📚 IB Physics: Newton’s Law of Gravitation and Its Applications | IB物理：万有引力定律及应用
Gravitation is a fundamental interaction that governs the motion of planets, satellites, and falling apples alike. In the IB Physics syllabus, Newton’s law of universal gravitation serves as the cornerstone for understanding orbital mechanics, gravitational fields, and energy conservation in space.
Newton’s law states that every point mass attracts every other point mass with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
The force is always attractive, acting along the line joining the two masses.
力始终为吸引力，作用方向沿两质量连线。
For extended spherical objects, r is measured from the centre of one sphere to the centre of the other.
对于球对称的物体，r从一个球心量到另一个球心。
The inverse-square relation means that doubling the distance reduces the force to one quarter.
平方反比关系意味着距离加倍时，力减小为原来的四分之一。
2. Gravitational Field Strength | 引力场强度
A gravitational field is a region of space where a mass experiences a gravitational force. The field strength g is defined as the force per unit mass placed in the field.
引力场是空间中质量会受到引力作用的区域。引力场强度g定义为置于场中单位质量所受到的引力。
g = F / m = GM / r²
This equation shows that the field strength at a distance r from the centre of a mass M depends only on M and r, not on the test mass. Near the Earth’s surface, g ≈ 9.8 N kg⁻¹.
g is a vector quantity, directed toward the centre of the attracting mass.
g为矢量，方向指向吸引质量中心。
At the surface of a spherical mass, g = GM/R², where R is the radius of the mass.
在球形质量表面，g = GM/R²，其中R为质量半径。
Inside a uniform spherical shell, the field is zero (shell theorem).
在均匀球壳内部，场强为零（球壳定理）。
3. Gravitational Potential Energy | 引力势能
In a uniform gravitational field near the Earth’s surface, gravitational potential energy is given by U = mgh. However, for large distances, the field is not uniform, and a more general expression is required.
The negative sign indicates that the gravitational potential energy is zero at infinity and becomes more negative as the masses get closer. This means work must be done against the gravitational force to separate the masses.
The reference point for gravitational potential energy is taken at r → ∞.
引力势能的参考点取在r → ∞处。
Bounded systems have negative total energy, while unbounded systems have positive or zero energy.
束缚系统的总能量为负，非束缚系统的总能量为正或为零。
4. Gravitational Potential | 引力势
Gravitational potential V is the gravitational potential energy per unit mass at a point in a field.
引力势V是引力场中某一点处单位质量的引力势能。
V = -GM / r
The unit of gravitational potential is J kg⁻¹. It is a scalar quantity, and the potential difference between two points equals the work done per unit mass in moving between them.
引力势的单位为J·kg⁻¹。它是标量，两点之间的势差等于单位质量在两点间移动时所做的功。
Quantity
Expression
Unit
Gravitational potential V
V = -GM/r
J kg⁻¹
Gravitational potential energy U
U = mV = -GMm/r
J
Field strength g
g = -dV/dr
N kg⁻¹
5. Kepler’s Laws of Planetary Motion | 开普勒行星运动定律
Johannes Kepler derived three empirical laws describing planetary motion around the Sun.
开普勒在观测数据的基础上总结出了描述行星绕太阳运动的三条经验定律。
First law: Planets move in elliptical orbits with the Sun at one focus.
第一定律：行星沿椭圆轨道运行，太阳位于椭圆的一个焦点上。
Second law: A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
第二定律：行星与太阳的连线在相等时间内扫过相等的面积。
Third law: The square of the orbital period is proportional to the cube of the semi-major axis.
第三定律：轨道周期的平方与半长轴的立方成正比。
T² ∝ a³
For an approximately circular orbit, a ≈ r, and Kepler’s third law can be derived from Newtonian mechanics together with the law of gravitation.
对于近似圆形的轨道，a ≈ r，开普勒第三定律可由牛顿力学与万有引力定律共同推导出来。
6. Orbital Motion and Circular Orbits | 轨道运动与圆轨道
A satellite in a circular orbit experiences a centripetal force provided by the gravitational attraction of the central body.
在圆形轨道上运行的卫星，其向心力由中心天体的引力提供。
GMm / r² = mv² / r
Cancelling m and solving for v gives the orbital speed at radius r.
消去m并解出v，可得半径为r处的轨道速度。
v = √(GM / r)
This shows that orbital speed decreases with increasing orbital radius. A higher orbit moves more slowly than a lower orbit.
这表明轨道速度随轨道半径增大而减小。高轨道的运动速度比低轨道更慢。
7. Orbital Speed and Period | 轨道速度与周期
The orbital period T is the time taken for one complete revolution. Combining v = 2πr/T with v = √(GM/r) yields a direct relation between T and r.
This is a powerful form of Kepler’s third law. It allows astronomers to determine the mass of a central object by measuring the orbit of a satellite around it.
这是开普勒第三定律的有力形式。天文学家可以通过测量周围卫星的轨道来确定中心天体的质量。
Orbit altitude
r from Earth’s centre
Orbital speed
Period
Low Earth orbit (200 km)
6.57 × 10⁶ m
7.8 km s⁻¹
≈ 88 minutes
Geostationary orbit (35,800 km)
4.22 × 10⁷ m
3.1 km s⁻¹
24 hours
8. Escape Velocity | 逃逸速度
Escape velocity is the minimum speed required for an object to escape a gravitational field and never return, reaching zero velocity at infinity.
逃逸速度是物体克服引力场永不返回所需的最小速度，它在无穷远处速度为零。
½mv² = GMm / R
Solving for v gives the escape speed from the surface of a mass M with radius R.
解出v即得到从质量为M、半径为R的质量表面逃逸的速度。
vₑₛ꜀ = √(2GM / R)
For the Earth, vₑₛ꜀ ≈ 11.2 km s⁻¹. Note that escape velocity is independent of the mass of the escaping object.
地球的逃逸速度约为11.2 km·s⁻¹。注意逃逸速度与逃逸物体的质量无关。
9. Energy in Orbits | 轨道能量
A satellite in a circular orbit has both kinetic energy K and gravitational potential energy U. The total mechanical energy is the sum of the two.
在圆轨道上运行的卫星同时具有动能K和引力势能U。机械能总量为两者之和。
E = K + U = ½mv² – GMm / r
Using v² = GM/r, we find:
利用v² = GM/r，可得：
E = -GMm / 2r
The total energy is negative, confirming that the satellite is bound to the central mass.
总能量为负，确认卫星被束缚在中心质量附近。
As r increases, E becomes less negative, meaning the total energy increases.
随着r增大，E的负值减小，即总能量增大。
To move to a higher orbit, a satellite must gain energy from its engines.
要进入更高的轨道，卫星必须从发动机获得能量。
10. Geostationary Satellites | 地球同步卫星
A geostationary satellite orbits above the equator with a period equal to the Earth’s rotational period (24 hours). It remains fixed relative to a point on the Earth’s surface.
Orbital radius r ≈ 4.22 × 10⁷ m, about 35,800 km above the ground.
轨道半径r ≈ 4.22 × 10⁷ m，即距地面约35,800 km。
The orbit must lie in the equatorial plane.
轨道必须位于赤道平面内。
The orbital speed is approximately 3.1 km s⁻¹.
轨道速度约为3.1 km·s⁻¹。
Geostationary satellites are used for telecommunications, weather monitoring, and broadcasting.
同步卫星用于通信、气象监测和广播。
11. Weightlessness and Apparent Weight | 失重与表观重量
Astronauts in orbit experience apparent weightlessness, even though gravity is still significant at their altitude. This occurs because both the astronauts and their spacecraft are in free fall around the Earth.
In orbit, the centripetal acceleration v²/r equals g at that altitude, so the normal reaction force N becomes zero. The condition for apparent weightlessness is therefore:
在轨道上，向心加速度v²/r等于该高度处的g，因此支持力N变为零。表观失重的条件是：
v² / r = g
Gravity is still present; astronauts are not beyond the Earth’s gravitational field.
引力仍然存在；宇航员并未脱离地球引力场。
True weightlessness would occur only at infinite distance or in the absence of gravitational fields.
真正的完全失重只会在无穷远处或不存在引力场的区域出现。
12. Limitations and General Relativity | 局限性及广义相对论
Newton’s law of gravitation is remarkably accurate for most everyday and astronomical situations, but it fails under extreme conditions such as strong gravitational fields or motions near the speed of light.
牛顿万有引力定律在大多数日常和天文场景中非常精确，但在强引力场或接近光速运动的极端条件下会失效。
Einstein’s general relativity describes gravity as the curvature of spacetime, correctly explaining phenomena such as gravitational lensing, the precession of Mercury’s orbit, and black holes.
爱因斯坦的广义相对论把引力描述为时空的弯曲，能够正确解释引力透镜、水星近日点进动以及黑洞等现象。
For IB Physics, Newton’s formulation remains the required model for solving orbital and gravitational problems.
在IB物理中，牛顿表述仍是解决轨道和引力问题所要求的模型。
General relativity is a conceptual extension that appears in the optional relativity topic.
广义相对论在相对论选修专题中作为概念性扩展出现。
Published by TutorHao | Physics Revision Series | aleveler.com
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Circular motion is one of the most frequently tested topics in IB Physics, appearing in both SL and HL papers. This article consolidates the essential dynamics of uniform circular motion — the definitions, formulas, derivations, and common exam traps — so you can approach any problem with confidence.
1. Angular Displacement and Angular Velocity | 角位移与角速度
When a particle moves along a circular path, its position can be described by the angle θ swept out from a reference line. The SI unit of angular displacement is the radian (rad), defined as the ratio of arc length s to radius r: θ = s / r. One complete revolution corresponds to 2π radians.
当质点沿圆周运动时，其位置可以用相对于参考线扫过的角度θ来描述。角位移的国际单位是弧度（rad），定义为弧长s与半径r之比：θ = s / r。一整圈对应2π弧度。
Angular velocity ω is the rate of change of angular displacement:
ω = Δθ / Δt (unit: rad s⁻¹)
For uniform circular motion, ω is constant. Note that angular velocity is technically a vector pointing along the axis of rotation, but in IB you usually work with its magnitude.
角速度ω是角位移的变化率：
ω = Δθ / Δt（单位：rad s⁻¹）
在匀速圆周运动中，ω为常量。严格地说，角速度是沿着转轴方向的矢量，但在IB考试中通常只使用其大小。
2. Period, Frequency and Angular Speed | 周期、频率与角速度
The period T is the time taken for one complete revolution. The frequency f is the number of revolutions per second. They are related by:
f = 1 / T
周期T是完成一整圈所需的时间。频率f是每秒完成的转数。二者关系为：
f = 1 / T
Since one revolution corresponds to an angular displacement of 2π radians, the angular speed is:
ω = 2π / T = 2πf
因为一整圈对应2π弧度的角位移，角速度大小为：
ω = 2π / T = 2πf
The linear speed v of a particle at radius r is simply v = rω. This relation connects rotational and translational quantities, and you will use it constantly.
半径为r处的质点线速度v为v = rω。这个关系连接了转动量与平动量，在解题中会频繁使用。
3. Centripetal Acceleration | 向心加速度
Even when speed is constant, a particle in circular motion is accelerating because its direction changes continuously. For uniform circular motion, the acceleration is always directed toward the centre of the circle, hence the name centripetal (centre-seeking).
The magnitude of centripetal acceleration is given by two equivalent forms:
a_c = v² / r = ω²r
向心加速度的大小有两种等价形式：
a_c = v² / r = ω²r
These two forms are interchangeable via v = rω. Choose the one that matches the quantities you know. For example, if you know speed and radius, use v²/r; if you know angular speed and radius, use ω²r.
Consider a particle moving with constant speed v around a circle. In a small time interval Δt, it sweeps out angle Δθ. The change in velocity Δv points approximately toward the centre, and its magnitude is vΔθ. Dividing by Δt gives a = vω = v²/r.
By Newton’s second law, the centripetal acceleration requires a net force directed toward the centre. This is the centripetal force:
F_c = ma_c = mv² / r = mω²r
根据牛顿第二定律，向心加速度需要指向圆心的合力提供。这就是向心力：
F_c = ma_c = mv² / r = mω²r
It is crucial to understand that centripetal force is not a new type of force. It is the net force arising from real interactions — tension, gravity, friction, normal reaction, or a combination — that happens to point toward the centre. Always ask: which physical force (or component) provides the centripetal force in this situation?
Below are the most common IB scenarios for horizontal circular motion. In each, you must identify the source of the centripetal force.
以下是IB考试中最常见的水平圆周运动情景。每种情景中都必须辨认向心力的来源。
Scenario | 情景
Centripetal force source | 向心力来源
Key equation | 关键方程
Car turning on flat road | 汽车在平路转弯
Friction between tyres and road | 轮胎与路面间的摩擦力
μmg = mv²/r
Ball on a string (horizontal) | 细绳拉球（水平）
Tension | 绳的张力
T = mv²/r
Conical pendulum | 圆锥摆
Horizontal component of tension | 张力的水平分量
T sinθ = mv²/r
Satellite in orbit | 卫星绕地运行
Gravitational attraction | 万有引力
GMm/r² = mv²/r
Maximum speed on a flat curve | 平路弯道的最大速度
For a car of mass m on a flat curve of radius r, the maximum speed before skidding occurs when the required centripetal force equals the maximum static friction:
μmg = mv_max² / r → v_max = √(μgr)
质量为m的汽车在半径为r的平路弯道上，不打滑的最大速度出现在所需向心力等于最大静摩擦力时：
μmg = mv_max² / r → v_max = √(μgr)
6. Vertical Circular Motion | 竖直圆周运动
Vertical circular motion is trickier because gravity changes the speed of the object, so the motion is generally not uniform. IB questions typically focus on the top and bottom points of a vertical loop, where forces are vertical.
At any point in vertical circular motion, the net force toward the centre equals mv²/r. At the top of a loop of radius r:
N + mg = mv²/r (if track provides normal reaction)
在竖直圆周运动中任意一点，指向圆心的合力等于mv²/r。在半径为r的圆环顶部：
N + mg = mv²/r（若轨道提供支持力）
At the bottom of the loop:
N − mg = mv²/r
在圆环底部：
N − mg = mv²/r
The minimum speed at the top to maintain contact (N = 0) is found from mg = mv²/r, giving v_min = √(gr). For a ball on a string in a vertical circle, the string becomes slack at the top; the same condition applies for the minimum speed.
