Tag: Physics

📚 A-Level Physics: Capacitors – Key Points Review | A-Level 物理：电容 考点精讲

In A-Level Physics, capacitors are essential components that store electrical energy in an electric field. Understanding their behaviour in direct current (DC) circuits, including charging and discharging processes, is crucial for exam success. This article provides a comprehensive review of key capacitor concepts, formulas, and common pitfalls.

在 A-Level 物理中，电容器是储存电场中电能的关键元件。理解它们在直流电路中的行为，包括充电和放电过程，对于考试至关重要。本文全面回顾关键的电容器概念、公式以及常见误区。

1. Capacitance Definition and Formula | 电容定义与公式

Capacitance (C) is defined as the charge (Q) stored per unit potential difference (V) across the plates: C = Q / V. The SI unit is the farad (F), where 1 F = 1 C V⁻¹. In practice, capacitance values are usually expressed in microfarads (µF, 10⁻⁶ F), nanofarads (nF, 10⁻⁹ F) or picofarads (pF, 10⁻¹² F).

电容（C）定义为极板间储存的电荷（Q）与电势差（V）之比：C = Q / V。国际单位是法拉（F），1 F = 1 C V⁻¹。实际应用中，电容值通常以微法（µF, 10⁻⁶ F）、纳法（nF, 10⁻⁹ F）或皮法（pF, 10⁻¹² F）表示。

For a fixed charge, the potential difference is inversely proportional to capacitance (V ∝ 1/C). Always check that units are consistent: charge in coulombs, voltage in volts, to obtain capacitance in farads.

对于固定电荷，电势差与电容成反比（V ∝ 1/C）。始终检查单位是否一致：电荷用库仑，电压用伏特，得到的电容以法拉为单位。

2. Parallel Plate Capacitor | 平行板电容器

For an ideal parallel-plate capacitor, the capacitance depends on the plate area (A), plate separation (d) and the permittivity (ε) of the material between the plates: C = εA / d. Greater area and smaller separation give higher capacitance.

对于理想的平行板电容器，电容取决于极板面积（A）、极板间距（d）和极板间材料的介电常数（ε）：C = εA / d。面积越大、间距越小，电容越大。

In a vacuum, ε = ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹. When a dielectric is inserted, ε = ε₀εᵣ, where εᵣ is the relative permittivity (dimensionless). Typical εᵣ values are 2–10 for plastic, and around 80 for water.

真空中 ε = ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹。插入电介质时，ε = ε₀εᵣ，其中 εᵣ 为相对介电常数（无量纲）。塑料的典型 εᵣ 值为 2–10，水约为 80。

3. Dielectrics and Dielectric Strength | 电介质与介电强度

A dielectric increases capacitance because its polarized molecules align with the applied electric field, reducing the effective field between the plates for a given charge. The capacitance with dielectric is C = εᵣ C₀, where C₀ is the vacuum capacitance.

电介质能增大电容，因为其极化分子与外电场对齐，减弱了给定电荷下极板间的有效电场。有电介质的电容为 C = εᵣ C₀，其中 C₀ 为真空电容。

However, every dielectric has a maximum electric field it can withstand before breakdown, called dielectric strength. Exceeding this value causes the dielectric to become conducting and permanently damages the capacitor.

然而，每种电介质都有一个能承受的最大电场，称为介电强度。超过该值会导致电介质导电，永久损坏电容器。

4. Energy Stored in a Capacitor | 电容器储存的能量

Energy is stored in the electric field between the plates when a capacitor is charged. The stored energy can be calculated using three equivalent formulas: E = ½ QV = ½ CV² = ½ Q²/C. This energy is released when the capacitor discharges, for example in a camera flash or a defibrillator.

给电容器充电时，能量储存在极板间的电场中。储存的能量可用三个等价公式计算：E = ½ QV = ½ CV² = ½ Q²/C。放电时该能量释放，例如在相机闪光灯或除颤器中。

Derivation note: Work done to move a small charge dq across a potential V is V dq, with V = q/C. Integrating from 0 to Q yields ½ Q²/C. This is a common exam derivation.

推导说明：将微小电荷 dq 移动通过电势 V 所做的功为 V dq，而 V = q/C。从 0 积分到 Q 即得 ½ Q²/C。这是常见的考试推导。

5. Charging and Discharging Circuits | 充放电电路

When a capacitor is connected in series with a resistor and a DC supply, the charge builds up according to q(t) = Q₀ (1 – e^–t/RC) and the voltage follows V(t) = V₀ (1 – e^–t/RC). The current is initially V₀/R and decays exponentially: I(t) = I₀ e^–t/RC.

当电容器与电阻和直流电源串联连接时，电荷按 q(t) = Q₀ (1 – e^–t/RC) 增长，电压遵循 V(t) = V₀ (1 – e^–t/RC)。电流初始为 V₀/R，随后指数衰减：I(t) = I₀ e^–t/RC。

During discharging (no supply), the capacitor acts as a source. The charge, voltage and current all decrease exponentially: q(t) = Q₀ e^–t/RC, V(t) = V₀ e^–t/RC, I(t) = I₀ e^–t/RC, where Q₀, V₀ and I₀ are initial values.

放电过程中（无电源），电容器充当电源。电荷、电压和电流均呈指数衰减：q(t) = Q₀ e^–t/RC、V(t) = V₀ e^–t/RC、I(t) = I₀ e^–t/RC，其中 Q₀、V₀ 和 I₀ 为初始值。

6. Time Constant and Exponential Decay | 时间常数与指数衰减

The time constant τ = RC (in seconds) is a crucial parameter. During charging, after one time constant, the capacitor reaches 63% of its final charge or voltage. During discharging, it falls to 37% of its initial value (since e⁻¹ ≈ 0.37).

时间常数 τ = RC（单位为秒）是一个关键参数。充电时，经过一个时间常数，电容器达到最终电荷或电压的 63%。放电时，它跌至初始值的 37%（因为 e⁻¹ ≈ 0.37）。

After about 5τ, the capacitor is considered fully charged or discharged (over 99%). Doubling either R or C doubles the time constant, slowing the process. This is often tested experimentally or in graph interpretation.

约 5τ 后，电容器被认为已充满电或放完电（超过 99%）。将 R 或 C 加倍会使时间常数加倍，过程变慢。这在实验或图像解读中常被考查。

7. Graphical Analysis of RC Circuits | RC电路的图像分析

Plots of charge, voltage or current against time are exponential curves. The initial gradient of the q–t or V–t graph during charging is Q₀/τ or V₀/τ. The current–time graph has the same shape with initial value I₀ = V₀/R.

电荷、电压或电流随时间的变化图是指数曲线。充电过程中 q–t 或 V–t 图的初始梯度为 Q₀/τ 或 V₀/τ。电流–时间图具有相同形状，初始值 I₀ = V₀/R。

To determine τ from experimental data, plot ln(q) or ln(V) against t. The result is a straight line with gradient –1/τ. This linearisation is a frequent practical skill.

要从实验数据求出 τ，可绘制 ln(q) 或 ln(V) 关于 t 的图像，得到斜率为 –1/τ 的直线。这种线性化处理是常见的实验技能。

8. Capacitors in Series and Parallel | 电容的串联与并联

Capacitors in parallel share the same voltage. The total capacitance is the sum: C_total = C₁ + C₂ + C₃ + …. This increases the overall charge storage for a given voltage.

并联电容器两端电压相同。总电容为各电容之和：C_total = C₁ + C₂ + C₃ + …。在给定电压下，这增加了总储存电荷量。

Capacitors in series each hold the same charge. The reciprocal sum rule applies: 1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + …. The total capacitance is always smaller than the smallest individual capacitor.

