, then mark it using the official mark scheme. Categorise your mistakes: were they due to missing knowledge, misinterpretation of the question, mathematical slip-ups, or insufficient explanation? This targeted feedback loop is far more efficient than passive reading.
Build a ‘command word’ glossary. AQA uses terms like ‘Explain’, ‘Describe’, ‘Calculate’, and ‘Suggest’ with specific expectations. For example, ‘Explain’ demands a step-by-step cause-and-effect account, whereas ‘Describe’ needs factual recall without reasoning. Reviewing past-paper mark allocations per command word sharpens your exam technique.
Analysing AQA Physics past papers reveals a highly structured exam that rewards precise mathematical working, clear conceptual explanations, and strong practical data skills. The key to success is not just doing more papers, but systematically learning from each attempt and plugging the specific gaps that the mark scheme exposes. With this evidence-based revision strategy, you can walk into the exam hall confident that you understand exactly what the examiner is looking for.