7. Worked Example: Car on a Flat Curve | 例题：平路转弯的汽车
Question: A car of mass 1200 kg travels around a flat circular track of radius 50 m. The coefficient of friction between tyres and road is 0.6. Calculate the maximum speed at which the car can turn without skidding.
Solution: The maximum centripetal force available is the maximum static friction:
F_max = μmg = 0.6 × 1200 × 9.8 = 7056 N
Set this equal to mv²/r:
1200 × v² / 50 = 7056 → v² = 294 → v = 17.1 m s⁻¹
解答：能提供的最大向心力为最大静摩擦力：
F_max = μmg = 0.6 × 1200 × 9.8 = 7056 N
令其等于mv²/r：
1200 × v² / 50 = 7056 → v² = 294 → v = 17.1 m s⁻¹
So the maximum speed is approximately 17 m s⁻¹. For any speed below this, friction supplies just enough force — note that static friction is self-adjusting up to its maximum value.
因此最大速度约为17 m s⁻¹。只要速度低于此值，摩擦力就会自动提供刚好够用的向心力——注意静摩擦力在达到最大值之前是自适应的。
8. Worked Example: Conical Pendulum | 例题：圆锥摆
Question: A mass of 0.5 kg is attached to a light string of length 1.2 m and swings in a horizontal circle with the string making an angle of 30° with the vertical. Find the period of the motion.
Solution: The radius of the horizontal circle is r = L sinθ. The vertical component of tension balances weight: T cosθ = mg. The horizontal component provides the centripetal force: T sinθ = mω²r.
解答：水平圆周的半径为r = L sinθ。张力的竖直分量与重力平衡：T cosθ = mg。水平分量提供向心力：T sinθ = mω²r。
Dividing the two equations eliminates T:
tanθ = ω²r / g = ω²L sinθ / g
Since sinθ cancels:
ω² = g / (L cosθ)
两式相除消去T：
tanθ = ω²r / g = ω²L sinθ / g
约去sinθ得：
ω² = g / (L cosθ)
Thus T = 2π√(L cosθ / g) = 2π√(1.2 × cos30° / 9.8) = 2π√(1.039 / 9.8) = 2.05 s. Note that the mass does not affect the period — only the length and angle matter.
For a banked curve with angle θ to the horizontal, the normal reaction has a horizontal component that contributes to the centripetal force. When designed for a speed v₀ with no reliance on friction, the banking angle satisfies:
This design speed is the ideal speed for the curve. If the actual speed differs from v₀, friction acts either up or down the slope to supply the extra or reduce the required centripetal force.
10. Common Misconceptions and Exam Tips | 常见误区与考试技巧
Misconception: Centripetal force is an extra force. Fact: It is the net force pointing toward the centre; label it in force diagrams as the resultant, not as a separate force.
误区：向心力是额外的力。 事实：它是指向圆心的合力；在受力图中应标为合力而非单独的力。
Misconception: An object moving in a circle with constant speed has no acceleration. Fact: Its direction changes, so it has centripetal acceleration of magnitude v²/r.
误区：匀速圆周运动没有加速度。 事实：速度方向在改变，因此存在大小为v²/r的向心加速度。
Misconception: Centripetal force does work on the object. Fact: The force is perpendicular to displacement, so work done is zero and kinetic energy stays constant in uniform circular motion.
误区：向心力对物体做功。 事实：向心力与位移垂直，做功为零，匀速圆周运动中动能保持不变。
Tip: Always draw a free-body diagram first. Then resolve forces radially and identify which components provide the centripetal force.
技巧：先画受力分析图，再沿半径方向分解力，找出提供向心力的分量。
Tip: When using the formula sheet, check whether you are given v or ω. Convert using v = rω if necessary.
技巧：使用公式表时，先确认题目给出的是v还是ω。必要时用v = rω换算。
11. Key Equations Summary | 关键公式总结
Quantity | 物理量
Equation | 公式
Angular speed | 角速度
ω = Δθ/Δt = 2π/T = 2πf
Linear speed | 线速度
v = rω
Centripetal acceleration | 向心加速度
a = v²/r = ω²r
Centripetal force | 向心力
F = mv²/r = mω²r = mωv
Banking angle | 弯道倾角
tanθ = v²/(rg)
Minimum speed at top of loop | 圆环顶部最小速度
v = √(gr)
Mastering circular motion dynamics means knowing not just the formulas but also the physical reasoning behind them. In an exam, always start with a force analysis, then apply Newton’s second law radially. With practice, these problems become quick marks.
Published by TutorHao | IB Physics Revision Series | aleveler.com
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📚 IB Physics: Relationship Between Wavelength, Frequency and Wave Speed | IB物理：波长、频率与波速关系及应用
Wave motion is a fundamental concept in IB Physics. Understanding how wavelength, frequency and wave speed are related allows students to describe sound, light, water waves and electromagnetic radiation with one simple equation.
1. What Are Wavelength, Frequency and Wave Speed? | 什么是波长、频率与波速？
Wavelength (λ) is the distance between two consecutive identical points on a wave, such as crest-to-crest or compression-to-compression. In SI units it is measured in metres (m).
Frequency (f) is the number of complete oscillations passing a fixed point per second. It is measured in hertz (Hz), where 1 Hz = 1 s⁻¹.
频率（f）是每秒通过固定点的完整振动次数，单位为赫兹（Hz），1 Hz = 1 s⁻¹。
The period (T) is the time for one complete oscillation, and is related to frequency by T = 1/f.
周期（T）是一次完整振动所需时间，与频率互为倒数，即 T = 1/f。
Wave speed (v) is the distance travelled by a wave disturbance per unit time, measured in metres per second (m s⁻¹).
波速（v）是波扰动在单位时间内传播的距离，单位为米每秒（m s⁻¹）。
2. The Wave Equation v = f × λ | 波的方程 v = f × λ
In one wavelength, the wave travels a distance λ in time T, so the speed is v = λ / T. Since T = 1/f, this becomes v = f × λ.
在一个波长内，波在时间 T 内传播距离 λ，因此速度为 v = λ / T。由 T = 1/f，可得 v = f × λ。
v = f × λ
This equation applies to all types of periodic waves, including transverse waves on strings and longitudinal sound waves.
该方程适用于所有周期性波，包括弦上的横波和声波等纵波。
3. Understanding From Wavefronts | 从波前理解
Imagine a wave source emitting crests at regular intervals. The distance between adjacent crests is λ, and each crest takes time T to reach the next position. The pattern moves forward at the wave speed.
想象一个波源以固定时间间隔发出波峰。相邻波峰之间的距离为 λ，每个波峰经过时间 T 前进一个波长。整个图样以波速向前移动。
If the source vibrates faster, frequency increases and adjacent crests become closer, so wavelength decreases when speed is fixed.
若波源振动更快，频率增大，相邻波峰间隔变小；在波速不变时，波长会减小。
4. Wave Speed Depends on the Medium | 波速取决于介质
For mechanical waves, wave speed is set by the properties of the medium, not by the frequency or wavelength. On a string, speed increases with greater tension and decreases with greater linear density.
对于机械波，波速由介质性质决定，而非频率或波长。在弦上，张力越大波速越大；线密度越大波速越小。
For sound, speed depends on temperature and medium. In air at 20 °C it is about 343 m s⁻¹; in water it is about 1480 m s⁻¹.
声速与温度和介质有关。在 20 °C 空气中约为 343 m s⁻¹；在水中约为 1480 m s⁻¹。
Since the medium fixes v, changing the source frequency changes the wavelength according to λ = v / f.
由于介质决定 v，改变波源频率会改变波长，即 λ = v / f。
5. Electromagnetic Waves in Vacuum | 真空中的电磁波
Electromagnetic waves are transverse waves that can travel through a vacuum. They all travel at the same speed in vacuum, c = 3.00 × 10⁸ m s⁻¹.
电磁波是可以在真空中传播的横波。所有电磁波在真空中的速度相同，c = 3.00 × 10⁸ m s⁻¹。
The general equation becomes c = f × λ. Since c is constant, a higher frequency corresponds to a shorter wavelength.
波速方程化为 c = f × λ。由于 c 恒定，频率越高，波长越短。
This explains why radio waves have long wavelengths and gamma rays have very short wavelengths.
这解释了为什么无线电波波长较长，而 γ 射线波长非常短。
6. Applications in Sound and Music | 声学与音乐中的应用
Musical instruments produce specific frequencies. A guitar string with fixed tension and length produces a fundamental frequency determined by the wave speed and the string length.
乐器产生特定频率。在张力和长度一定时，吉他弦发出的基频由波速和弦长决定。
In a pipe organ, the resonant wavelength fits into the pipe length, so changing the effective length changes the frequency heard.
在管风琴中，共振波长与管长匹配，因此改变有效管长会改变听到的频率。
7. Applications in Communication and Medicine | 通信与医学中的应用
Mobile phones, Wi-Fi and radio use electromagnetic waves of different frequencies. Antenna designs depend on the corresponding wavelength.
手机、Wi-Fi 和无线电使用不同频率的电磁波。天线设计取决于对应的波长。
Ultrasound imaging uses high-frequency mechanical waves. Higher frequency gives better resolution but reduces penetration depth.
超声成像使用高频机械波。频率越高分辨率越好，但穿透深度会减小。
Laser light with a smaller wavelength can focus to a smaller spot, which is essential in eye surgery and optical data storage.
波长更小的激光可以聚焦到更小的光斑，这对眼科手术和光存储非常重要。
8. Worked Example 1 – Sound Wave | 例题 1：声波
A sound wave has a frequency of 680 Hz in air where the wave speed is 340 m s⁻¹. Find its wavelength.
空气中一列声波的频率为 680 Hz，波速为 340 m s⁻¹。求波长。
λ = v / f = 340 / 680 = 0.50 m
The wavelength is 0.50 m.
波长为 0.50 m。
9. Worked Example 2 – Light Wave | 例题 2：光波
A laser emits visible light with wavelength 500 nm in vacuum. Calculate its frequency.
一束激光在真空中的波长为 500 nm。计算其频率。
f = c / λ = (3.00 × 10⁸) / (500 × 10⁻⁹) = 6.00 × 10¹⁴ Hz
The frequency is approximately 6.00 × 10¹⁴ Hz.
该光波的频率约为 6.00 × 10¹⁴ Hz。
10. Common Misconceptions and Exam Tips | 常见误区与考试技巧
Misconception: Changing frequency changes wave speed. In fact, wave speed is determined by the medium for a fixed state.
误区：改变频率会改变波速。事实上，在介质状态不变时，波速由介质决定。
Use standard units: nm should be converted to metres before substitution.
使用标准单位：代入公式前应把纳米换算成米。
When comparing waves, state whether the medium is the same or different.
比较波时，需说明介质是否相同。
11. Quick Practice Questions | 快速练习
Question / 问题
Answer / 答案
A wave has λ = 2.0 m and f = 5.0 Hz. What is v? / 一列波的 λ = 2.0 m，f = 5.0 Hz，v 是多少？
v = 10 m s⁻¹
A wave has v = 1500 m s⁻¹ and λ = 0.75 m. Find f. / 一列波的 v = 1500 m s⁻¹，λ = 0.75 m，求 f。
f = 2000 Hz
What is the wavelength of a 100 MHz radio wave in air? / 空气中频率为 100 MHz 的无线电波波长是多少？
λ = 3.0 m
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📚 IB Physics: Basic Concepts of Magnetic Fields and Ampere Force | IB物理：磁场基本概念与安培力
Magnetic fields are one of the most fundamental topics in IB Physics, forming the basis for electromagnetism, induction, and modern technology. This article covers the essential definitions, the concept of magnetic flux density, field line representations, and the Ampere force experienced by current-carrying conductors in magnetic fields.
Magnetic fields are produced by moving electric charges, i.e., electric currents. Permanent magnets also produce magnetic fields due to the aligned motion of electrons within their atoms. Every magnetic field is fundamentally a relativistic effect of moving charges.
In IB Physics, two key sources of magnetic fields are considered: current-carrying wires and permanent magnets. The magnetic field is a vector field, denoted by B, and its direction at any point is defined as the direction in which the north pole of a small compass needle points.
在IB物理中，磁场的主要来源有两种：通电导线和永磁体。磁场是矢量场，用 B 表示。某一点的磁场方向定义为小磁针N极在该点静止时的指向。
B = F_max / (q v sin θ)
This definition relates the magnetic field strength to the maximum force experienced by a moving test charge q at speed v. The unit of B is the tesla (T), where 1 T = 1 N·s/(C·m) = 1 N/(A·m).
该定义将磁感应强度与运动试探电荷q在速度v下所受的最大力联系起来。B的单位是特斯拉（T），1 T = 1 N·s/(C·m) = 1 N/(A·m)。
2. Magnetic Flux Density vs. Magnetic Flux | 磁感应强度与磁通量
Magnetic flux density (B) is a measure of how dense the magnetic field lines are, representing the strength of the field. Magnetic flux (Φ) is the total number of field lines passing through a given area, calculated by the dot product of B and the area vector.
Here, θ is the angle between the direction of the magnetic field and the normal to the surface area A. The SI unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T·m².
A common exam question asks whether flux is zero when the field is parallel to the surface. Since θ = 90°, cos 90° = 0, so Φ = 0. This emphasises that flux depends on the orientation of the surface relative to the field.
Magnetic field lines are a visual tool used to represent the direction and strength of a magnetic field. The lines point away from the north pole and towards the south pole outside a magnet, forming closed loops through the interior from south to north.
Lines never intersect; the field has a unique direction at each point.
Lines are closer together where the field is stronger.
Lines always form closed loops; there are no magnetic monopoles.
The tangent to a field line at any point gives the direction of B at that point.
磁感线永不相交；每一点的磁场方向唯一。
磁感线越密集，表示磁场越强。
磁感线总是闭合回路；不存在磁单极子。
磁感线上任意一点的切线方向即为该点的B方向。
∮ B · dA = 0
Gauss’s law for magnetism states that the net magnetic flux through any closed surface is zero, confirming that magnetic field lines are closed loops. This is a fundamental contrast to electric field lines, which begin and end on charges.