串联电容器每个储存的电荷相同。适用倒数求和规则：1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + …。总电容总是小于最小的单个电容值。

9. Practical Applications of Capacitors | 电容器的实际应用

Capacitors are used in smoothing circuits after rectification to reduce ripple in DC power supplies. In timing circuits, the RC time constant controls delays or oscillations (e.g., blinking lights). They also deliver intense pulses of energy in camera flashes and defibrillators.

电容器用于整流后的平滑电路，以减少直流电源的纹波。在定时电路中，RC 时间常数控制延迟或振荡（如闪烁灯）。它们还在相机闪光灯和除颤器中提供强能量脉冲。

Beyond DC circuits, capacitors play a role in AC filtering, coupling/decoupling signals, and in touch screens where capacitance changes upon contact. Supercapacitors store far greater energy for battery backup systems.

除直流电路外，电容器还用于交流滤波、信号耦合/去耦合，以及触摸屏（接触时电容改变）。超级电容器储存大得多的能量，用于备用电池系统。

10. Exam Tips and Common Pitfalls | 考试技巧与常见误区

Always convert capacitance units to farads (e.g., 10 µF = 10 × 10⁻⁶ F) before substituting into formulas. When using E = ½ CV², ensure V is in volts; a common error is leaving kV unchecked. For series/parallel combinations, first simplify the circuit step by step.

在代入公式前，始终将电容单位转换为法拉（例如 10 µF = 10 × 10⁻⁶ F）。使用 E = ½ CV² 时确保 V 以伏特为单位；常见错误是未将千伏换算。对于串并联组合，先逐步简化电路。

Do not confuse charging and discharging equations. In graphs, read off time constants carefully using the 63%/37% method or gradients. Sketch exponential curves clearly, showing the smooth approach to asymptotes. A quick ln plot can verify exponential behaviour in experiments.

不要混淆充电和放电方程。图像中，使用 63%/37% 方法或梯度仔细读取时间常数。绘制指数曲线时要清晰，显示平滑趋近渐近线。实验中可用快速对数作图验证指数行为。

Remember: if the question involves energy transfer during charging, 50% of the energy delivered by the battery is stored in the capacitor; the rest is dissipated as heat in the resistance – a key conceptual point often tested.

记住：如果题目涉及充电过程中的能量转移，电池提供的能量有 50% 储存在电容器中，其余以热的形式耗散在电阻上——这是一个常考的关键概念点。

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📚 A-Level Physics Unit 4: Deriving Key Formulas from the Jan 19 Insert | A-Level 物理 Unit 4：2019年1月公式表关键公式推导

Welcome to this in-depth derivation guide for the essential equations listed in the A-Level Physics Unit 4 formula insert (January 2019). Understanding the origins of these formulas not only deepens your conceptual grasp but also arms you with the ability to tackle unseen problems with confidence. We will walk through step-by-step reasoning for each key relationship.

欢迎阅读这篇深入推导指南，分别呈现 A-Level 物理 Unit 4 公式表（2019年1月）所列的关键方程。理解公式的由来不仅深化概念理解，还能让你自信地应对陌生问题。我们将逐步推演每一个重要关系。

1. Momentum and Impulse | 动量与冲量

Momentum p is defined as the product of mass and velocity: p = mv. This is a vector quantity with the same direction as velocity. Impulse measures the effect of a force acting over a time interval and is defined as the change in momentum: Impulse = Δp = FΔt for a constant force. The relationship follows directly from Newton’s second law F = dp/dt; integrating over time yields the impulse-momentum theorem.

动量 p 定义为质量与速度的乘积：p = mv，是与速度同向的矢量。冲量衡量力在一段时间内作用的效果，定义为动量的变化：冲量 = Δp = FΔt（恒力情况下）。这一关系直接来自牛顿第二定律 F = dp/dt；对时间积分便得到冲量-动量定理。

2. Centripetal Acceleration | 向心加速度

Consider an object moving in a circle of radius r at constant speed v. In a short time Δt, the velocity vector rotates through a small angle Δθ. The change in velocity Δv points roughly toward the centre and has magnitude vΔθ. Since angular speed ω = Δθ/Δt, the acceleration magnitude is a = Δv/Δt = vω. Using v = ωr, we obtain the familiar forms a = v²/r = ω²r. This acceleration is always directed towards the centre, hence the term ‘centripetal’.

考虑物体以恒定速率 v 沿半径为 r 的圆周运动。在短时间 Δt 内，速度矢量转过小角 Δθ。速度变化量 Δv 大致指向圆心，其大小为 vΔθ。因为角速度 ω = Δθ/Δt，加速度大小为 a = Δv/Δt = vω。代入 v = ωr，得到熟知的形式 a = v²/r = ω²r。此加速度始终指向圆心，故称之为‘向心’加速度。

3. Coulomb’s Law and Electric Field | 库仑定律与电场

Coulomb’s law gives the force between two point charges: F = kQ₁Q₂/r², where k = 1/(4πε₀). The electric field strength E at a point is the force per unit positive charge: E = F/q. For a point charge Q, the radial field has magnitude E = kQ/r². The direction is away from a positive charge and towards a negative charge. In a uniform field between parallel plates, E = V/d.

库仑定律给出两个点电荷之间的作用力：F = kQ₁Q₂/r²，其中 k = 1/(4πε₀)。电场强度 E 定义为单位正电荷所受的力：E = F/q。对于点电荷 Q，其径向电场的大小为 E = kQ/r²，方向背离正电荷而指向负电荷。在平行板间的匀强电场中，E = V/d。

4. Electric Potential around a Point Charge | 点电荷周围的电势

Electric potential V at a point is the work done per unit charge to move a small test charge from infinity to that point. The force on the test charge q is F = qE, with E = kQ/r² directed radially. The work done against the electric force is dW = -F dr = -qE dr. Integrating from infinity to a distance r gives V = W/q = ∫_∞^r (-E dr) = ∫_∞^r (-kQ/r² dr) = kQ/r. This result shows the potential is positive for positive Q and decreases as 1/r.

电势 V 是指将单位正电荷从无穷远处移到该点时所做的功。试探电荷 q 所受电场力为 F = qE，其中 E = kQ/r² 沿径向。反抗电场力做功 dW = -F dr = -qE dr。从无穷远积分至距离 r 可得 V = W/q = ∫_∞^r (-E dr) = ∫_∞^r (-kQ/r² dr) = kQ/r。结果说明正电荷的电势为正，且以 1/r 递减。

5. Capacitance of a Parallel Plate Capacitor | 平行板电容器的电容

A parallel plate capacitor with plate area A and separation d stores charge Q and develops a voltage V. Using Gauss’s law, the uniform electric field between the plates is E = σ/ε₀ = Q/(Aε₀). The potential difference is V = Ed = Qd/(Aε₀). Capacitance is defined as C = Q/V, so substituting gives C = ε₀A/d. This formula highlights the geometric dependence: larger area increases capacitance, while greater separation reduces it.

面积为 A、间距为 d 的平行板电容器储存电荷 Q 并产生电压 V。利用高斯定律，两板间的匀强电场为 E = σ/ε₀ = Q/(Aε₀)。电势差 V = Ed = Qd/(Aε₀)。电容定义为 C = Q/V，代入即得 C = ε₀A/d。该公式体现出几何依赖性：板面积越大则电容越大，间距越大则电容越小。

6. Energy Stored in a Capacitor | 电容器储存的能量

Charging a capacitor requires work because each additional element of charge dq is moved against the growing voltage V = q/C. The work done dW = V dq = (q/C) dq. Integrating from 0 to the final charge Q gives total stored energy W = ∫₀^Q (q/C) dq = ½ Q²/C. Using C = Q/V, this can be rewritten as W = ½ Q V = ½ C V². This energy is stored in the electric field between the plates.

给电容器充电需要做功，因为每增加微小电荷 dq 都要反抗不断升高的电压 V = q/C。做功 dW = V dq = (q/C) dq。从 0 积分到最终电荷 Q，得总储能 W = ∫₀^Q (q/C) dq = ½ Q²/C。利用 C = Q/V，可改写为 W = ½ Q V = ½ C V²。这部分能量储存在两极板间的电场中。

7. Magnetic Force on a Current-Carrying Wire | 电流在磁场中的受力

When a wire of length L carrying current I is placed perpendicular to a uniform magnetic field B, each charge carrier experiences a magnetic force F = Bqv. For a steady current, the total moving charge in the wire is q = It, and the drift velocity relates to length by L = v t. Combining, the total force is F = B (It) v = B I (v t) = B I L. If the wire is at an angle θ to the field, the perpendicular component gives F = B I L sinθ. This is the foundation for motor effect calculations.

当长度为 L、载有电流 I 的导线垂直于匀强磁场 B 放置时，每个载流子受到磁力 F = Bqv。对于稳恒电流，导线中运动的总电荷 q = It，且漂移速度与长度的关系为 L = v t。综合得总力 F = B (It) v = B I (v t) = B I L。若导线与磁场成 θ 角，则垂直分量给出 F = B I L sinθ。这是电动机效应计算的基础。

8. Magnetic Flux and Faraday’s Law | 磁通量与法拉第定律

Magnetic flux Φ through a surface is defined as Φ = B A cosθ, where θ is the angle between the field lines and the normal to the surface. Faraday’s law states that the induced emf in a coil is equal to the negative rate of change of flux linkage: ε = -N dΦ/dt. The negative sign embodies Lenz’s law, which says the induced current opposes the change producing it. A common derivation for a straight wire moving perpendicularly through a field gives the motional emf ε = B L v, consistent with the flux cutting picture.

通过某一面的磁通量 Φ 定义为 Φ = B A cosθ，其中 θ 是磁感线与面法线的夹角。法拉第定律指出，线圈中感生电动势等于磁通匝链数变化率的负值：ε = -N dΦ/dt。负号体现了楞次定律——感生电流总在阻碍产生它的变化。对于直导线垂直切割磁感线的常见推导给出动生电动势 ε = B L v，与磁通切割图像一致。

9. The Transformer Equation | 变压器公式

An ideal transformer operates with two coils wound on a common iron core so that the changing flux from the primary coil links fully with the secondary coil. The rate of flux change is the same for both, so the induced emf per turn is identical: ε_p/N_p = ε_s/N_s. Since the terminal voltages are nearly equal to the induced emfs, we get V_p/V_s = N_p/N_s. Assuming 100% efficiency, power conservation gives V_p I_p = V_s I_s, leading to I_s/I_p = N_p/N_s.

理想变压器在共同铁芯上绕有两个线圈，使初级线圈的变化磁通完全交链到次级线圈。两线圈磁通变化率相同，因而每匝感生电动势相等：ε_p/N_p = ε_s/N_s。由于端电压近似等于感生电动势，可得 V_p/V_s = N_p/N_s。假设效率100%，功率守恒给出 V_p I_p = V_s I_s，继而导出 I_s/I_p = N_p/N_s。

10. Elastic Collisions in One Dimension | 一维弹性碰撞

In a perfectly elastic collision, both momentum and kinetic energy are conserved. For two objects of masses m₁ and m₂ with initial velocities u₁ and u₂, the conservation laws are: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ and ½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂². When the target is initially stationary (u₂ = 0), solving simultaneously yields v₁ = (m₁ – m₂)u₁/(m₁ + m₂) and v₂ = (2 m₁)u₁/(m₁ + m₂). These special cases are frequently tested in particle physics and mechanics problems.

完全弹性碰撞中，动量和动能均守恒。设有质量 m₁ 和 m₂ 的两物体，初速度分别为 u₁ 和 u₂，守恒律为：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ 以及 ½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂²。当靶最初静止 (u₂ = 0)，联立解得 v₁ = (m₁ – m₂)u₁/(m₁ + m₂) 和 v₂ = (2 m₁)u₁/(m₁ + m₂)。这些特殊情况常在粒子物理与力学问题中考查。

11. Radioactive Decay Law and Half-Life | 放射性衰变定律与半衰期

The activity of a radioactive sample is proportional to the number of undecayed nuclei N: A = λN, where λ is the decay constant. The underlying differential equation is dN/dt = -λN. Separating variables and integrating gives ln(N/N₀) = -λt, or N = N₀ e^-λt. The half-life T_½ is the time taken for half the nuclei to decay; setting N = N₀/2 yields e^-λT_½ = ½, so T_½ = ln2/λ. The same form applies to activity and mass.

放射性样品的活度与尚未衰变的原子核数 N 成正比：A = λN，其中 λ

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📚 A-Level Physics: Nuclear Physics Key Points and Revision | A-Level 物理：核物理考点精讲

Nuclear physics digs into the heart of matter, exploring the structure of the atomic nucleus, the forces that hold it together, and the transformations that release enormous amounts of energy. From radioactivity and half-life to fission, fusion and E=mc², these concepts form a core part of the A-Level Physics syllabus and link directly to real-world applications such as nuclear power and medical imaging.

核物理深入物质的核心，探索原子核的结构、将其凝聚在一起的力以及释放巨大能量的核转变。从放射性、半衰期到裂变、聚变和 E=mc²，这些概念构成 A-Level 物理课程的核心部分，并与核电、医学成像等现实应用直接相连。

1. Structure of the Nucleus and Isotopes | 原子核结构与同位素

Every atom contains a tiny, dense nucleus made up of nucleons — positively charged protons and uncharged neutrons. The number of protons is the atomic number Z, which defines the element, while the total number of nucleons is the mass number A. A nuclide is written as ^A_ZX, for example ¹²₆C. Isotopes are atoms of the same element with the same Z but different A, such as carbon-12 and carbon-14. They have identical chemical properties but different nuclear stability.

每个原子都含有一个微小致密的原子核，由核子——带正电的质子和不带电的中子组成。质子数即原子序数 Z，决定了元素的种类，而核子总数是质量数 A。核素表示为 ^A_ZX，例如 ¹²₆C。同位素是质子数相同但质量数不同的原子，比如碳‑12 和碳‑14。它们化学性质相同，但核稳定性不同。

2. The Strong Nuclear Force | 强核力

The nucleus does not fly apart despite electrostatic repulsion between protons because a powerful attractive force — the strong nuclear force — acts between all nucleons. This force is very short-range (∼10⁻¹⁵ m), attractive at separations around 1–3 fm, and repulsive at extremely small distances to prevent nucleon collapse. It is charge-independent, meaning it acts equally between proton-proton, neutron-neutron and proton-neutron pairs. In larger nuclei, extra neutrons are needed to add strong-force attraction without increasing electrostatic repulsion, which explains the neutron-to-proton ratio trend along the stability line.

尽管质子间存在静电斥力，原子核并未飞散，这是因为所有核子之间还存在一种强大的吸引力——强核力。该力为极短程力（约 10⁻¹⁵ m），在 1–3 fm 的间距上表现为引力，而在极近距离上表现为斥力，以防止核子塌缩。强核力与电荷无关，质‑质、中‑中和质‑中对间的作用强度相同。在较大的核中，需要额外的中子来增加强核力吸引而不增大静电斥力，这就解释了沿稳定线变化的中子‑质子比趋势。

3. Mass Defect and Binding Energy | 质量亏损与结合能

The mass of a nucleus is always less than the sum of the masses of its separate nucleons. This difference is called the mass defect (Δm). The missing mass has been converted into the binding energy that holds the nucleus together, according to E = Δmc². Binding energy is the work required to separate a nucleus into its individual protons and neutrons. It is usually expressed in MeV; 1 atomic mass unit (u) is equivalent to 931.5 MeV of energy. For example, the mass defect for helium-4 is about 0.0304 u, giving a total binding energy of roughly 28.3 MeV.