4. Force on a Current-Carrying Conductor | 电流导体在磁场中的受力
A current-carrying wire placed in a magnetic field experiences a force known as the Ampere force. This force is the macroscopic result of the magnetic force acting on the individual moving charge carriers inside the wire.
where F is the force in newtons, B is the magnetic flux density in teslas, I is the current in amperes, L is the length of the conductor in the field in metres, and θ is the angle between the wire (current direction) and the magnetic field.
If θ = 90°, the force is maximum: F_max = BIL. If θ = 0°, the wire is parallel to the field and experiences no force. This angular dependence is crucial in many IB questions involving rotating coils or angled wires.
To determine the direction of the Ampere force, use Fleming’s left-hand rule. Hold your left hand so that the thumb, index finger, and middle finger are mutually perpendicular.
安培力的方向使用弗莱明左手定则判断。将左手拇指、食指和中指相互垂直。
Index finger: direction of the magnetic field (B).
Middle finger: direction of the current (I).
Thumb: direction of the force (F).
食指：磁场方向（B）。
中指：电流方向（I）。
拇指：安培力方向（F）。
F = I L × B
The cross product form reminds us that F is perpendicular to both I and B. The magnitude is given by |F| = I L B sin θ, and the direction follows the right-hand rule for cross products, which is equivalent to Fleming’s left-hand rule when applied to charge carriers.
矢量叉积形式 F = I L × B 提醒我们F同时垂直于电流方向和磁场方向。其大小由 |F| = I L B sin θ 给出，方向遵循叉积的右手定则，这等价于对载流子应用弗莱明左手定则。
6. Interactions Between Parallel Current-Carrying Wires | 平行电流导线间的相互作用
Two parallel current-carrying wires each produce a magnetic field that exerts a force on the other wire. This interaction is one of the most elegant applications of the Ampere force.
两根平行的通电导线各自产生磁场，并对另一根导线施力。这种相互作用是安培力最经典的应用之一。
F/L = (μ₀ I₁ I₂) / (2π d)
where μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space, I₁ and I₂ are the currents, and d is the separation between the wires.
If the currents flow in the same direction, the wires attract each other.
If the currents flow in opposite directions, the wires repel each other.
电流方向相同时，导线相互吸引。
电流方向相反时，导线相互排斥。
This principle is used to define the ampere in the SI system: one ampere is the constant current that produces a force of 2 × 10⁻⁷ N per metre of length between two infinitely long parallel wires 1 metre apart in vacuum.
A direct current (DC) electric motor converts electrical energy into mechanical energy using the Ampere force. A rectangular coil placed in a magnetic field experiences forces on its sides that produce a torque, causing the coil to rotate.
Here, τ is the torque, N is the number of turns of the coil, A is the area of the coil, and θ is the angle between the field and the normal to the coil plane. The torque is maximum when the coil plane is parallel to the field (θ = 90°).
In IB examinations, students are expected to explain how the split-ring commutator reverses the current direction every half-turn to maintain continuous rotation, and why the torque varies with the angle of rotation.
Problem: A straight wire of length 0.40 m carries a current of 5.0 A. It is placed in a uniform magnetic field of flux density 0.20 T at an angle of 30° to the field. Calculate the magnitude of the force on the wire.
F = 0.20 × 5.0 × 0.40 × sin 30° = 0.20 × 5.0 × 0.40 × 0.5 = 0.20 N
The direction of the force is perpendicular to both the wire and the magnetic field. Use the left-hand rule to identify the exact direction in three dimensions.
安培力的方向同时垂直于导线和磁场。使用左手定则确定其三维空间中的具体方向。
Common mistake: students forget the sin θ term and simply multiply BIL when the wire is not perpendicular to the field. Always check the angle given in the problem.
Students often confuse magnetic and electric forces. The table below summarises their essential differences.
学生经常混淆磁力和电力。下表总结了它们的基本区别。
Property
Electric Force
Magnetic Force
Acts on
Any charged particle
Only moving charged particles
Does work on particle
Yes, changes kinetic energy
No, force ⊥ velocity
Direction
Parallel/anti-parallel to E
Perpendicular to B and v
Field lines
Start and end on charges
Always closed loops
A magnetic field does no work on a moving charge since the force is always perpendicular to the velocity. This means the speed of a charged particle in a uniform magnetic field remains constant, although its direction changes.
The following points summarise the most frequent traps in IB Physics exams on this topic.
以下要点总结了IB物理考试中本主题最常见的陷阱。
Always convert all quantities to SI units before substitution.
Check whether θ in F = BIL sin θ is the angle between the wire and the field, not the complement.
For flux, use the angle between the normal to the surface and the field, not between the surface and the field.
Use the left-hand rule for the force on a current, but the right-hand rule for the force on a moving positive charge; a negative charge reverses the direction.
Do not say that the magnetic force does work; it can only change direction, not speed.
代入公式前，务必将所有量换算为SI单位。
检查 F = BIL sin θ 中的θ是导线与磁场的夹角，而不是其余角。
对于磁通量，使用表面法线与磁场的夹角，而不是表面与磁场的夹角。
电流受力用左手定则；正电荷受力用右手定则；负电荷方向相反。
不能说磁场力做功；它只能改变方向，不能改变速率。
In vector calculus notation, remember that dF = I dL × B. The differential form is useful when the conductor is curved or the field is non-uniform, which appears in higher-level IB problems.
在矢量微积分表示中，dF = I dL × B。微分形式在处理弯曲导体或非均匀磁场时很有用，这在IB高阶问题中会出现。
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📚 IB Physics: Heating Effect of Current and Joule’s Law | IB物理：电流热效应与焦耳定律
When an electric current flows through a conductor, electrical energy is converted into thermal energy. This phenomenon, known as the heating effect of current, is governed by Joule’s Law and forms a cornerstone of circuit analysis in IB Physics. Understanding this effect is essential for explaining everything from the glowing filament of a lamp to the design of safety fuses in household appliances.
Electrical power is defined as the rate at which electrical energy is transferred. For a component with potential difference V across it and current I through it, the power P is given by:
电功率定义为电能转移的速率。对于两端电压为V、通过电流为I的元件，功率P由下式给出：
P = VI
This relationship holds universally for any electrical component, whether it is a resistor, a motor, or a semiconductor diode. The SI unit of power is the watt (W), equivalent to one joule per second (J s⁻¹).
Total electrical energy E transferred in time t is then:
在时间t内转移的总电能E为：
E = VIt
In IB Physics, you must be able to distinguish between power (a rate) and energy (a quantity). Power is measured in watts, while energy is measured in joules. A common exam question asks students to calculate the energy consumed by an appliance over a given time period using the formula E = VIt.
2. Joule’s Law: Statement and Formula | 焦耳定律：表述与公式
Joule’s Law states that the heat H produced in a conductor is directly proportional to the square of the current I, the resistance R of the conductor, and the time t for which the current flows:
焦耳定律指出：导体中产生的热量H与电流I的平方、导体的电阻R以及通电时间t成正比：
H = I²Rt
This law was experimentally established by James Prescott Joule in the 1840s. It is a direct consequence of the work done by the electric field in moving charge carriers through a resistive medium. Each collision between electrons and lattice ions transfers kinetic energy to the lattice, manifesting as macroscopic thermal energy.
The power dissipated as heat, often called the ohmic heating power, is:
以热量形式耗散的功率，通常称为欧姆热功率，为：
P = I²R
Using Ohm’s law (V = IR), this can also be written as P = V²/R. All three forms — P = VI, P = I²R, and P = V²/R — are equivalent for ohmic conductors, but each is convenient in different contexts.
Consider a charge ΔQ moving through a potential difference V. The work done on the charge is ΔW = VΔQ. Since current I = ΔQ/Δt, the work done per unit time (i.e., power) becomes:
For a purely resistive component, applying Ohm’s law V = IR gives P = I²R. Over a time interval t, the total heat generated is H = Pt = I²Rt. This derivation connects the macroscopic energy transfer to microscopic charge motion, a key conceptual step in IB Paper 1 and Paper 2 questions.
It is important to note that this derivation assumes an ideal resistor where all electrical energy is converted to heat. In real devices such as motors, some energy is converted to mechanical work, so the simple Joule heating formula applies only to the resistive part of the circuit.
4. Resistance and Temperature Dependence | 电阻与温度的关系
For metallic conductors, resistance increases with temperature. As current flows and heats the conductor, the lattice ions vibrate more vigorously, increasing the probability of electron-ion collisions. This leads to a higher resistance at higher temperatures.
The relationship is approximately linear over limited temperature ranges:
在有限的温度范围内，这一关系近似为线性：
R = R₀(1 + αΔT)
where R₀ is the resistance at a reference temperature, α is the temperature coefficient of resistance, and ΔT is the temperature change. For tungsten (used in light bulb filaments), α ≈ 0.0045 °C⁻¹.
An important consequence is that the current through a filament lamp is not proportional to voltage: as the filament heats up, its resistance increases, so the I-V characteristic becomes curved. This non-ohmic behaviour is a classic IB data-analysis question. When the lamp is first switched on, its resistance is low, leading to a brief high current surge — this explains why bulbs often blow at the moment of switching on.
5. Series and Parallel Resistors: Heating Comparison | 串联与并联电阻的发热比较
In a series circuit, the same current flows through all resistors. Since P = I²R, the resistor with the highest resistance dissipates the most power. This is why in a series string of Christmas lights, the brighter bulbs have higher resistance.
In a parallel circuit, the same voltage appears across each resistor. Here P = V²/R is more convenient: the resistor with the lowest resistance dissipates the most power. A toaster with multiple heating elements in parallel allows each element to operate at full mains voltage.
This comparison is frequently tested in IB Paper 1 multiple-choice questions. A reliable strategy is to identify which variable is common to all resistors first, then choose the appropriate power formula.
Joule’s Law has numerous practical applications across everyday life and industry. Electric heaters, kettles, toasters, and hair dryers all use resistive elements designed to maximize heat output. The heating element is typically a nichrome wire (an alloy of nickel and chromium) with high resistivity and a high melting point.
Incandescent light bulbs operate on this principle: the tungsten filament is heated to approximately 2500 °C, at which temperature it emits visible light. However, only about 5% of the energy is converted to light; the remaining 95% is dissipated as heat, making incandescent bulbs extremely inefficient.
Fuses are protective devices that exploit the heating effect. A fuse contains a thin wire with a low melting point. If the current exceeds a rated value, the wire heats up quickly and melts, breaking the circuit and protecting downstream components. The fuse rating is chosen so that I²R heating causes the fuse to melt before damage occurs to the appliance.
In the transmission of electrical power, Joule heating represents an unavoidable loss. To minimize I²R losses in power lines, electricity is transmitted at very high voltages and low currents. Step-up transformers raise the voltage to hundreds of kilovolts, reducing current and hence reducing I²R losses quadratically.
Efficiency η is defined as the ratio of useful output power to total input power:
效率η定义为有用输出功率与总输入功率之比：
η = Puseful / Pinput × 100%
For a heater, nearly all input power becomes useful heat, so efficiency approaches 100%. For a light bulb, efficiency is low because most energy is wasted as heat. For an electric motor, efficiency depends on overcoming friction and resistive losses in the windings.
IB Physics questions often ask students to calculate efficiency given input voltage, current, resistance, and useful output data. For example, a motor operating at 12 V and drawing 2 A has an input power of 24 W. If it lifts a mass at a rate corresponding to 18 W of mechanical power, its efficiency is (18/24) × 100% = 75%.
Example 1: A 230 V electric kettle has a heating element of resistance 23 Ω. Calculate the current, the power, and the energy converted to heat in 2 minutes.
例题1：一只230V的电水壶的加热元件电阻为23Ω。计算电流、功率以及2分钟内转化为热能的能量。
Using Ohm’s law, I = V/R = 230/23 = 10 A. The power is P = VI = 230 × 10 = 2300 W. The energy converted in t = 120 s is E = Pt = 2300 × 120 = 276,000 J = 276 kJ.
Example 2: Two resistors of 4 Ω and 6 Ω are connected in parallel across a 12 V battery. Which resistor produces more heat per second?
例题2：两个阻值分别为4Ω和6Ω的电阻并联连接在12V电池两端。哪个电阻每秒产生的热量更多？
In parallel, the voltage is the same across both resistors. Using P = V²/R: for the 4 Ω resistor, P = 144/4 = 36 W; for the 6 Ω resistor, P = 144/6 = 24 W. The 4 Ω resistor produces more heat because lower resistance draws a higher current.
Example 3: A 2 A current flows through a 5 Ω resistor for 10 minutes. Calculate the heat dissipated and the rise in temperature if the mass of the resistor is 50 g and its specific heat capacity is 400 J kg⁻¹ K⁻¹ (assuming no heat loss).
Misconception 1: “Higher resistance always means more heat.” This is only true in series circuits where current is constant. In parallel circuits, lower resistance leads to greater current and therefore more heat.
Misconception 2: “P = V²/R and P = I²R are always interchangeable.” They are only equivalent for ohmic conductors that obey V = IR. For non-ohmic devices such as diodes or thermistors, these formulas give different results and must be applied with caution.
Misconception 3: “The filament lamp obeys Ohm’s law.” A cold filament has lower resistance; as it heats up, resistance increases. The I-V graph is a curve, not a straight line, so the lamp does not obey Ohm’s law across its operating range.
Misconception 4: “Power lost in transmission lines can be reduced by increasing the voltage.” While this is true, students sometimes think higher voltage means higher current. In fact, for a given transmitted power, voltage and current are inversely related (P = VI), so raising voltage lowers current and reduces I²R losses dramatically.
A standard IB experiment investigates the relationship between heat generated and current. An insulated calorimeter contains a known mass of water and an immersion heater. Different currents are passed through the heater for a fixed time, and the temperature rise of the water is measured.
Using Q = mcΔT and equating this to H = I²Rt, we can plot ΔT against I². The graph should be a straight line through the origin, confirming the I² dependence in Joule’s Law. Uncertainties in temperature measurement (±0.5 K) and current (±0.01 A) should be propagated to evaluate the final uncertainty.
Another common investigation uses a rheostat to vary the current through a fixed resistor immersed in oil. The temperature rise of the oil is recorded against time for different resistor values, allowing students to verify both the I² and R dependence of heating.
When tackling heating effect problems, always begin by identifying whether the circuit is series or parallel to determine which variable (current or voltage) is constant. Then choose the most appropriate form of the power formula.
Pay close attention to units: energy in joules, power in watts, time in seconds. If time is given in minutes or hours, convert to seconds first. Also remember that 1 kWh = 3.6 × 10⁶ J — this conversion frequently appears in energy billing questions.
For non-ohmic components, never assume V = IR holds over the entire range. Use the instantaneous values from the I-V graph. When asked to estimate power at a specific point, find V and I at that point and multiply them.
In data-based questions, the gradient of a ΔT vs I² graph directly gives R/(mc), which can be used to determine the resistance if the mass and specific heat capacity of the calorimeter system are known. Always include error bars and discuss whether the line of best fit passes through the origin within experimental uncertainty.
12. Connecting to the IB Physics Syllabus | 联系IB物理大纲
This topic sits within Topic 5 (Electricity and Magnetism) of the IB Physics syllabus, specifically under the subtopic of electric circuits. It also extends into Topic 8 (Energy Production) when discussing transmission losses, and Topic 2 (Mechanics/Thermal Physics) when linking heat to temperature change via specific heat capacity.
In the new IB syllabus (first assessment 2025), students are expected to derive Joule’s law from energy conservation principles and apply it to practical situations including electrical heating and safety devices. The ability to solve multi-step problems combining V = IR, P = VI, and Q = mcΔT is an essential skill for achieving top marks in Paper 2.
Understanding the heating effect of current is not just about memorising formulas — it requires a deep conceptual grasp of energy transformation chains. Mastering this topic builds a solid foundation for more advanced studies in electromagnetism, thermodynamics, and electrical engineering at the university level.
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📚 IB Physics: Electromotive Force and Internal Resistance Analysis | IB物理：电池电动势与内阻分析
In any real battery, the chemical reactions that produce electrical energy are not perfectly efficient. Inside the battery, charge carriers encounter resistance from the electrolyte and electrodes, meaning that when current flows, some electrical energy is converted to heat before the charges ever leave the terminals. Understanding this internal resistance is essential for predicting how a battery actually behaves in a circuit.
1. Understanding EMF and Terminal Voltage | 理解电动势与端电压
Electromotive force (EMF), denoted by E, is the total energy supplied by the battery per unit charge passing through it. It is measured in volts and represents the theoretical maximum potential difference the battery can provide when no current flows. The terminal voltage V is the actual potential difference measured across the battery terminals when it delivers current.
电动势（EMF）用 E 表示，是电池在单位电荷通过时所提供的总能量。它以伏特为单位，表示电池在没有电流流动时能够提供的理论最大电势差。端电压 V 则是在电池输出电流时，实际测得的两端子之间的电势差。
When no current flows, the terminal voltage equals the EMF because there is no voltage drop across the internal resistance. As soon as the battery is connected to a load, the terminal voltage falls below the EMF. The difference between E and V is exactly the energy lost inside the battery per unit charge.
当没有电流时，端电压等于电动势，因为内阻上没有电压降。一旦电池接入负载，端电压就会低于电动势。E 与 V 之差恰好是单位电荷在电池内部损失的能量。
2. Internal Resistance and the Circuit Model | 内阻与电路模型
A real battery can be modelled as an ideal EMF source in series with a small resistor, r, known as the internal resistance. This resistance is not a separate physical component but a lumped representation of the battery’s internal opposition to charge flow. The larger the current drawn, the greater the voltage drop across r.
实际电池可以建模为一个理想电动势源与一个小电阻 r 串联，这个电阻就是内阻。它并不是一个独立的物理元件，而是电池内部对电荷流动阻碍的集中表示。输出电流越大，内阻 r 上的电压降就越大。
For a circuit consisting of the battery and an external load resistance R, the total resistance in the circuit is R + r. The current is therefore given by I = E / (R + r). This model is extremely powerful: it allows us to treat a real battery exactly like an ideal source plus a known resistor.
对于包含电池和外接负载电阻 R 的电路，回路总电阻为 R + r。因此电流为 I = E / (R + r)。这个模型非常强大：它让我们能够将真实电池视为理想电源与一个已知电阻的串联组合。
3. The Relationship Between EMF, Terminal Voltage, Current and Internal Resistance | 电动势、端电压、电流和内阻的关系
From the circuit model, the terminal voltage V is the EMF minus the voltage drop across the internal resistance. This gives the fundamental equation:
根据电路模型，端电压 V 等于电动势减去内阻上的电压降。由此得到基本方程：
V = E − I r
This linear relationship is the key to understanding battery behaviour. When the current I is zero, V equals E. As the current increases, the terminal voltage decreases linearly with a gradient of −r. The greater the internal resistance, the more sharply the voltage drops when current is drawn.
这个线性关系是理解电池行为的关键。当电流 I 为零时，V 等于 E。随着电流增大，端电压以 −r 为斜率线性下降。内阻越大，输出电流时电压下降得越显著。
For example, a battery with E = 9.0 V and r = 0.5 Ω, delivering 2.0 A, has a terminal voltage of V = 9.0 − (2.0)(0.5) = 8.0 V. The missing 1.0 V represents energy converted to heat inside the battery itself.
例如，一个 E = 9.0 V、r = 0.5 Ω 的电池，输出 2.0 A 电流时，端电压为 V = 9.0 − (2.0)(0.5) = 8.0 V。损失的 1.0 V 代表电池内部转化为热量的能量。
4. Measuring EMF and Internal Resistance | 测量电动势和内阻
To determine E and r experimentally, you can measure the terminal voltage V and the current I for several different load resistances. Using the equation V = E − I r, a graph of V against I gives a straight line with a y-intercept equal to E and a gradient equal to −r.
为了实验测定 E 和 r，可以针对不同的负载电阻测量端电压 V 和电流 I。利用方程 V = E − I r，画出 V 对 I 的图像，得到一条直线，其纵截距等于 E，斜率等于 −r。
A typical setup uses a variable resistor as the load, an ammeter in series, and a voltmeter connected directly across the battery terminals. By adjusting the variable resistor, you record pairs of (V, I) values. Ensure the switch is closed only while taking readings to avoid polarisation and temperature changes in the battery.
Alternatively, you can measure the open-circuit voltage with a high-resistance voltmeter to obtain E, then use a single known load to find r. However, the graphical method is preferred because it averages out random errors and reveals any systematic drift.
When plotting V on the vertical axis and I on the horizontal axis, the equation V = E − I r produces a straight line of the form y = mx + c. The y-intercept is E, the x-intercept is E/r, and the absolute value of the gradient is r. This graph is a standard IB Physics analysis task.
以 V 为纵轴、I 为横轴作图时，方程 V = E − I r 给出形如 y = mx + c 的直线。纵截距为 E，横截距为 E/r，斜率的绝对值为 r。这张图是 IB 物理标准的分析任务。
Note that the x-intercept corresponds to the short-circuit current, I_s = E/r, which is the maximum current the battery could supply if the external resistance were zero. In practice, this condition is rarely achieved and can damage the battery, but the intercept can still be extrapolated from the graph.
When drawing the best-fit line, ignore outliers and remember that the theoretical line should be straight. A curved graph indicates that r changes with current, often due to heating or electrolyte polarization.
画最佳拟合线时忽略异常点，理论线应为直线。若图像弯曲，说明 r 随电流变化，通常由发热或电解质极化引起。
6. Short Circuit and Maximum Current | 短路与最大电流
If the terminals of a battery are connected directly by a wire of negligible resistance, the external resistance approaches zero. The current is then limited only by the internal resistance, giving I_max = E/r. This is called the short-circuit current.
During a short circuit, the terminal voltage drops to nearly zero because nearly all of the EMF is used to drive current through the internal resistance. The power dissipated inside the battery is P = I²r, which can cause rapid overheating, venting, or even rupture in real batteries.
短路时端电压几乎降至零，因为几乎所有电动势都用于在内阻上驱动电流。电池内部耗散功率为 P = I²r，可能导致电池快速过热、泄气甚至破裂。
In practical terms, a fresh AA alkaline battery with E ≈ 1.5 V and r ≈ 0.15 Ω could theoretically supply 10 A, but typical short-circuit currents are lower due to additional contact resistance. Always use a current-limiting resistor when testing batteries.
实际中，一节新的 AA 碱性电池 E ≈ 1.5 V、r ≈ 0.15 Ω，理论上可提供 10 A，但由于接触电阻，实际短路电流更低。测试电池时务必使用限流电阻。
7. Power Dissipation and Efficiency | 功率损耗与效率
The total power produced by the battery is P_total = E I. Part of this power is delivered to the external load, P_out = V I, and the remainder is dissipated inside the battery as heat, P_loss = I²r. The sum of these two equals the total power: E I = V I + I²r.
电池产生的总功率为 P_total = E I。其中一部分输出给外部负载，P_out = V I，其余部分在电池内部以热量形式耗散，P_loss = I²r。两者之和等于总功率：E I = V I + I²r。
The efficiency of the battery is the ratio of useful output power to total power:
电池的效率是有用输出功率与总功率之比：
η = V / E = R / (R + r)
When the load resistance R is very large compared to r, the terminal voltage approaches E and the efficiency approaches 100%. Conversely, when R is small, much of the energy is wasted inside the battery, and efficiency is low.
当负载电阻 R 远大于 r 时，端电压接近 E，效率接近 100%。反之，当 R 很小时，大量能量在电池内部被浪费，效率很低。
8. Matching External Resistance for Maximum Power Transfer | 外阻匹配与最大功率传输
For a given battery, the power delivered to the external load is P_out = I²R, where I = E / (R + r). As R varies, P_out has a maximum value. Some IB exam questions ask you to derive or state that maximum power is transferred when the load resistance equals the internal resistance.
对于给定电池，输出到外部负载的功率为 P_out = I²R，其中 I = E / (R + r)。当 R 变化时，P_out 存在最大值。一些 IB 考题要求推导或说明：当负载电阻等于内阻时，传输功率最大。
The condition R = r gives P_max = E² / (4r). Under this condition, the terminal voltage is exactly half of the EMF, and the efficiency is 50%. This result is called the maximum power transfer theorem.
满足 R = r 时，P_max = E² / (4r)。此时端电压恰好等于电动势的一半，效率为 50%。这一结论称为最大功率传输定理。
It is important to distinguish between maximum power transfer and maximum efficiency. In many real applications, such as power distribution, we want high efficiency rather than maximum power. Deliberately matching the load to r is rarely desirable for battery-powered devices.
9. Effects of Internal Resistance on Real Batteries | 内阻对真实电池的影响
Internal resistance is not constant. It depends on temperature, state of charge, and the chemical composition of the battery. As a battery discharges, its internal resistance often increases because the concentration of active reactants decreases and reaction products accumulate on the electrodes.
At low temperatures, chemical reactions slow down and internal resistance increases significantly. This is why a car battery may struggle to start an engine on a cold morning. Conversely, high temperatures reduce internal resistance but can accelerate unwanted side reactions and reduce battery lifetime.
For rechargeable batteries, internal resistance is also influenced by the charge-discharge cycle history. Overcharging or deep discharging can damage the electrode structure, permanently raising r. Regulators and battery management systems monitor r as a health indicator.
对于充电电池，内阻还受到充放电循环历史的影响。过充或深放电会损伤电极结构，使 r 永久增大。电源管理芯片常将 r 作为健康状态的监测指标。
10. Experimental Techniques and Sources of Error | 实验技巧与误差来源
In the IB laboratory, measuring E and r requires careful attention to reduce systematic errors. The voltmeter should have a very high resistance so that the current it draws is negligible compared with the circuit current. The ammeter should have a very low resistance to avoid introducing additional series resistance.
在 IB 实验室中，测量 E 和 r 需要仔细减小系统误差。电压表内阻应非常高，使其分走的电流相对回路电流可忽略。电流表内阻应非常低，以避免引入额外的串联电阻。
Contact resistance at the variable resistor and connecting wires can also distort results. Use thick connecting leads and ensure all connections are clean and tight. The battery itself may heat up when large currents flow, changing r during the experiment, so keep currents small and readings quick.
滑动变阻器和导线连接点的接触电阻也会使结果失真。使用粗导线并确保所有连接清洁、紧固。当大电流流过时电池本身会发热，使 r 在实验过程中改变，因此应保持小电流并快速读数。
Repeated readings and taking the best-fit line of V versus I help reduce random errors. Extrapolating to find E is more reliable than single measurements, provided the linear relationship holds. If the graph shows curvature, you should note that r is not constant and limit your conclusions to the range of currents tested.
重复读数并对 V 与 I 作最佳拟合线可减少随机误差。外推求 E 比单次测量更可靠，前提是线性关系成立。若图像出现弯曲，应说明 r 并非恒定，并只在你测试的电流范围内下结论。
11. Worked Examples | 例题分析
Example 1: A battery has E = 6.0 V and r = 0.80 Ω. It is connected to a resistor of 4.2 Ω. Find (a) the circuit current, (b) the terminal voltage, (c) the power dissipated in the internal resistance.
(a) I = E / (R + r) = 6.0 / (4.2 + 0.8) = 6.0 / 5.0 = 1.2 A
(a) I = E / (R + r) = 6.0 / (4.2 + 0.8) = 6.0 / 5.0 = 1.2 A
(b) V = E − I r = 6.0 − (1.2)(0.80) = 6.0 − 0.96 = 5.04 V
(b) V = E − I r = 6.0 − (1.2)(0.80) = 6.0 − 0.96 = 5.04 V
(c) P_loss = I²r = (1.2)² × 0.80 = 1.44 × 0.80 = 1.152 W ≈ 1.2 W
(c) P_loss = I²r = (1.2)² × 0.80 = 1.44 × 0.80 = 1.152 W ≈ 1.2 W
Example 2: When a cell supplies 0.50 A, the terminal voltage is 1.45 V. When it supplies 2.0 A, the terminal voltage is 1.30 V. Find E and r.
例 2：某电池输出 0.50 A 时端电压为 1.45 V；输出 2.0 A 时端电压为 1.30 V。求 E 和 r。
1.45 = E − 0.50r, 1.30 = E − 2.0r
Subtracting the equations: 0.15 = 1.5r → r = 0.10 Ω. Then E = 1.45 + 0.50 × 0.10 = 1.50 V.