原子核的质量总是小于其各个独立核子的质量之和。这个差值称为质量亏损（Δm）。根据 E = Δmc²，亏损的质量已转化为将核子结合在一起的结合能。结合能是将一个原子核完全分离成独立质子和中子所需做的功，常以 MeV 表示；1 个原子质量单位 (u) 相当于 931.5 MeV 的能量。例如，氦‑4 的质量亏损约为 0.0304 u，对应的总结合能大约为 28.3 MeV。

4. Binding Energy per Nucleon and Stability | 每个核子的结合能与稳定性

Dividing the total binding energy by the number of nucleons gives the binding energy per nucleon. A graph of this quantity against mass number A shows a broad peak around iron-56, the most stable nucleus. For light nuclei, fusion dramatically increases binding energy per nucleon; for very heavy nuclei, fission increases it. A high binding energy per nucleon means a more tightly bound, lower energy state, so nuclei tend to move towards the iron peak through radioactive decay or nuclear reactions.

将总结合能除以核子数得到每个核子的平均结合能。该量随质量数 A 变化的曲线在铁‑56 附近呈现一个宽阔的峰顶，表明铁‑56 是最稳定的核。对于轻核，聚变可大幅提高每个核子的结合能；对于很重的核，裂变则能提高该值。每个核子结合能越高，表示原子核束缚越紧、能量越低，因此原子核会通过放射性衰变或核反应向铁峰移动。

5. Radioactive Decay: Alpha, Beta and Gamma | 放射性衰变：α、β 和 γ 辐射

Unstable nuclei spontaneously emit radiation to become more stable. The three main types are alpha (α), beta (β) and gamma (γ). Their properties are summarised below:

不稳定的原子核会自发辐射以变得更加稳定，主要类型有 α、β 和 γ 三种。它们的性质总结如下：

Beta decay also involves an antineutrino (ν̄) to conserve energy and momentum. For beta-plus (β⁺) decay, a positron ⁰₊₁e and a neutrino are emitted.

β 衰变还伴随反中微子 (ν̄) 以保持能量和动量守恒。对于 β⁺ 衰变，会发射正电子 ⁰₊₁e 和一个中微子。

6. Decay Equations and Conservation Laws | 衰变方程与守恒定律

Nuclear decay equations must obey conservation of nucleon number and charge. In alpha decay, the parent nucleus loses two protons and two neutrons, so A decreases by 4 and Z by 2. For example:

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

核衰变方程必须遵守核子数守恒和电荷守恒。在 α 衰变中，母核失去两个质子和两个中子，因此 A 减 4，Z 减 2。例如：

²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

In β⁻ decay, a neutron changes into a proton, so A stays the same, Z increases by 1, and an electron plus an antineutrino are emitted:

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄

在 β⁻ 衰变中，一个中子转变成质子，因此 A 不变，Z 增加 1，同时发射一个电子和一个反中微子：

¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ν̄

Gamma emission usually follows alpha or beta decay when the daughter nucleus is left in an excited state; it involves no change in A or Z.

γ 辐射通常在子核处于激发态时伴随 α 或 β 衰变发生，不改变 A 和 Z。

7. Activity and the Decay Constant | 活度与衰变常量

The activity A of a radioactive sample is the number of decays per unit time. It is proportional to the number of undecayed nuclei N present:

A = −dN/dt = λN

放射性样品的活度 A 是单位时间内发生衰变的核数目。它与尚未衰变的核数目 N 成正比：

A = −dN/dt = λN

where λ is the decay constant, a probability of decay per nucleus per second. The SI unit of activity is the becquerel (Bq), where 1 Bq = 1 decay per second. A large λ corresponds to a rapidly decaying source.

其中 λ 是衰变常量，表示每个核每秒发生衰变的概率。活度的国际单位是贝克勒尔 (Bq)，1 Bq = 每秒 1 次衰变。λ 越大，源衰变得越快。

8. Half-Life and Exponential Decay | 半衰期与指数衰减

Radioactive decay is a random process described by an exponential law:

N = N₀ e^−λt

放射性衰变是一个随机过程，遵循指数规律：

N = N₀ e^−λt

The half-life T_1/2 is the time taken for the number of undecayed nuclei (and the activity) to halve. Setting N = N₀/2 gives:

T_1/2 = ln2 / λ

半衰期 T_1/2 是未衰变核数（及活度）减少一半所需的时间。令 N = N₀/2 可得：

T_1/2 = ln2 / λ

The same exponential form applies to activity and mass. A useful rule: after n half-lives, the remaining fraction is (1/2)ⁿ.

同样的指数形式适用于活度和质量。一个常用的经验法则：经过 n 个半衰期后，剩余的比例为 (1/2)ⁿ。

9. Nuclear Fission | 核裂变

Nuclear fission occurs when a heavy nucleus, such as uranium-235, captures a neutron and splits into two smaller daughter nuclei, releasing energy and two or three neutrons. A typical reaction is:

²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n + energy

核裂变是指重核（如铀‑235）俘获一个中子后分裂成两个较小的子核，同时释放能量和两到三个中子。一个典型的反应方程为：

²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3 ¹₀n + 能量

The released neutrons can trigger further fissions, creating a chain reaction. In a nuclear reactor, control rods absorb excess neutrons to keep the reaction steady, and a moderator slows down neutrons to increase the probability of fission. The energy released comes from the increase in binding energy per nucleon; the fission fragments are more tightly bound than the original heavy nucleus.

释放出的中子可诱发更多的裂变，从而形成链式反应。在核反应堆中，控制棒吸收过剩的中子以维持反应平稳，慢化剂则减慢中子速度以提高裂变概率。释放的能量源于每个核子结合能的增加：裂变碎片的结合比原来的重核更紧密。

10. Nuclear Fusion | 核聚变

Nuclear fusion is the joining together of light nuclei to form a heavier nucleus, releasing energy because the product has a higher binding energy per nucleon. For example, in the Sun, hydrogen nuclei fuse into helium through the proton-proton chain. A simple representation of fusion is:

²₁H + ³₁H → ⁴₂He + ¹₀n + energy

核聚变是轻核结合成一个较重核的过程，由于产物每个核子的结合能更高，因而释放能量。例如，在太阳中，氢核通过质子‑质子链反应聚变成氦核。聚变的一个简化表达式为：

²₁H + ³₁H → ⁴₂He + ¹₀n + 能量

Fusion requires extremely high temperatures (millions of kelvin) to give nuclei enough kinetic energy to overcome the Coulomb barrier. On Earth, this is achieved in experimental reactors like tokamaks, but sustained, controlled fusion for power generation remains a challenge.

聚变需要极高的温度（数百万开尔文），以使核获得足够的动能来克服库仑势垒。在地球上，托卡马克等实验装置可以达到这些温度，但实现持续、可控的聚变发电仍是一项挑战。

11. Mass–Energy Equivalence: E = mc² | 质能等价：E = mc²

Einstein’s famous equation states that mass and energy are interchangeable. In nuclear reactions, the total mass of the products is slightly smaller than that of the reactants; this ‘lost’ mass Δm is converted into kinetic energy of the products. The energy released is given by:

E = Δm c²

爱因斯坦著名的方程指出质量与能量可以相互转化。在核反应中，产物的总质量略小于反应物的总质量；这个“亏损”的质量 Δm 转化为产物的动能。释放的能量由下式给出：

E = Δm c²

With 1 u = 1.661 × 10⁻²⁷ kg and c = 3.00 × 10⁸ m s⁻¹, it follows that 1 u is equivalent to 931.5 MeV. This conversion factor is vital for calculating energy changes in both fission and fusion.