两式相减：0.15 = 1.5r → r = 0.10 Ω。于是 E = 1.45 + 0.50 × 0.10 = 1.50 V。
12. Common Misconceptions and Exam Tips | 常见误解与考试提示
A frequent error is confusing EMF with terminal voltage. EMF is the energy supplied per unit charge, while terminal voltage is the measured potential difference under load. Another misconception is that a battery always outputs a constant voltage; in reality, the terminal voltage decreases as current increases.
In exam questions, always draw the circuit model with r inside the battery symbol. Label E and r explicitly. When using V = E − I r, make sure the current I is the same through the external circuit and the battery. For open-circuit questions, set I = 0 so that V = E.
在考试题中，务必画出带内阻 r 的电池电路模型，明确标出 E 和 r。使用 V = E − I r 时，确保电流 I 同时流过外部电路和电池内部。对于断路问题，令 I = 0，则 V = E。
Finally, be careful with units: resistance in ohms, current in amperes, voltage in volts. Check whether the question asks for terminal voltage, EMF, or power, as these quantities are related but not interchangeable. Reading the question carefully is the first step to full marks.
Published by TutorHao | Physics Revision Series | aleveler.com
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📚 IB Physics: Core Concepts and Key Points of Electric Fields | IB物理：电场核心概念与考点
Electric fields are a cornerstone of the IB Physics syllabus, bridging the gap between mechanics and electromagnetism. This article consolidates the core definitions, laws, and problem-solving techniques you need to master for both Standard Level (SL) and Higher Level (HL) exams.
Charge is a fundamental property of matter, measured in coulombs (C). Like charges repel, opposite charges attract. The net charge of an isolated system is conserved.
电荷是物质的基本属性，单位是库仑（C）。同种电荷相互排斥，异种电荷相互吸引。孤立系统的净电荷守恒。
Coulomb’s law states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
库仑定律指出，两个点电荷之间的静电力与它们电荷量的乘积成正比，与它们之间距离的平方成反比。
F = k·|q₁q₂| / r², where k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²
F = k·|q₁q₂| / r²，其中 k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C²
Note that the force is a vector: it acts along the line joining the two charges. In IB problems, always sketch the direction of forces before calculating their magnitudes.
注意力是矢量：其方向沿两电荷连线。在IB解题中，先画出力的方向再进行定量计算。
2. Electric Field Strength | 电场强度
Electric field strength E at a point is defined as the force per unit positive charge placed at that point. It is a vector quantity with units N/C or equivalently V/m.
电场中某点的电场强度E定义为该点处单位正电荷所受的力。它是矢量，单位为N/C，也可写作V/m。
E = F / q
E = F / q
For a point charge Q, the field strength at distance r is:
对于点电荷Q，在距离r处产生的场强为：
E = kQ / r²
E = kQ / r²
Field strength depends only on the source charge and distance, not on the test charge. A positive test charge experiences a force in the direction of the field; a negative test charge experiences a force opposite to the field.
Electric field lines provide a visual representation of the field. The tangent to a field line at any point gives the direction of E at that point. The density of lines indicates the magnitude of the field.
Field lines start on positive charges and end on negative charges.
电场线始于正电荷，终止于负电荷。
Field lines never cross each other.
电场线永不相交。
In a uniform field, the lines are parallel and equally spaced.
在匀强电场中，电场线平行且间距相等。
For a uniform electric field between two parallel plates, the field strength is constant everywhere, and the field lines are straight and parallel.
两块平行板之间的匀强电场中，场强处处相同，电场线为平行直线。
4. Electric Potential Energy and Electric Potential | 电势能与电势
Electric potential energy (U) is the energy a charge possesses due to its position in an electric field. Work done by an external force to move a charge in a field changes its potential energy.
电势能（U）是电荷在电场中因位置而具有的能量。外力移动电荷做功会改变其电势能。
Electric potential V at a point is the potential energy per unit positive charge placed at that point:
电场中某点的电势V定义为该点处单位正电荷所具有的电势能：
V = U / q
V = U / q
Potential is a scalar quantity, measured in volts (V = J/C). For a point charge Q, the potential at distance r is:
电势是标量，单位是伏特（V = J/C）。对于点电荷Q，距离r处的电势为：
V = kQ / r
V = kQ / r
Unlike field strength, which falls off as 1/r², potential falls off as 1/r. The zero of potential is conventionally taken at infinity.
与场强按1/r²衰减不同，电势按1/r衰减。通常取无穷远处为电势零点。
5. Potential Difference and Work Done | 电势差与做功
The potential difference (V) between two points is the work done per unit charge in moving a charge between those points:
两点之间的电势差（V）等于将单位电荷在这两点之间移动时所做的功：
W = q·ΔV
W = q·ΔV
In a uniform field with plate separation d and potential difference ΔV, the field strength is:
在间距为d、电势差为ΔV的匀强电场中，场强为：
E = ΔV / d
E = ΔV / d
This relationship is frequently tested in IB papers. Remember that d is the distance along the field direction, not along any arbitrary path.
这一关系在IB考试中经常考查。注意d是沿电场方向的距离，而不是任意路径长度。
Work done by the electric field when a charge moves from point A to point B is independent of the path taken; it depends only on the potential difference between the endpoints.
电场力做功与路径无关，只与起点和终点之间的电势差有关。
6. Equipotential Surfaces | 等势面
Equipotential surfaces are surfaces on which every point has the same electric potential. No work is done in moving a charge along an equipotential surface.
等势面是电势处处相等的曲面。电荷沿等势面移动时电场力不做功。
Key properties of equipotential surfaces include:
等势面的主要性质包括：
They are always perpendicular to electric field lines.
等势面始终与电场线垂直。
The electric field is directed from high potential to low potential.
电场方向从高电势指向低电势。
For a point charge, the equipotential surfaces are concentric spheres centered on the charge.
对于点电荷，等势面是以电荷为球心的同心球面。
In a uniform field, the equipotential surfaces are parallel planes perpendicular to the field lines.
匀强电场中的等势面是垂直于电场线的平行平面。
When drawing field lines, ensure they meet equipotential surfaces at right angles. This visual check is a quick way to verify your diagrams in exams.
画电场线时，确保电场线与等势面垂直相交。这是考试中快速检查作图是否正确的一个技巧。
7. Motion of Charged Particles in Electric Fields | 带电粒子在电场中的运动
A charged particle placed in a uniform electric field experiences a constant force. If the particle is stationary or moving parallel to the field, it undergoes linear acceleration.
带电粒子在匀强电场中受到恒定电场力。如果粒子静止或初速度方向与电场平行，则做匀加速直线运动。
For a particle of charge q and mass m in a field E, the acceleration is:
对于质量为m、电荷量为q的粒子在场强E中的加速度为：
a = qE / m
a = qE / m
If the initial velocity is perpendicular to the field, the particle follows a parabolic trajectory, analogous to projectile motion under gravity. This is a common HL exam scenario.
Use the principle of conservation of energy: for a charge accelerating from rest through a potential difference ΔV, the kinetic energy gained is:
运用能量守恒定律：从静止开始经过电势差ΔV加速的电荷，其获得的动能为：
½mv² = q·ΔV
½mv² = q·ΔV
This energy approach is often simpler than kinematic equations when dealing with potential differences.
在处理电势差问题时，能量法通常比运动学方程更简便。
8. Parallel Plate Capacitors | 平行板电容器
A parallel plate capacitor consists of two conducting plates separated by a dielectric. Its capacitance C is defined as the charge stored per unit potential difference:
平行板电容器由两块被电介质隔开的导体板构成。其电容C定义为单位电势差所储存的电荷量：
C = Q / V
C = Q / V
For a parallel plate capacitor:
对于平行板电容器：
C = ε₀·A / d
C = ε₀·A / d
where A is the plate area and d is the plate separation. Capacitance is measured in farads (F), with typical values ranging from pF to μF.
其中A是极板面积，d是极板间距。电容的单位为法拉（F），实际常用皮法（pF）到微法（μF）量级。
When a capacitor is connected to a battery, the potential difference remains constant and the charge changes with capacitance. When disconnected, the charge remains constant and the potential difference changes.
电容器连接电源时，电势差不变，电荷随电容变化；断开电源后，电荷量不变，电势差随电容变化。
9. Energy Stored in a Capacitor | 电容器储存的能量
The energy stored in a charged capacitor can be expressed in three equivalent forms:
带电电容器储存的能量可以用三种等价形式表示：
E = ½QV = ½CV² = Q²/(2C)
E = ½QV = ½CV² = Q²/(2C)
The energy is stored in the electric field between the plates. The energy density (energy per unit volume) in a vacuum field is:
能量储存在两板之间的电场中。真空电场的能量密度（单位体积的能量）为：
u = ½ε₀E²
u = ½ε₀E²
This elegant result shows that energy is a property of the field itself, not of the charges directly. IB questions often ask which formula to use based on the quantities that remain constant during charging or discharging.
A dielectric is an insulating material that reduces the electric field between the plates for a given charge. Introducing a dielectric with dielectric constant κ increases capacitance by a factor of κ:
电介质是绝缘材料，在电荷一定时能减弱两板间的电场。引入相对介电常数为κ的电介质后，电容增大κ倍：
C = κ·ε₀·A / d
C = κ·ε₀·A / d
The dielectric constant is a dimensionless number greater than 1 for most materials. For vacuum, κ = 1 exactly.
相对介电常数是无量纲量，大多数材料的κ大于1，真空的κ精确等于1。
In IB exams, you should be able to analyze two scenarios: (a) capacitor connected to a battery (V constant), and (b) capacitor isolated (Q constant). When a dielectric is inserted into an isolated capacitor, the voltage drops and the energy stored decreases; when inserted into a connected capacitor, the charge and energy increase.
12. Key Formulas and Quick Revision Table | 核心公式与快速复习表
The table below summarizes the essential formulas for the electric field topic. It is not exhaustive, but it covers the highest-yield equations for your exam.
下表汇总了电场主题的核心公式。虽然并非面面俱到，但覆盖了考试中最高频的方程。
Quantity
Formula
Notes
Coulomb’s force
F = kq₁q₂/r²
k = 8.99 × 10⁹ N·m²/C²
Field strength (definition)
E = F/q
Valid for any field
Field strength (point charge)
E = kQ/r²
Radial field
Field strength (uniform)
E = ΔV/d
d along field direction
Potential (point charge)
V = kQ/r
Zero at infinity
Work done
W = qΔV
Path independent
Capacitance
C = Q/V = ε₀A/d
Add dielectric: multiply by κ
Stored energy
E = ½QV = ½CV²
Choose convenient form
Students who memorize these formulas along with their conditions of validity will find most IB electric field questions straightforward. The most common exam errors stem from misapplying a formula outside its valid range.
Published by TutorHao | Physics Revision Series | aleveler.com
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📚 IB Physics: Core Concepts and Exam Points in Electromagnetism | IB物理：电磁学核心概念与考点梳理
Electromagnetism is one of the most conceptually rich and exam-relevant topics in IB Physics. It unifies electric and magnetic phenomena through the concept of fields, and it underpins countless applications from capacitors to generators. This article provides a structured review of the core ideas, key equations, and common exam traps you must master for both SL and HL.
Electric charge is a fundamental property of matter. Like charges repel, unlike charges attract. The SI unit of charge is the coulomb (C), and the elementary charge is e = 1.60 × 10⁻¹⁹ C.
电荷是物质的基本属性。同种电荷相斥，异种电荷相吸。电荷的国际单位是库仑（C），元电荷为 e = 1.60 × 10⁻¹⁹ C。
Coulomb’s law gives the force between two point charges:
库仑定律给出了两个点电荷之间的作用力：
F = k|q₁q₂| / r² = (1 / 4πε₀) · |q₁q₂| / r²
Here, k ≈ 8.99 × 10⁹ N·m²·C⁻², and ε₀ = 8.85 × 10⁻¹² C²·N⁻¹·m⁻² is the permittivity of free space. The force acts along the line joining the charges.
IB exam tip: Always state whether the force is attractive or repulsive; do not just give a magnitude.
IB考试提示：在描述库仑力时一定要说明是引力还是斥力，不能只写大小。
2. Electric Field and Field Lines | 电场与电场线
An electric field is a region where a charge experiences a force. The electric field strength E at a point is defined as the force per unit positive charge: E = F/q. Its unit is N·C⁻¹ or V·m⁻¹.
电场是电荷在其中会受到力的空间区域。电场强度 E 定义为每单位正电荷所受的力：E = F/q，单位是 N·C⁻¹ 或 V·m⁻¹。
For a point charge Q, the field strength at distance r is:
对于点电荷 Q，距离 r 处的场强为：
E = kQ / r²
Electric field lines start on positive charges and end on negative charges. The density of lines indicates the strength of the field. In a uniform field, the lines are parallel and equally spaced.
电场线从正电荷出发，终止于负电荷。电场线的疏密表示场强大小。在匀强电场中，电场线平行且间距相等。
For a uniform field between two parallel plates separated by distance d with potential difference V:
对于间距为 d、电势差为 V 的两平行板之间的匀强电场：
E = V / d
Remember: Field strength is a vector; use superposition for multiple charges.
注意：场强是矢量；多个电荷时需用叠加原理。
3. Electric Potential and Potential Difference | 电势与电势差
Electric potential V at a point is the work done per unit charge in bringing a positive test charge from infinity to that point. For a point charge Q:
电场中某点的电势 V 是将单位正电荷从无穷远处移到该点所做的功。对于点电荷 Q：
V = kQ / r
Potential difference (voltage) ΔV between two points is the work done per unit charge moving a charge between those points: W = qΔV.
两点之间的电势差（电压）ΔV 是单位电荷在两点间移动时所做的功：W = qΔV。
Equipotential surfaces are surfaces of constant potential. In a uniform field, they are planes perpendicular to the field lines. No work is done moving a charge along an equipotential surface.
等势面是电势相等的面。在匀强电场中，等势面是与电场线垂直的平面。电荷沿等势面移动时电场力不做功。
Common misconception: Potential is zero at infinity is a convention; only differences matter in calculations.
常见误区：电势“无穷远处为零”只是约定；实际计算中只有电势差才有意义。
4. Capacitance and Energy Storage | 电容与储能
A capacitor stores charge and energy. Capacitance C is defined as C = Q/V, where Q is the magnitude of charge on either plate and V is the potential difference between the plates. The unit is the farad (F).
电容器储存电荷和能量。电容 C 定义为 C = Q/V，其中 Q 是任一极板上的电荷量，V 是两极板间的电势差。单位是法拉（F）。
For a parallel-plate capacitor in a vacuum, capacitance depends on geometry:
真空中的平行板电容器，其电容取决于几何结构：
C = ε₀A / d
Where A is the plate area and d is the separation. If a dielectric of relative permittivity εᵣ fills the gap, multiply by εᵣ.
其中 A 是极板面积，d 是极板间距。若两极板间充满相对介电常数为 εᵣ 的电介质，则电容要乘以 εᵣ。
The energy stored in a charged capacitor is:
充电电容器储存的能量为：
E = ½ QV = ½ CV² = Q² / (2C)
HL requirement: Understand how inserting a dielectric changes C, Q, V, and stored energy in constant-voltage vs. isolated-capacitor cases.
HL要求：理解在恒压或孤立电容器情形下，插入电介质如何改变 C、Q、V 和储能。
5. Electric Current and Ohm’s Law | 电流与欧姆定律
Electric current is the rate of flow of charge. The average current is I = ΔQ/Δt. Conventional current direction is from positive to negative, opposite to electron flow.
电流是电荷流动的速率。平均电流为 I = ΔQ/Δt。规定电流方向是从正极到负极，与电子运动方向相反。
Ohm’s law states that for an ohmic conductor at constant temperature, the potential difference V across it is proportional to the current I through it:
欧姆定律指出，对于温度恒定的欧姆导体，其两端电压 V 与通过它的电流 I 成正比：
V = IR
Resistance R depends on the material and geometry: R = ρL / A, where ρ is resistivity, L is length, and A is cross-sectional area. Resistivity is temperature-dependent; for metals it increases with temperature.
I–V characteristic curves: a straight line through the origin for ohmic conductors; curves for filament lamps, diodes, and thermistors.
I–V 特性曲线：欧姆导体为过原点的直线；白炽灯、二极管、热敏电阻则呈现曲线。
6. DC Circuits and Kirchhoff’s Laws | 直流电路与基尔霍夫定律
Real circuits consist of resistors, cells, and connecting wires. A cell has an internal resistance r, so the terminal voltage is less than the emf when current flows: V_terminal = ε − Ir.
Kirchhoff’s laws are powerful tools for complex circuits:
基尔霍夫定律是分析复杂电路的有力工具：
Kirchhoff’s current law (KCL): The sum of currents entering any junction equals the sum leaving it (charge conservation).
基尔霍夫电流定律（KCL）：流入任一节点的电流之和等于流出该节点的电流之和（电荷守恒）。
Kirchhoff’s voltage law (KVL): The sum of emfs around any closed loop equals the sum of potential drops (energy conservation).
基尔霍夫电压定律（KVL）：沿任一闭合回路，电动势之和等于电势降之和（能量守恒）。
7. Magnetic Fields and Magnetic Force | 磁场与磁场力
Magnetic fields are produced by moving charges or permanent magnets. The magnetic field strength (magnetic flux density) B is measured in tesla (T). Field lines point from north to south outside a magnet.
磁场由运动的电荷或永磁体产生。磁感应强度（磁通密度）B 的单位是特斯拉（T）。磁场线在磁体外部从 N 极指向 S 极。
A charge q moving with velocity v perpendicular to a uniform magnetic field B experiences a force:
当电荷 q 以速度 v 垂直于匀强磁场 B 运动时，受到的磁场力为：
F = qvB
For an arbitrary angle θ between v and B, F = qvB sinθ. The direction is given by the right-hand rule for a positive charge. This force is always perpendicular to the velocity, so it does no work and changes only the direction of motion.
若 v 与 B 的夹角为 θ，则 F = qvB sinθ。方向由右手定则确定（针对正电荷）。该力始终垂直于速度，因此不做功，只改变运动方向。
For a current-carrying wire of length L in a uniform magnetic field, the force is:
对于处在匀强磁场中、长度为 L 的通电导线，所受磁场力为：
F = BIL sinθ
Common exam question: circular motion of a charged particle in a perpendicular magnetic field, with radius r = mv / (qB).
常见考题：带电粒子在垂直磁场中做匀速圆周运动，半径 r = mv / (qB)。
8. Electromagnetic Induction and Faraday’s Law | 电磁感应与法拉第定律
Electromagnetic induction occurs when the magnetic flux through a circuit changes. Magnetic flux Φ through an area A in a field B is defined as Φ = BA cosθ, where θ is the angle between the field direction and the normal to the area. The unit is the weber (Wb).
当穿过回路的磁通量发生变化时，就会产生电磁感应。磁通量 Φ 定义为 Φ = BA cosθ，其中 θ 是磁场方向与面积法线方向的夹角，单位是韦伯（Wb）。
Faraday’s law states that the induced emf is equal to the negative rate of change of magnetic flux linkage:
法拉第定律指出，感应电动势等于磁通链变化率的负值：
ε = −N (ΔΦ / Δt)
Here N is the number of turns, and NΦ is the flux linkage. Lenz’s law gives the direction: the induced current opposes the change that produced it. The negative sign in Faraday’s law reflects Lenz’s law.
其中 N 是线圈匝数，NΦ 是磁通链。楞次定律给出感应电流的方向：感应电流总是阻碍引起它的磁通量变化。法拉第定律中的负号正体现了楞次定律。
SL/HL distinction: HL requires using the derivative form ε = −d(NΦ)/dt and explaining motional emf ε = BvL.
Alternating current (AC) varies sinusoidally with time: I = I₀ sin(ωt) and V = V₀ sin(ωt). The root-mean-square (rms) values are used for power calculations:
Average power dissipated in a resistor is P = V_rms I_rms = I_rms²R = V_rms²/R.
电阻上消耗的平均功率为 P = V_rms I_rms = I_rms²R = V_rms²/R。
An ideal transformer steps voltage up or down using mutual induction:
理想变压器利用互感升压或降压：
V_s / V_p = N_s / N_p
For an ideal transformer, input power equals output power: V_p I_p = V_s I_s. Power losses in real transformers are reduced by laminated iron cores, thick low-resistance wires, and efficient designs to minimize eddy currents and hysteresis.
10. Common Exam Traps and How to Avoid Them | 常见考试陷阱与应对策略
Many students lose marks on electromagnetism not because they lack knowledge, but because of small conceptual slips. Here are the most frequent traps.
许多学生在电磁学上失分并非因为知识不足，而是因为一些小的概念性疏漏。以下是最高频的陷阱。
Trap 1: Confusing electric field E (N/C) with electric potential V (J/C). They are related by E = ΔV/d only in uniform fields.
陷阱1：混淆电场强度 E（N/C）与电势 V（J/C）。只有在匀强电场中它们才满足 E = ΔV/d。
Trap 2: Forgetting that the magnetic force does no work. A charged particle in a magnetic field changes direction but not speed.
陷阱2：忘记磁场力不做功。带电粒子在磁场中只改变方向，不改变速率。
Trap 3: Using the wrong rms vs. peak values. Power must be calculated with rms values in AC circuits.
陷阱3：混用有效值与峰值。交流电路中计算功率必须使用有效值。
Trap 4: Ignoring internal resistance. Terminal voltage is not the same as emf except at open circuit.
陷阱4：忽略内阻。除了断路情况，路端电压不等同于电动势。
Trap 5: Misapplying Lenz’s law. Always ask: “Does the induced current oppose the change in flux?” Then find the direction.
陷阱5：错误应用楞次定律。始终问自己：“感应电流是否阻碍了磁通量的变化？”然后判断方向。
To score high, practice drawing field lines, labeling directions, and writing symbolic answers before substituting numbers. Review past paper questions on circuits and induction until the patterns become automatic.
Published by TutorHao | Physics Revision Series | aleveler.com
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📚 IB Physics: Core Concepts of Simple Harmonic Motion & Period Calculation | IB物理：简谐运动核心概念与周期计算
Simple Harmonic Motion (SHM) is one of the most fundamental topics in IB Physics, appearing in both Standard Level (Topic 4) and Higher Level (Topic 9) syllabi. It bridges mechanics and wave phenomena, requiring students to understand the interplay between displacement, velocity, acceleration, and restoring forces. This guide breaks down the essential definitions, conditions, energy transformations, and — most importantly — the period calculations that appear repeatedly in IB examinations.
Simple harmonic motion is defined as the oscillatory motion in which the acceleration of the object is directly proportional to its displacement from equilibrium and always directed toward the equilibrium position. This means the acceleration is always opposite in direction to the displacement.
The mathematical condition for SHM can be written as:
a = −ω²x
where a is acceleration, x is displacement from equilibrium, and ω is the angular frequency (measured in rad s⁻¹). The negative sign indicates that acceleration always points toward equilibrium — a restoring effect.
其中 a 为加速度，x 为相对于平衡位置的位移，ω 为角频率（单位 rad s⁻¹）。负号表示加速度始终指向平衡位置——即回复效应。
For an oscillating system to exhibit SHM, two key conditions must be satisfied:
Restoring force: The net force must be proportional to displacement and directed toward equilibrium (F = −kx, Hooke’s Law).
回复力：合力必须与位移成正比且指向平衡位置（F = −kx，胡克定律）。
Linear range: The system must obey this linear relationship within its range of oscillation; for a pendulum this requires small angles (typically θ < 10°).
No damping: In ideal SHM, energy is conserved and oscillations continue indefinitely (energy losses ignored).
无阻尼：理想简谐运动中能量守恒，振动无限持续（忽略能量损失）。
2. Key Kinematic Quantities | 关键运动学量
Three primary quantities describe SHM: displacement, velocity, and acceleration. Each varies sinusoidally with time, and their phases are offset relative to one another.
For an object starting at maximum displacement (amplitude A) at t = 0, the displacement equation is:
x = A cos(ωt)
If the object starts at equilibrium and moves in the positive direction, use the sine form: x = A sin(ωt). Both forms are acceptable in IB answers; just be consistent with initial conditions.
若物体在 t = 0 时从最大位移（振幅 A）处开始，位移方程为：x = A cos(ωt)。若物体从平衡位置出发并向正方向运动，则使用正弦形式：x = A sin(ωt)。IB答题中两种形式均可，但需与初始条件保持一致。
Differentiating the displacement equation yields velocity:
v = −Aω sin(ωt)
The maximum speed occurs as the object passes through equilibrium and is given by v_max = Aω. At this point the velocity is nonzero while the restoring force and acceleration are zero.
This confirms that acceleration is proportional to displacement but in the opposite direction. The maximum acceleration, which occurs at the turning points, is a_max = ω²A.
The angular frequency ω connects the period of oscillation to the physical properties of the system. For any SHM system, the period T is the time for one complete oscillation and is related to ω by:
角频率 ω 将振动周期与系统的物理性质联系起来。对于任何简谐运动系统，周期 T 是一次完整振动所需的时间，与 ω 的关系为：
T = 2π / ω
Note that in ideal SHM, the period is independent of the amplitude. This is a hallmark property of SHM — whether you pull a pendulum back 2° or 8°, the period remains essentially the same (within the small-angle approximation).
One of the most common IB exam traps is confusing ω with frequency f. They are related by ω = 2πf, where f is the ordinary frequency measured in hertz (Hz). Thus:
IB考试中最常见的陷阱之一是混淆 ω 与频率 f。它们的关系为 ω = 2πf，其中 f 是普通频率，单位为赫兹（Hz）。因此：
T = 1/f = 2π/ω
4. Period of a Mass–Spring System | 弹簧振子的周期
For a mass m attached to a spring with spring constant k, the restoring force is given by Hooke’s Law: F = −kx. Combining this with Newton’s Second Law (F = ma), we obtain ma = −kx, which simplifies to a = −(k/m)x.
对于质量为 m、连接在劲度系数为 k 的弹簧上的物体，回复力由胡克定律给出：F = −kx。结合牛顿第二定律（F = ma），可得 ma = −kx，化简为 a = −(k/m)x。
Comparing this with the general SHM condition a = −ω²x, we identify ω² = k/m, giving:
将此与一般简谐运动条件 a = −ω²x 比较，可得 ω² = k/m，从而：
T = 2π√(m/k)
Key observations:
Heavier mass → longer period: More inertia means the system oscillates more slowly.
质量越大 → 周期越长：惯性越大，系统振动越慢。
Stiffer spring → shorter period: A larger k value means a stronger restoring force and faster oscillation.
弹簧越硬 → 周期越短：k 值越大意味着回复力越强，振动越快。
Period is independent of amplitude and gravitational field: A mass–spring system on the Moon has the same period as on Earth (assuming ideal conditions).
周期与振幅和重力场无关：在月球上的弹簧振子与在地球上的周期相同（理想条件下）。
This formula is essential for both SL and HL examinations. In practical labs, this relationship is often verified by plotting T² against m, which yields a straight line with slope 4π²/k.
该公式在SL和HL考试中都至关重要。在实验课中，通常通过绘制 T² 对 m 的图线来验证此关系，所得直线的斜率为 4π²/k。
5. Period of a Simple Pendulum | 单摆的周期
For a simple pendulum of length L with a small-amplitude oscillation, the restoring force is the component of gravity tangential to the motion: F = −mg sin θ. For small angles, sin θ ≈ θ, and using arc length s = Lθ, we can show that:
对于摆长为 L、小角度振动的单摆，回复力是重力沿运动切向方向的分量：F = −mg sin θ。在小角度下，sin θ ≈ θ，利用弧长 s = Lθ，可以证明：
a = −(g/L)x
Therefore ω² = g/L, and the period is:
因此 ω² = g/L，周期为：
T = 2π√(L/g)
Critical points to remember for IB exams:
IB考试中需记住的关键要点：
Period depends only on L and g — not on mass. A heavier bob does not change the period.
周期只取决于 L 和 g —— 与质量无关。更重的摆锤不会改变周期。
Small-angle approximation is required. For angles beyond approximately 10°, sin θ ≠ θ and the motion deviates from true SHM; the period becomes slightly longer.
g can be determined experimentally by measuring T and L: g = 4π²L/T². This is a standard IB Physics practical investigation.