因为 1 u = 1.661 × 10⁻²⁷ kg, c = 3.00 × 10⁸ m s⁻¹，可得 1 u 相当于 931.5 MeV。这个转换因子对于计算裂变和聚变中的能量变化至关重要。

12. Applications of Radioactivity: Carbon Dating | 放射性的应用：碳定年法

Radioactive isotopes have many practical uses. One key application is carbon-14 dating. Cosmic rays produce neutrons in the upper atmosphere that convert nitrogen-14 into radioactive carbon-14:

¹⁴₇N + ¹₀n → ¹⁴₆C + ¹₁H

放射性同位素有许多实际应用，其中一个重要应用是碳‑14 定年法。宇宙射线在高空大气中产生中子，将氮‑14 转化为放射性碳‑14：

¹⁴₇N + ¹₀n → ¹⁴₆C + ¹₁H

Living organisms constantly exchange carbon with their environment, maintaining a steady proportion of carbon-14 to carbon-12. When an organism dies, the intake stops and the carbon-14 decays with a half-life of about 5730 years. By measuring the remaining activity, the time since death can be estimated. This technique is widely used in archaeology and geology for organic materials up to around 50 000 years old.

活体生物与环境不断进行碳交换，使体内碳‑14 与碳‑12 的比例保持恒定。生物死亡后，碳的摄入停止，碳‑14 按照约 5730 年的半衰期衰变。通过测量剩余的活度，便可估算死亡以来的时间。该方法广泛应用于考古学和地质学，可测定距今约 5 万年以内的有机物质。

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📚 Resistance in A-Level Physics: Key Concepts | A-Level 物理：电阻考点精讲

Resistance is one of the most fundamental concepts in electricity, describing how a component opposes the flow of electric current. For A-Level Physics, a solid understanding of resistance is essential not only for circuit analysis but also for grasping deeper principles like energy dissipation and material properties. This article will walk you through every essential topic, from Ohm’s law and resistivity to internal resistance and potential dividers, ensuring you are well-prepared for exam questions and practical assessments.

电阻是电学中最基本的概念之一，它描述了元件对电流流动的阻碍作用。在A-Level物理中，扎实掌握电阻知识不仅对电路分析至关重要，也是理解能量耗散和材料特性等深层次原理的基础。本文将带你逐一梳理核心考点，从欧姆定律、电阻率到内阻和分压器，帮助你从容应对考试与实验评估。

1. Defining Resistance and Ohm’s Law | 电阻的定义与欧姆定律

Resistance (R) is defined as the ratio of the potential difference (V) across a component to the current (I) flowing through it: R = V / I. The unit of resistance is the ohm (Ω), named after Georg Ohm. A component has a resistance of 1 Ω if a potential difference of 1 V drives a current of 1 A through it.

电阻（R）定义为元件两端的电势差（V）与流过它的电流（I）之比：R = V / I。电阻的单位是欧姆（Ω），以纪念乔治·欧姆。如果一个元件两端电压为1 V时通过的电流为1 A，那么它的电阻就是1 Ω。

Ohm’s law states that, for a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across it, provided physical conditions remain unchanged. This linear relationship means resistance is constant; such conductors are called ohmic conductors. The law can be written as V = I R, and its graph is a straight line passing through the origin.

欧姆定律指出，对于温度恒定的金属导体，在物理条件不变的情况下，通过导体的电流与其两端的电势差成正比。这种线性关系意味着电阻是恒定的，这类导体称为欧姆导体。定律可写作V = I R，其图像是一条过原点的直线。

It is crucial to understand that resistance is not defined by Ohm’s law; the ratio V/I always gives resistance, even for non-ohmic components. Ohm’s law only tells us that this ratio remains constant under certain conditions.

必须注意，电阻并非由欧姆定律定义；V/I 这个比值始终给出电阻值，即使对于非欧姆元件也是如此。欧姆定律只是告诉我们，在特定条件下这一比值保持恒定。

2. Resistivity: A Property of the Material | 电阻率：材料的固有属性

While resistance depends on the dimensions of a conductor, resistivity (ρ) characterises the material itself. The resistance of a uniform wire is given by R = ρ L / A, where L is the length and A is the cross-sectional area. Resistivity has the unit ohm-metre (Ω m).

电阻取决于导体的尺寸，而电阻率（ρ）则表征材料本身的性质。均匀导线的电阻由公式 R = ρ L / A 给出，其中 L 为长度，A 为横截面积。电阻率的单位是欧姆·米（Ω m）。

For a given material, longer wires have higher resistance (R ∝ L), while thicker wires have lower resistance (R ∝ 1/A). Resistivity is temperature-dependent; for metals, it increases with temperature because greater lattice vibrations impede electron flow. Semiconductors, however, often show a decrease in resistivity as temperature rises due to more charge carriers becoming available.

对于给定材料，导线越长电阻越大（R ∝ L），而导线越粗电阻越小（R ∝ 1/A）。电阻率取决于温度；对金属而言，温度升高时电阻率增大，因为更剧烈的晶格振动阻碍了电子流动。而半导体随着温度升高，由于有更多载流子被释放，其电阻率通常下降。

Conductivity (σ) is the reciprocal of resistivity: σ = 1/ρ. It is measured in siemens per metre (S m⁻¹) and is useful when comparing how well different materials conduct electricity.

电导率（σ）是电阻率的倒数：σ = 1/ρ，单位为西门子每米（S m⁻¹），在比较不同材料的导电性能时十分有用。

3. Temperature Dependence of Resistance | 电阻的温度依赖性

For most metallic conductors, resistance increases with temperature. The approximate relationship over limited temperature ranges is R = R₀ [1 + α (T – T₀)], where α is the temperature coefficient of resistance. A positive α means resistance rises with temperature, which is typical of pure metals. Materials like carbon or semiconductors have negative temperature coefficients.

对于大多数金属导体，电阻随温度升高而增大。在有限的温度范围内，近似关系为 R = R₀ [1 + α (T – T₀)]，其中 α 是电阻温度系数。正的 α 表示电阻随温度升高而增大，这是纯金属的典型特征。碳或半导体等材料则具有负温度系数。

This effect is crucial in practical devices. A thermistor, for example, uses a semiconductor material whose resistance drops significantly as temperature increases, making it suitable for temperature sensing. An incandescent light bulb filament displays a high resistance when hot, which explains the brief surge of current when first switched on.

这一效应在实际设备中非常关键。例如，热敏电阻使用半导体材料，其电阻随温度升高而显著下降，适合用于温度传感。白炽灯泡的灯丝在炽热时电阻很高，这就解释了刚接通电源时的短暂电流冲击。

Superconductivity is a special state where certain materials exhibit exactly zero resistivity below a critical temperature. This phenomenon is not part of all A-Level specifications but can appear as an application of the temperature–resistance relationship.

超导是一种特殊状态，某些材料在低于临界温度时电阻率恰好为零。这一现象并不在所有A-Level大纲中要求，但可作为电阻-温度关系的一个应用实例出现。

4. Current–Voltage (I–V) Characteristics | 电流-电压特性曲线

Exam boards expect you to sketch and interpret I–V graphs for various components. For an ohmic conductor at constant temperature, the I–V graph is a straight line through the origin, with slope equal to 1/R. For a filament lamp, the line starts straight but curves as the filament heats up, showing increasing resistance.

考试局要求你能够绘制并解释不同元件的 I-V 特性曲线。对于温度恒定的欧姆导体，I-V 图像是一条过原点的直线，斜率等于 1/R。白炽灯灯丝的图像开始为直线，随后随着灯丝升温而弯曲，显示电阻增大。

A diode has an extremely high resistance in reverse bias (almost no current) and conducts readily in forward bias once the threshold voltage (about 0.6 V for silicon) is exceeded. Its I–V graph shows zero current for negative voltages and a sharply rising current beyond the threshold. A thermistor’s resistance decreases as current increases due to self-heating, producing a non-linear curve bending upwards.