可通过实验测定 g：通过测量 T 和 L 计算：g = 4π²L/T²。这是IB物理标准实验探究课题。
A common misconception is that the pendulum period changes when the mass is increased. It does not — the relation T = 2π√(L/g) contains no mass term. However, changing the length of the string or moving the pendulum to a location with different gravitational field strength absolutely will change the period.
In ideal SHM, mechanical energy is conserved. Energy continuously transforms between kinetic energy (KE) and potential energy (PE), with the total remaining constant.
在理想简谐运动中，机械能守恒。能量在动能（KE）和势能（PE）之间持续转化，总能量保持恒定。
At the turning points (x = ±A): all energy is potential.
在转向点（x = ±A）：所有能量均为势能。
E_total = ½kA²
At the equilibrium position (x = 0): all energy is kinetic, and the speed is maximum.
在平衡位置（x = 0）：所有能量均为动能，速度达到最大值。
E_total = ½mv_max² = ½kA²
At any intermediate displacement x, the energies are:
在任意中间位移 x 处，能量分别为：
KE = ½k(A² − x²)
PE = ½kx²
For a pendulum, the same principle applies but with gravitational potential energy replacing elastic potential energy. At maximum height (amplitude), the bob has maximum gravitational PE; at the lowest point, all energy is kinetic.
Exam tip: When asked to sketch energy–displacement graphs, remember that PE is a parabola (∝ x²), KE is an inverted parabola, and total energy is a horizontal line — all of which remain positive throughout the motion.
Phase describes the position within the oscillation cycle at a given time. Two oscillating systems are said to be in phase when they reach their maximum displacements in the same direction at the same time.
In SHM, displacement, velocity, and acceleration have specific phase relationships:
在简谐运动中，位移、速度和加速度之间存在特定的相位关系：
Velocity leads displacement by π/2 (90°): when displacement is maximum, velocity is zero; when displacement is zero, velocity is maximum.
速度超前位移 π/2（90°）：位移最大时速度为零；位移为零时速度最大。
Acceleration is π (180°) out of phase with displacement: they are in opposite directions at all times.
加速度与位移反相 π（180°）：它们始终方向相反。
Acceleration is π/2 (90°) behind velocity: maximum acceleration occurs at the turning points where velocity is zero.
加速度落后速度 π/2（90°）：最大加速度出现在转向点，此时速度为零。
Graphically, these phase relationships appear as shifted sine/cosine curves. Being able to read these graphs and connect them to physical motion is a core IB skill.
SHM is the one-dimensional projection of uniform circular motion. Consider an object moving in a circle of radius A with constant angular speed ω. If you project its position onto a diameter, the projection executes SHM.
简谐运动是匀速圆周运动的一维投影。考虑一个物体在半径为 A 的圆上以恒定角速度 ω 运动。如果将其位置投影到直径上，投影点做简谐运动。
This geometric interpretation explains several important relationships:
这种几何解释阐明了几个重要的关系：
The amplitude of SHM equals the radius of the reference circle: A.
简谐运动的振幅等于参考圆的半径：A。
The angular frequency of SHM equals the angular speed of the circular motion: ω.
简谐运动的角频率等于圆周运动的角速度：ω。
The period of SHM equals the time for one full revolution: T = 2π/ω.
简谐运动的周期等于旋转一周所需的时间：T = 2π/ω。
The reference circle is particularly useful for solving problems involving initial conditions. If a particle starts at some arbitrary position and phase, sketching the reference circle helps determine whether to use sine or cosine and how to adjust the phase constant φ in the general equation x = A cos(ωt + φ).
参考圆在解决涉及初始条件的问题时特别有用。如果粒子从某个任意位置和相位开始运动，画出参考圆有助于判断应使用正弦还是余弦函数，以及如何调整一般方程 x = A cos(ωt + φ) 中的相位常数 φ。
For IB Paper 2 problems that combine SHM with circular motion (e.g., a particle on a rotating turntable viewed from the side), this conceptual link is often the key to finding the correct answer.
对于结合简谐运动与圆周运动的IB Paper 2题（例如从侧面观察转盘上粒子的运动），这种概念联系往往是找到正确答案的关键。
9. Damping and Resonance | 阻尼与共振
Real-world oscillating systems lose energy over time due to friction, air resistance, or other dissipative forces. This gradual loss of amplitude is called damping. While ideal SHM assumes no energy loss, IB Physics requires an understanding of how damping affects real systems.
Light damping: Amplitude decreases gradually over many cycles; the period remains approximately constant.
欠阻尼：振幅在多个周期内逐渐减小；周期近似保持不变。
Critical damping: The system returns to equilibrium in the shortest possible time without oscillating.
临界阻尼：系统在不振动的情况下以最短时间回到平衡位置。
Heavy (over) damping: The system returns to equilibrium very slowly, without oscillating.
过阻尼：系统缓慢回到平衡位置，不发生振动。
Resonance occurs when the driving frequency matches the natural frequency of the system, causing a dramatic increase in amplitude. The natural frequency f₀ is related to the natural period by f₀ = 1/T. For a mass–spring system, f₀ = (1/2π)√(k/m); for a pendulum, f₀ = (1/2π)√(g/L).
IB exam questions often ask you to interpret a resonance curve (amplitude vs. driving frequency). Key features include: the peak occurs at f₀, and increased damping lowers the peak amplitude while slightly broadening the curve.
IB考题通常要求解读共振曲线（振幅 vs. 驱动频率）。关键特征包括：峰值出现在 f₀ 处；阻尼增大时峰值振幅降低，曲线稍微变宽。
10. Worked Examples | 典型例题精解
Example 1: Mass–Spring System | 例1：弹簧振子系统
A 0.25 kg mass is attached to a spring with k = 100 N m⁻¹. Calculate the period and frequency of oscillation.
一个 0.25 kg 的物体连接在劲度系数 k = 100 N m⁻¹ 的弹簧上。计算振动周期和频率。
Solution | 解答：
T = 2π√(m/k) = 2π√(0.25 / 100) = 2π√(0.0025) = 2π × 0.05 = 0.314 s
f = 1/T = 1/0.314 = 3.18 Hz
Example 2: Pendulum on the Moon | 例2：月球上的单摆
A pendulum has a period of 2.0 s on Earth (g = 9.81 m s⁻²). What is its period on the Moon where g = 1.62 m s⁻²?
一个单摆在地球上（g = 9.81 m s⁻²）的周期为 2.0 s。在月球上（g = 1.62 m s⁻²）其周期为多少？
Note how a weaker gravitational field produces a longer period — the pendulum swings more slowly on the Moon. This example illustrates why gravitational field strength is a crucial parameter for pendulum systems but completely irrelevant for ideal mass–spring systems.
This example reinforces the relationship between period, angular frequency, and the kinematic quantities. Always convert period to angular frequency first before attempting to find velocities or accelerations.
此例强化了周期、角频率与运动学量之间的关系。在求速度或加速度之前，务必先将周期转换为角频率。
11. Common IB Exam Mistakes | 常见IB考试错误
Over years of marking IB Physics papers, certain errors appear repeatedly. Being aware of these can significantly boost your exam performance.
在多年批改IB物理试卷的过程中，某些错误反复出现。了解这些错误可以显著提升你的考试成绩。
Forgetting the small-angle approximation for pendulums: The formula T = 2π√(L/g) is ONLY valid for θ < 10°. IB questions may specify "small oscillations" — if not stated, note this assumption in your answer.
Confusing frequency f and angular frequency ω: f is in Hz, ω is in rad s⁻¹; they differ by a factor of 2π. When using T = 2π√(m/k), do not insert k/m directly into f.
混淆频率 f 与角频率 ω：f 单位为 Hz，ω 单位为 rad s⁻¹；两者相差 2π 倍。使用 T = 2π√(m/k) 时，不要将 k/m 直接代入 f。
Sign errors with acceleration direction: The restoring force always points toward equilibrium. In vector problems, carefully define your positive direction first.
加速度方向符号错误：回复力始终指向平衡位置。在矢量问题中，先明确定义正方向。
Using x = A sin(ωt) without checking initial conditions: Verify where the object starts at t = 0 before selecting the sine or cosine form.
未检查初始条件就使用 x = A sin(ωt)：在选用正弦或余弦形式前，需先确认 t = 0 时物体的位置。
Omitting units: Period must be in seconds, angular frequency in rad s⁻¹, and amplitude in the same length unit throughout the calculation.
遗漏单位：周期必须以秒为单位，角频率以 rad s⁻¹ 为单位，振幅在整个计算过程中保持相同的长度单位。
12. Exam Strategy Checklist | 备考策略清单
To succeed in SHM questions on both Paper 1 and Paper 2, build a systematic approach:
要在Paper 1和Paper 2的简谐运动题目中取得成功，需要建立系统化的解题方法：
Identify the system type: Is it a mass–spring (T = 2π√(m/k)) or a pendulum (T = 2π√(L/g))? This determines which formula set applies.
Extract all given quantities: Write down m, k, L, g, A, T, or ω as given in the problem. Convert to SI units early.
提取所有已知量：写下题目中给出的 m、k、L、g、A、T 或 ω。尽早转换为SI单位。
Check assumptions: For pendulums, verify small angles; for ideal SHM, confirm no damping is assumed.
检查假设条件：对于单摆，确认小角度；对于理想简谐运动，确认假设无阻尼。
Draw a diagram or reference circle: Visualizing the motion helps determine phase and initial conditions.
画图或参考圆：可视化运动有助于判断相位和初始条件。
For graphical questions: Identify which quantity is plotted on each axis, then relate slopes and intercepts to ω, k, or g.
对于图像题：确认每个坐标轴上的物理量，然后将斜率和截距与 ω、k 或 g 联系起来。
Interpret the problem physically: Before calculating, predict the answer qualitatively — will the period increase or decrease based on changing parameters?
从物理角度理解问题：在计算之前，定性地预测答案——参数变化后周期会增大还是减小？
Remember: the period equations for SHM are among the most tested formulas in the IB Physics syllabus. Mastery of when and how to apply them, accompanied by a solid grasp of the sinusoidal relationships between x, v, and a, will position you strongly for both multiple-choice and extended-response questions.
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📚 IB Physics: Core Concepts of Waves and Wave Graph Analysis | IB物理：波动核心概念与波的图像分析
Waves are everywhere in physics, from sound and light to quantum matter waves. Understanding the core concepts of waves and the ability to interpret wave graphs are essential skills for IB Physics students.
A wave is a disturbance that travels through a medium or space, transporting energy and momentum from one point to another without any bulk movement of the medium itself.
For example, if you drop a stone into a still pond, circular ripples spread outward. The water molecules mainly move up and down; the wave itself travels horizontally.
A wavefront is an imaginary line connecting points that are in the same phase; a ray is a line perpendicular to the wavefront and points in the direction of propagation.
波前是连接同相各点的假想线；射线是与波前垂直且指向传播方向的线。
2. Transverse and Longitudinal Waves | 横波与纵波
Waves are classified by the direction of particle oscillation relative to the direction of wave propagation.
波按照质点振动方向与波传播方向的关系进行分类。
Transverse wave: particles oscillate perpendicular to the direction of travel. Examples include waves on a string, electromagnetic waves, and water surface waves.
横波：质点振动方向与传播方向垂直。例如绳波、电磁波和水面波。
Longitudinal wave: particles oscillate parallel to the direction of travel. Examples include sound waves and seismic P-waves.
纵波：质点振动方向与传播方向平行。例如声波和地震 P 波。
In a longitudinal wave, regions of compression and rarefaction travel along the medium.
在纵波中，一系列疏密相间的区域（压缩区与稀疏区）沿介质传播。
3. Key Quantities: Amplitude, Wavelength, Period, Frequency | 关键物理量：振幅、波长、周期、频率
To describe a wave fully we need several quantities that characterise both its size and its rhythm.
为了完整地描述一个波，我们需要几个既能刻画其大小又能刻画其节奏的物理量。
Amplitude (A): the maximum displacement of a particle from its equilibrium position. It is related to the energy carried by the wave.
振幅 (A)：质点偏离平衡位置的最大位移。它与波携带的能量有关。
Wavelength (λ): the distance between two consecutive points in the same phase, e.g. crest to crest or trough to trough.
波长 (λ)：两个相邻同相点之间的距离，例如波峰到波峰或波谷到波谷。
Period (T): the time required for one complete oscillation of a particle, or for the wave to advance by one wavelength.
周期 (T)：质点完成一次全振动所需的时间，或波前进一个波长所需的时间。
Frequency (f): the number of complete oscillations per unit time; f = 1/T, with unit hertz (Hz).
频率 (f)：单位时间内完成全振动的次数；f = 1/T，单位是赫兹 (Hz)。
f = 1/T
The frequency of a wave is determined by the source and does not change when the wave enters a different medium.
波的频率由波源决定，当波进入不同介质时频率不会改变。
4. The Wave Equation: v = fλ | 波速公式：v = fλ
Imagine following a fixed point on the wave, such as a wave crest. In one period T, the crest travels exactly one wavelength λ.
设想跟踪波上的某个固定点，例如一个波峰。在一个周期 T 内，该波峰恰好前进一个波长 λ。
Therefore the wave speed is:
因此波速为：
v = λ/T = fλ
Since f = 1/T, we also write v = fλ.
因为 f = 1/T，所以也可写作 v = fλ。
For example, a sound wave of frequency 440 Hz in air at 20 °C has a speed of about 343 m s⁻¹, so its wavelength is λ = v/f ≈ 0.78 m.
例如，在 20 °C 空气中，频率为 440 Hz 的声波速度约为 343 m s⁻¹，因此其波长 λ = v/f ≈ 0.78 m。
Important: v depends on the medium, while f is fixed by the source. When a wave crosses into a medium with a different speed, λ changes accordingly.
重要：波速取决于介质，而频率由波源决定。当波进入波速不同的介质时，波长会相应地改变。
5. Two Essential Wave Graphs: y–x and y–t | 两种关键波形图：y–x 图与 y–t 图
IB Physics often tests your ability to switch between two types of wave graph.
IB 物理经常考查你在两类波形图之间进行切换的能力。
Displacement–position graph (y–x graph): a “snapshot” of the wave at one instant. The horizontal axis is distance along the wave, and the vertical axis is the displacement of particles from equilibrium.