二极管在反向偏压下电阻极高（几乎无电流），在正向偏压超过阈值电压（硅管约0.6 V）后则很容易导通。其 I-V 图像显示负电压时电流为零，超过阈值后电流急剧上升。热敏电阻由于自身发热，随着电流增大电阻下降，产生的非线性曲线向上弯曲。

Understanding these graphs helps in designing circuits and in recognising whether a component obeys Ohm’s law. Remember that the ratio V/I at any point still defines the resistance, even if the graph is curved.

理解这些图像有助于设计电路，并判断一个元件是否遵循欧姆定律。请记住，即使图像是弯曲的，任意点的V/I比值仍然定义为该点对应的电阻。

5. Resistors in Series and Parallel | 串联与并联电阻

When resistors are connected in series, the total resistance is the sum of individual resistances: R_total = R₁ + R₂ + R₃ + … . The same current flows through each resistor, but the total potential difference is divided among them in proportion to their resistances.

电阻串联时，总电阻等于各电阻之和：R_total = R₁ + R₂ + R₃ + … 。每个电阻流过相同的电流，但总电势差按电阻比例分配在各电阻上。

In a parallel combination, the reciprocal of the total resistance is the sum of the reciprocals of individual resistances: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … . The potential difference across each branch is the same, while the total current is the sum of the branch currents. The total resistance in a parallel circuit is always less than the smallest individual resistance.

电阻并联时，总电阻的倒数等于各电阻倒数之和：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … 。各支路两端电势差相同，总电流则是各支路电流之和。并联电路的总电阻总是小于其中最小的单个电阻。

Being able to reduce complex networks of resistors to a single equivalent resistance is a crucial skill. Often, you will need to combine series and parallel rules step by step. Drawing a simplified diagram after each step reduces errors.

能够将复杂的电阻网络简化为单一等效电阻是一项关键技能。通常你需要逐步结合串联和并联规则。每简化一步后画出简化图，可以减少错误。

6. The Potential Divider and Potentiometer | 分压器与电位器

A potential divider is one of the most practical circuits built with resistors. It consists of two resistors in series across a voltage supply. The output voltage is taken from the midpoint between them and is given by V_out = V_in × (R₂ / (R₁ + R₂)), where R₂ is the resistor across which V_out is measured.

分压器是最实用的电阻应用电路之一。它由两个串联电阻跨接在电源两端构成。输出电压从两电阻中点取出，计算公式为 V_out = V_in × (R₂ / (R₁ + R₂))，其中 R₂ 是测量输出电压时所跨接的电阻。

This circuit can supply any fraction of the input voltage, from 0 V to nearly V_in, simply by using a variable resistor (potentiometer) or by choosing appropriate fixed resistor values. Potential dividers are widely used to provide a reference voltage, to connect sensors like LDRs and thermistors to a processing unit, or to compare an unknown voltage with a known one.

这一电路通过使用可变电阻（电位器）或选择合适的固定电阻值，能提供从0 V到接近V_in的任意比例电压。分压器被广泛用于提供参考电压、将光敏电阻或热敏电阻等传感器连接到处理单元，或比较未知电压与已知电压。

When a sensor such as a thermistor replaces one of the resistors, the output voltage changes with temperature. In an LDR potential divider, the output increases when light intensity decreases. These applications are frequently examined, often requiring you to explain or sketch how V_out varies with the environmental condition.

当用热敏电阻等传感器代替其中一个电阻时，输出电压会随温度变化。在光敏电阻分压器中，光照减弱时输出电压增大。这些应用在考试中频繁出现，往往要求你解释或描绘V_out如何随环境条件变化。

7. Electromotive Force (EMF) and Internal Resistance | 电动势与内阻

A real power supply, such as a battery, is not perfect; it has an internal resistance (r) that causes the terminal potential difference to drop when current flows. The electromotive force (ε) is the energy per unit charge supplied by the source, measured in volts. It is equal to the terminal p.d. when no current is drawn.

真实的电源，比如电池，并不完美；它具有内阻（r），当有电流流过时，端电压会下降。电动势（ε）是电源每单位电荷提供的能量，单位为伏特。当没有电流输出时，电动势等于端电压。

The relationship is given by ε = V + I r, where V is the terminal p.d. and I r is the ‘lost volts’ across the internal resistance. Rewriting as V = ε – I r shows that a graph of V against I is a straight line with gradient -r and y-intercept ε. This is a classic experiment where you vary a variable resistor and record voltage and current readings.

这一关系表示为 ε = V + I r，其中 V 是端电压，I r 是落在内阻上的“损耗电压”。改写为 V = ε – I r 表明，V 对 I 的图像是一条斜率为 -r、y截距为 ε 的直线。这是一个经典实验，通过改变可变电阻并记录电压和电流读数来完成。

Internal resistance causes power to be dissipated inside the battery itself (P = I² r), which heats the battery and reduces efficiency. Understanding ε and r helps explain why a battery’s voltage seems to ‘sag’ under heavy load and why a fresh battery has a lower internal resistance than an old one.

内阻会导致电池内部耗散功率（P = I² r），使电池发热并降低效率。理解电动势和内阻有助于解释为什么电池端电压在大负载下会“骤降”，以及为什么新电池的内阻比旧电池小。

8. Kirchhoff’s Laws in Resistive Circuits | 基尔霍夫定律在电阻电路中的应用

Kirchhoff’s first law (current law) states that the total current entering a junction equals the total current leaving it. This is a consequence of charge conservation. In a resistive network, it allows you to find unknown branch currents by setting up equations. The second law (voltage law) states that the sum of the e.m.f.s around any closed loop equals the sum of the p.d.s across the components in that loop.

基尔霍夫第一定律（电流定律）指出，进入一个节点的总电流等于离开该节点的总电流。这是电荷守恒的结果。在电阻网络中，可以据此列出方程求解未知支路电流。第二定律（电压定律）指出，沿任一闭合回路的电动势总和等于该回路中各元件电势差的总和。

These laws are essential for analysing circuits that cannot be reduced to simple series–parallel combinations, such as those with multiple e.m.f.s in different branches. You often combine them with Ohm’s law to solve for unknown resistances or currents. Sign conventions matter: assign a positive direction for current and treat a voltage source as positive if the loop goes from negative to positive terminal.

这些定律对于分析无法简化为简单串并联的电路（例如不同支路含多个电动势的电路）至关重要。你通常需要结合欧姆定律来求解未知电阻或电流。符号规则很关键：设定电流正方向，如果回路经过电动势时是从负极到正极，则该电动势取正值。

9. Electrical Power and Joule Heating | 电功率与焦耳热

When a current passes through a resistor, electrical energy is converted to thermal energy. The power dissipated is given by P = I V. Using Ohm’s law, this can be written as P = I² R or P = V² / R. These formulas let you calculate heat generated in circuit components and are vital for selecting resistors with appropriate power ratings.

电流通过电阻时，电能转化为热能。耗散的功率由 P = I V 给出。利用欧姆定律，可表示为 P = I² R 或 P = V² / R。这些公式可用于计算电路元件产生的热量，对于选择具有适当额定功率的电阻至关重要。

Joule heating has practical applications such as in electric heaters and fuse wires, where the heat is the desired outcome. However, in most electronic circuits, excessive heating reduces efficiency and can damage components. The power rating of a resistor (e.g., 0.25 W) indicates the maximum power it can dissipate safely without overheating.

焦耳热在实际中有广泛应用，如电热器和保险丝，其中热量正是所需要的。但在大多数电子电路中，过热会降低效率并可能损坏元件。电阻器的额定功率（如0.25 W）表示它在不过热的前提下可以安全耗散的最大功率。

Efficiency calculations often involve comparing the power delivered to a load with the total power supplied. For a battery with internal resistance r connected to an external load R, the power delivered to the load is maximum when R = r (the maximum power transfer theorem), though this is only 50% efficient because half the power is lost internally.

效率计算经常涉及比较负载获得的功率与总供电功率。对于一个内阻为 r 的电池连接外部负载 R，当 R = r 时负载获得最大功率（最大功率传输定理），但此时效率仅为50%，因为一半功率在内部损耗了。

10. Experimental Skills: Measuring Resistance | 实验技能：测量电阻

There are several standard methods to measure resistance in the A-Level laboratory. The simplest uses a voltmeter and an ammeter (V–I method) where R = V/I. This method requires correcting for meter resistance: in a circuit with an ammeter connected in series, the voltmeter should be placed across the component only, not across the ammeter as well, to avoid including the ammeter’s voltage drop.

A-Level实验室中有几种标准方法来测量电阻。最简单的是使用伏特表和安培表（伏安法），R = V/I。该方法需要修正电表内阻的影响：在电流表串联的电路中，电压表应只并联在待测元件两端，不要跨接电流表，以免计入电流表上的电压降。

A Wheatstone bridge offers a more precise way to determine an unknown resistance without needing current or voltage measurements. It consists of two potential dividers in parallel; when the bridge is balanced (zero current through a galvanometer connected between the midpoints), the ratio R₁/R₂ = R₃/Rₓ. The unknown resistance Rₓ can then be calculated accurately.

惠斯通电桥提供了一种更精确的测量未知电阻的方法，无需测量电流或电压。它由两个并联的分压器组成；当电桥平衡（连接在两个中点之间的检流计无电流通过）时，R₁/R₂ = R₃/Rₓ。这样就能准确计算出未知电阻 Rₓ。

A multimeter is a convenient instrument that can measure resistance directly by supplying a small current and measuring the voltage drop. However, you should understand its limitations, such as that the component must be removed from the circuit when measuring its resistance to avoid parallel paths affecting the reading.

万用表是一种方便的仪表，可直接测量电阻，其原理是提供一个小电流并测量电压降。但你应该了解它的局限性，比如测量电阻时必须将元件从电路中取下，以避免并联通路影响读数。

When investigating how resistance depends on length or cross-sectional area, using a metre bridge along with standard resistance wire is a common practical. Plotting R against L yields a straight line whose gradient can be used to determine the resistivity of the wire if the area is known.

在研究电阻如何取决于长度或横截面积时，使用滑线电桥和标准电阻线是一个常见实验。将 R 对 L 作图得到一条直线，如果已知横截面积，可通过斜率计算出电阻丝的电阻率。

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📚 A-Level Physics Unit 2 Formula Derivation (Jan21) | A-Level物理单元2公式推导（2021年1月）

Welcome to the ultimate guide for deriving the key formulas found in the A-Level Physics Unit 2 Insert for January 2021. Mastering these derivations not only deepens your understanding but also equips you to tackle any calculation problem with confidence. This article walks you through each equation step by step, from fundamental principles to the final expression, ensuring you see the logical flow behind every symbol.

欢迎阅读A-Level物理单元2（2021年1月）公式推导完整指南。掌握这些推导不仅能加深理解，也能让你充满信心地应对任何计算题。本文将逐步带你推导每一个关键公式，从基本原理到最终表达式，让你看清每个符号背后的逻辑脉络。

1. Introduction to the Unit 2 Insert | 单元2公式表简介

The A-Level Physics Unit 2 Insert (January 2021) provides a collection of essential formulas for mechanics, materials, and waves. While you can use the sheet in the exam, understanding where these formulas come from is crucial for applying them correctly. We will cover derivations for SUVAT equations, force and momentum, energy, elasticity, wave optics, and quantum phenomena.

A-Level物理单元2（2021年1月）的公式表提供了力学、材料和波动的核心公式。虽然考试时可以查阅，但理解这些公式的来源对于正确运用至关重要。我们将涵盖匀加速运动方程、力与动量、功与能、弹性、波动光学以及量子现象的推导。

2. Deriving the SUVAT Equations | 推导匀加速运动方程

The SUVAT equations relate displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) for uniformly accelerated motion. Start with the definition of acceleration: a = (v – u) / t. Rearranging gives the first equation:

SUVAT方程描述了匀加速运动中位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）的关系。从加速度的定义出发：a = (v – u) / t，整理即得第一式：

v = u + at

Since acceleration is constant, average velocity = (u + v)/2. Displacement is average velocity multiplied by time: s = ((u + v)/2) t. Substitute v = u + at into this expression:

由于加速度恒定，平均速度 = (u + v)/2。位移等于平均速度乘以时间：s = ((u + v)/2) t。将 v = u + at 代入：

s = ½ (u + u + at) t = ut + ½ at²

To eliminate time, solve v = u + at for t: t = (v – u)/a. Substitute into s = ((u + v)/2) t:

为消去时间，由 v = u + at 解出 t = (v – u)/a，代入 s = ((u + v)/2) t：

s = (u + v)(v – u) / (2a) = (v² – u²) / (2a) ⇒ v² = u² + 2as

These four equations (including s = vt – ½ at²) form the SUVAT set. They are derived purely from definitions of velocity and acceleration under constant a.

这四个方程（包括 s = vt – ½ at²）构成了SUVAT集合，完全由恒定加速度下的速度和位移定义导出。

3. Newton’s Second Law and Impulse | 牛顿第二定律与冲量

Newton’s second law states that the net force acting on an object equals the rate of change of its momentum: F = dp/dt. For constant mass, p = mv, so F = m (dv/dt) = ma. The impulse J delivered by a constant force over time Δt is J = F Δt, which equals the change in momentum:

牛顿第二定律指出，作用于物体的净力等于其动量变化率：F = dp/dt。对于质量恒定的情况，p = mv，因此 F = m (dv/dt) = ma。恒力在时间 Δt 内产生的冲量 J = F Δt，等于动量的变化：

F = m a

J = F Δt = Δp = m(v – u)

In collisions where force varies, the area under a force–time graph gives the impulse, still equal to Δp. This link between force, time, and momentum is fundamental in mechanics.

在变力碰撞中，力-时间图像下的面积即为冲量，仍等于 Δp。力、时间与动量之间的这种联系是力学的基础。

4. Work, Energy, and Power | 功、能与功率

Work done by a constant force F acting over a displacement s in the direction of the force is W = F s. If the force is at an angle θ to the displacement, W = F s cos θ. The work–energy principle states that the net work done equals the change in kinetic energy. For an object starting at speed u and accelerating to v, using F = ma and v² = u² + 2as:

恒力 F 沿位移 s 方向做的功为 W = F s。若力与位移夹角为 θ，则 W = F s cos θ。功-能原理表明，净功等于动能的变化。对一个从速度 u 加速到 v 的物体，利用 F = ma 和 v² = u² + 2as：

W = F s = (ma) × ((v² – u²)/(2a)) = ½ mv² – ½ mu²

This shows kinetic energy Eₖ = ½ mv². Power P is the rate of doing work: P = W/t = F v (for constant force and velocity in the same direction). The unit of power is the watt (W) = J s⁻¹.

这表明动能 Eₖ = ½ mv²。功率 P 是做功的速率：P = W/t = F v（力与速度同向且恒定时）。功率单位为瓦特（W）= J s⁻¹。

5. Hooke’s Law and Elastic Potential Energy | 胡克定律与弹性势能

For a spring obeying Hooke’s law, the extension x is directly proportional to the applied force: F = k x, where k is the spring constant. The elastic potential energy stored in a stretched or compressed spring is the area under the force–extension graph, which is a triangle for a linear spring:

对于满足胡克定律的弹簧，伸长量 x 与施加的力成正比：F = k x，其中 k 为劲度系数。拉伸或压缩弹簧所储存的弹性势能等于力-伸长图下的面积，对于线性弹簧为三角形面积：

Eₑₗ = ½ F x = ½ k x²

This derivation assumes the spring’s limit of proportionality is not exceeded. If a force–extension graph is non-linear, the energy stored is still the area under the curve but cannot be given by a simple ½ k x².

该推导假设弹簧未超过比例极限。若力-伸长图非线性，储存的能量仍为曲线下的面积，但不能用简单的 ½ k x² 表示。

6. Young’s Modulus and Stress-Strain | 杨氏模量与应力-应变

Stress σ is the force per unit cross-sectional area: σ = F / A. Strain ε is the extension per unit original length: ε = ΔL / L. Young’s modulus E quantifies the stiffness of a material, defined as the ratio of stress to strain in the linear region:

应力 σ 是单位横截面积上的力：σ = F / A。应变 ε 是单位原长的伸长量：ε = ΔL / L。杨氏模量 E 衡量材料的刚度，定义为线弹性区域内应力与应变之比：

E = σ / ε = (F/A) / (ΔL/L) = FL / (A ΔL)

Combining with Hooke’s law for a wire, the effective spring constant is k = EA / L. Substituting into F = k ΔL gives the same expression. The unit of Young’s modulus is N m⁻² or Pa.

结合金属丝的胡克定律，其等效劲度系数 k = EA / L。代入 F = k ΔL 可得一致表达式。杨氏模量的单位是 N m⁻² 或 Pa。

7. The Wave Equation: v = f λ | 波动方程：v = f λ

Waves transfer energy without transferring matter. The frequency f is the number of complete oscillations per second. The wavelength λ is the distance between consecutive points in phase. In one period T = 1/f, the wave advances by one wavelength. Therefore, the speed v is distance/time:

波传递能量而不传递物质。频率 f 是每秒完整振动的次数。波长 λ 是相邻同相位点之间的距离。在一个周期 T = 1/f 内，波前进一个波长。因此，波速 v = 距离/时间：

v = λ / T = f λ

This relationship holds for all types of waves: mechanical (sound, water) and electromagnetic. It is used extensively in analysing diffraction, interference, and standing waves.

此关系适用于所有类型的波：机械波（声波、水波）和电磁波。它广泛用于分析衍射、干涉和驻波。

8. Refractive Index and Snell’s Law | 折射率与斯涅尔定律

When light travels from medium 1 to medium 2, its speed changes, causing refraction. Snell’s law links the angles of incidence θ₁ and refraction θ₂ to the refractive indices. The refractive index n of a medium is n = c / v, where c is the speed of light in vacuum. Using wavefronts and the principle of least time or boundary conditions, we derive:

当光从介质1进入介质2时，其速度改变，从而发生折射。斯涅尔定律将入射角 θ₁ 和折射角 θ₂ 与折射率联系起来。介质的折射率 n = c / v，其中 c 是真空光速。利用波前和最短时间原理或边界条件，可以推导出：

n₁ sin θ₁ = n₂ sin θ₂

For light entering a block of refractive index n from air (n ≈ 1), the equation simplifies to sin θ₁ = n sin θ₂. The critical angle for total internal reflection occurs when θ₂ = 90°, giving sin θc = 1/n.

当光从空气（n ≈ 1）射入折射率为 n 的介质时，方程简化为 sin θ₁ = n sin θ₂。发生全内反射的临界角满足 θ₂ = 90°，得到 sin θc = 1/n。

9. Superposition and Standing Waves | 叠加与驻波

The principle of superposition states that when two or more waves meet, the resultant displacement is the vector sum of individual displacements. Standing waves are formed when two identical progressive waves travel in opposite directions. For a string fixed at both ends, the condition for a standing wave is that the length L equals an integer multiple of half-wavelengths:

叠加原理指出，当两个或更多波相遇时，合位移等于各波位移的矢量和。驻波由两列完全相同但反向传播的行波叠加形成。对于两端固定的弦，驻波条件为长度 L 等于半波长的整数倍：

L = n λ/2, where n = 1, 2, 3, …

The frequency of the nth harmonic is fₙ = n (v/2L), with v being the wave speed on the string. Similar conditions apply to pipes (open or closed), leading to different harmonic series.

第 n 次谐波的频率为 fₙ = n (v/2L)，v 为弦上的波速。类似的条件适用于管乐器（开管或闭管），得到不同的谐波序列。

10. Two-Source Interference and Young’s Double Slit | 双源干涉与杨氏双缝

Young’s double-slit experiment demonstrates the wave nature of light. Two coherent sources of light create an interference pattern. Constructive interference occurs when the path difference Δ between the two waves is an integer multiple of λ: Δ = nλ. Destructive interference occurs when Δ = (n + ½)λ. For small angles, the fringe spacing Δy on a screen at distance D is given by:

杨氏双缝实验证明了光的波动性。两个相干光源产生干涉图样。当两束光的路程差 Δ 为 λ 的整数倍时发生相长干涉：Δ = nλ。当 Δ = (n + ½)λ 时发生相消干涉。对于小角度，屏幕上距离 D 处的条纹间距 Δy 由下式给出：

Δy = λ D / d

This is derived by approximating sin θ ≈ tan θ ≈ θ for small θ, where d is the slit separation. The same formula applies to sound waves and microwaves in demonstration experiments.

其推导基于小角度近似 sin θ ≈ tan θ ≈ θ，d 为双缝间距。该公式同样适用于声波和微波的演示实验。

11. Photon Energy and the Photoelectric Effect | 光子能量与光电效应

The photoelectric effect provides evidence for the particle model of light. A single photon of frequency f carries energy E = h f, where h is Planck’s constant. When a photon hits a metal surface, its energy can be used to overcome the work function φ (the minimum energy needed to release an electron) and give the electron kinetic energy. Einstein’s photoelectric equation is:

光电效应为光的粒子模型提供了证据。一个频率为 f 的光子携带能量 E = h f，其中 h 为普朗克常数。当光子撞击金属表面时，其能量可用于克服功函数 φ（释放电子所需的最小能量）并赋予电子动能。爱因斯坦光电方程如下：

h f = φ + ½ m v²ₘₐₓ

The maximum kinetic energy Eₖ₍ₘₐₓ₎ can be measured by applying a stopping potential Vₛ: Eₖ₍ₘₐₓ₎ = e Vₛ, where e is the elementary charge. The threshold frequency f₀ is given by h f₀ = φ. This derivation shows that kinetic energy increases linearly with frequency, not intensity.

最大动能 Eₖ₍ₘₐₓ₎ 可通过施加遏止电压 Vₛ 测量：Eₖ₍ₘₐₓ₎ = e Vₛ，e 为元电荷。截止频率 f₀ 满足 h f₀ = φ。该推导表明，最大动能随频率线性增加，而与光强无关。

12. Key Takeaways and Exam Tips | 关键要点与应试技巧

When tackling Unit 2 questions, always start by identifying the fundamental principles at play. Write down the relevant derived equation from the insert, but be prepared to show how it follows from basic definitions if asked. Practice drawing labelled diagrams (e.g., force–extension, wavefronts) to support your derivations. Remember to check unit consistency and use standard form for very large or small numbers.

解答单元2题目时，首先确定所用的基本原理。写下公式表中的相关推导方程，但如果题目要求展示推导过程，务必能从基本定义开始展示。练习绘制标注清晰的示意图（如力-伸长图、波前图）来辅助推导。记得检查单位一致性，并对非常大或非常小的数字使用科学记数法。

The formulas derived here are interconnected. For instance, the conservation of energy ties together kinetic energy, potential energy, and work done. Mastering derivations gives you a safety net: even if you forget a formula, you can reconstruct it.

本文推导的公式相互关联。例如，能量守恒将动能、势能和做功联系起来。掌握推导能为你提供一张安全网：即使忘记了某个公式，你也能重新推导出来。

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