Displacement–time graph (y–t graph): shows how a single particle (or a single point of the medium) moves over time. The horizontal axis is time, and the vertical axis is displacement.
Many students confuse these two graphs. The key is to remember: the x-axis tells you what is fixed.
许多学生容易混淆这两张图。关键在于记住：横轴告诉你“什么量是固定的”。
6. How to Extract Information from the Graphs | 如何从图像中提取信息
From a y–x graph you can directly read the amplitude A and the wavelength λ. The period T cannot be read unless the wave speed v is given.
从 y–x 图中可以直接读出振幅 A 和波长 λ。除非给出了波速 v，否则无法直接读出周期 T。
From a y–t graph you can directly read the amplitude A and the period T. The wavelength λ cannot be read unless the wave speed v is known.
从 y–t 图中可以直接读出振幅 A 和周期 T。除非已知波速 v，否则无法直接读出波长 λ。
To relate the two graphs, use:
要关联这两张图，需要用到：
v = λ/T
Quantity
From y–x graph
From y–t graph
Amplitude A
Maximum vertical displacement
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📚 IB Physics: Core Formulas and Question Types for Work, Energy and Power | IB物理：功、能量与功率核心公式及题型归纳
Work, energy and power form one of the most frequently tested areas in IB Physics. This topic connects mechanics, circular motion and thermal physics, and it appears in both Paper 1 and Paper 2. In this article, we will summarise the core formulas, clarify common misconceptions, and classify typical exam question types to help you revise efficiently.
Work is done when a force acts on an object and the object moves in the direction of the force. The general equation is:
W = F × s × cos θ
Here, W is work measured in joules (J), F is the applied force in newtons (N), s is the displacement in metres (m), and θ is the angle between the force and displacement vectors.
当力作用在物体上，且物体沿力的方向发生位移时，力就对物体做了功。一般公式为：
W = F × s × cos θ
其中 W 是功，单位焦耳（J）；F 是力，单位牛顿（N）；s 是位移，单位米（m）；θ 是力与位移方向之间的夹角。
Study tip: If the force is parallel to displacement, then θ = 0° and W = Fs. If the force is perpendicular to displacement, then W = 0. For example, the normal force and gravity do no work when an object slides horizontally along a flat surface.
2. Kinetic Energy and the Work–Energy Theorem | 动能与动能定理
Kinetic energy is the energy an object possesses due to its motion. The formula is:
Eₖ = ½ m v²
where m is mass and v is speed. Kinetic energy is a scalar quantity and is always positive or zero.
动能是物体由于运动而具有的能量，公式为：
Eₖ = ½ m v²
其中 m 是质量，v 是速度大小。动能是标量，永远大于或等于零。
The work–energy theorem states that the net work done on an object equals its change in kinetic energy:
W_net = ΔEₖ = ½ m v² − ½ m u²
This theorem is extremely useful for solving problems involving variable forces or curved paths, because it avoids detailed kinematic analysis.
动能定理指出：合外力对物体所做的功等于物体动能的变化量：
W_net = ΔEₖ = ½ m v² − ½ m u²
这个定理在解决变力或曲线运动问题时非常有用，因为可以绕过复杂的运动学分析。
3. Gravitational Potential Energy | 重力势能
Near the Earth’s surface, gravitational potential energy is given by:
Eₚ = m g h
where g is gravitational field strength (about 9.8 N kg⁻¹ on Earth, but IB often uses 10 N kg⁻¹ for simplicity), and h is the height above a chosen reference level.
在地球表面附近，重力势能公式为：
Eₚ = m g h
其中 g 是重力场强度（地球上约9.8 N kg⁻¹，IB考试中常取10 N kg⁻¹），h 是相对于所选参考平面的高度。
Important: The reference level is arbitrary. What matters in calculations is the change in height, Δh, not the absolute value of h. When solving free-fall or pendulum questions, always define your zero level clearly at the start.
For an ideal spring obeying Hooke’s law, the elastic potential energy stored is:
Eₑ = ½ k x²
where k is the spring constant and x is the extension or compression from the natural length.
对于遵循胡克定律的理想弹簧，储存的弹性势能为：
Eₑ = ½ k x²
其中 k 是劲度系数，x 是相对原长的伸长量或压缩量。
In exam questions, remember that the force from a spring is F = kx, but the energy stored is ½kx². The factor of ½ appears because the force increases linearly from zero to kx during the stretching process.
Power is the rate at which energy is transferred or work is done. The average power is:
P = W / t = ΔE / t
where P is measured in watts (W = J s⁻¹).
功率是能量转移或做功的快慢。平均功率为：
P = W / t = ΔE / t
其中 P 的单位是瓦特（W = J s⁻¹）。
When a constant force acts on an object moving at constant velocity, instantaneous power can also be calculated as:
P = F × v
This form is commonly used for vehicle problems, where engine power is constant and the driving force decreases as speed increases.
当恒力作用于匀速运动的物体时，瞬时功率还可以表示为：
P = F × v
这个形式常用于交通工具问题：当发动机功率恒定时，速度增大则牵引力减小。
6. Efficiency in Energy Transfers | 能量转化中的效率
Efficiency is the ratio of useful output energy (or power) to total input energy (or power):
η = (useful output energy / total input energy) × 100%
or equivalently using power: η = P_out / P_in × 100%.
效率是有用输出能量（或功率）与总输入能量（或功率）之比：
η = (有用输出能量 / 总输入能量) × 100%
也可以用功率表示为：η = P_out / P_in × 100%。
In IB questions, efficiency is often combined with energy conservation. For example, an electric motor may transfer 80% of electrical energy into kinetic energy and the rest into thermal energy. Make sure you identify which energy is ‘useful’ in the context of the question.
When only conservative forces (such as gravity and ideal spring forces) do work, the total mechanical energy remains constant:
Eₖ + Eₚ = constant
This is a powerful tool for solving roller-coaster, pendulum, projectile and spring problems.
当只有保守力（如重力、理想弹簧弹力）做功时，系统总机械能保持不变：
Eₖ + Eₚ = constant
这是解决过山车、单摆、抛体和弹簧问题的重要工具。
If non-conservative forces (like friction or air resistance) are present, mechanical energy is not conserved. In such cases, use:
E_initial + W_non-conservative = E_final
where the work done by friction is negative.
如果存在非保守力（如摩擦力或空气阻力），机械能不守恒。此时应使用：
E_initial + W_non-conservative = E_final
其中摩擦力做功为负值。
8. Common Question Type 1: Work Done by a Constant Force | 题型一：恒力做功
In this type, you are given a force, a displacement and an angle. The key is to correctly resolve the force component along the displacement.
此类题目会给出力、位移和夹角。关键在于正确分解力在位移方向上的分量。
Example: A child pulls a sled with a force of 50 N at an angle of 30° above the horizontal for 20 m. Work done = 50 × 20 × cos 30° = 866 J. Note that the vertical component of the force does no work because displacement is horizontal.
9. Common Question Type 2: Energy Conservation in Vertical Motion | 题型二：竖直运动中的能量守恒
These problems often involve a mass sliding down a frictionless incline or falling from a height. You can equate the initial potential energy to the final kinetic energy.
这类问题通常涉及物体沿光滑斜面下滑或从高处落下。可将初始势能等于末态动能。
Sample approach: A ball of mass 2 kg is dropped from a height of 5 m. Just before hitting the ground, v = √(2gh) = √(2 × 9.8 × 5) = 9.9 m s⁻¹. You can also use kinematics to check, but energy is often faster.
典型思路：一个2 kg的小球从5 m高处自由下落。落地前瞬间速度 v = √(2gh) = √(2 × 9.8 × 5) ≈ 9.9 m s⁻¹。也可以用运动学公式验证，但能量法通常更快。
10. Common Question Type 3: Power and Variable Force | 题型三：功率与变力问题
Car and motor problems typically give a constant power output P, and you must find the maximum speed when the driving force equals the resistive force.
汽车或电动机问题通常给定恒定输出功率P，需要求当牵引力等于阻力时的最大速度。
Example: A car of mass 1000 kg experiences a constant resistive force of 400 N. If the engine delivers 20 kW, the maximum speed is v = P / F = 20000 / 400 = 50 m s⁻¹. At speeds lower than this, the excess power accelerates the car.
示例：一辆1000 kg的汽车受到恒定的400 N阻力。若发动机输出20 kW，则最大速度 v = P / F = 20000 / 400 = 50 m s⁻¹。当车速低于此值时，多余功率用于加速。
11. Common Question Type 4: Work Done by Friction and Thermal Energy | 题型四：摩擦力做功与热能
When an object slides along a rough surface, the work done against friction is converted into thermal energy. This is a common Paper 2 extended-response context.
物体沿粗糙表面滑动时，克服摩擦力做的功转化为热能。这是Paper 2简答题的常见情境。
Friction = μ × N, where μ is the coefficient of kinetic friction and N is the normal force.
摩擦力 f = μ × N，其中μ为动摩擦因数，N为正压力。
Work done by friction = f × s × cos 180° = −f × s.
摩擦力做功 = f × s × cos 180° = −f × s。
The thermal energy generated = f × s (positive value).
产生的热能等于f × s（取正值）。
For example, a block slides 3 m on a surface with μ = 0.2, mass 5 kg. The thermal energy produced = 0.2 × 5 × 9.8 × 3 = 29.4 J.
12. Common Question Type 5: Graphs and Area Interpretation | 题型五：图像与面积含义
In IB Physics, force–displacement graphs are used to represent work. The area under a force–displacement graph equals the work done by the force.
在IB物理中，力-位移图像用于表示做功。力-位移图线下方的面积等于力所做的功。
Similarly, the area under a power–time graph represents energy transferred. You should be comfortable calculating areas for rectangles, triangles and trapeziums.
同理，功率-时间图线下方的面积表示转移的能量。你需要熟练计算矩形、三角形和梯形的面积。
Another common graph is the potential energy versus position curve. The slope of an Eₚ–x graph gives the negative of the force: F = −dEₚ/dx. This is an AIHL/AAHL friendly concept but also appears in SL energy contexts.
13. Misconceptions and Quick Exam Tips | 常见误区与快速应试技巧
Work is a scalar, not a vector. Do not assign a direction to work.
功是标量，不是矢量。不要给功指明方向。
Kinetic energy depends on speed squared, so doubling speed quadruples kinetic energy.
动能与速度的平方成正比，速度加倍时动能变为原来的四倍。
In energy conservation problems, define the zero potential energy level before solving.
在能量守恒问题中，先定义零势能参考面再解题。
For cyclic processes like a ball returning to its starting height, the net work done by gravity over the complete cycle is zero.
对于小球回到起始高度的循环过程，重力在整个循环中所做的总功为零。
Do not confuse W = Fs with P = Fv. The first requires displacement; the second requires velocity.
不要混淆W = Fs和P = Fv。前者需要位移，后者需要速度。
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📚 IB Physics: Methods for Evaluating Uncertainties and Errors | IB 物理：不确定度与误差的评估方法
Measurement is central to physics, but no measurement is perfect. A value such as 5.02 g is incomplete unless we know how reliable it is: 5.02 ± 0.01 g. In IB Physics, evaluating uncertainties and errors is not an optional extra; it is part of the language used to communicate experimental results honestly.
测量是物理学的核心，但没有任何测量是完美的。像 5.02 g 这样的数值，如果不说明其可信程度，就是不完整的：应写成 5.02 ± 0.01 g。在 IB 物理中，评估不确定度和误差并非可有可无的步骤，而是诚实表达实验结果所必需的语言。
1. Why Uncertainties Matter | 为什么不确定度很重要
A result without an uncertainty cannot be compared with another result or with a theoretical prediction. If one experiment gives 9.81 m s⁻² and another gives 9.79 m s⁻², we cannot decide whether they agree unless each is accompanied by an uncertainty.
没有不确定度的结果无法与另一个结果或理论预测进行比较。如果一个实验得到 9.81 m s⁻²，另一个实验得到 9.79 m s⁻²，在没有各自不确定度的情况下，我们无法判断它们是否一致。
Uncertainty also tells us how much confidence we should place in a conclusion. It helps scientists decide whether a result is consistent with a law, or whether an unexpected outcome needs further investigation.
Error is the difference between a measured value and the true value. The true value may never be known exactly, so the actual error is often unknown.
误差 是测量值与真实值之间的差异。真实值可能永远无法精确得知，因此实际误差通常是未知的。
Uncertainty is a range of values within which the true value is likely to lie. It is not a mistake; it is a quantified estimate of doubt. For example, a length of 12.0 ± 0.2 cm suggests the true length is probably between 11.8 cm and 12.2 cm.
不确定度 是真实值可能落在的一个数值范围。它不是错误，而是对怀疑程度的量化估计。例如，长度 12.0 ± 0.2 cm 意味着真实长度可能在 11.8 cm 和 12.2 cm 之间。
3. Random and Systematic Errors | 随机误差与系统误差
Random errors cause readings to scatter above and below the true value. They may be caused by human reaction time, slight changes in conditions, or electrical noise. Random errors reduce precision and can be reduced by repeating measurements and averaging.
Systematic errors push all readings in the same direction. They are caused by zero errors, poorly calibrated instruments, or incorrect technique. Averaging cannot remove systematic errors, and they reduce accuracy.
4. Absolute, Relative and Percentage Uncertainty | 绝对、相对与百分比不确定度
The absolute uncertainty is recorded with the measured value and has the same unit. If a resistance is measured as 24.6 ± 0.4 Ω, then 0.4 Ω is the absolute uncertainty.
The relative uncertainty compares the uncertainty with the measured value. The percentage uncertainty is the relative uncertainty multiplied by 100 %:
相对不确定度将不确定度与测量值进行比较。百分比不确定度是相对不确定度乘以 100 %：
relative uncertainty = Δx / x
percentage uncertainty = (Δx / x) × 100 %
For example, if x = 4.8 ± 0.2 m, the relative uncertainty is 0.2 / 4.8 = 0.042, and the percentage uncertainty is about
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📚 IB Physics: Core Concepts of Kinematics and Graph Analysis Methods | IB物理：运动学核心概念与图像分析方法
Kinematics is the foundation of IB Physics. It describes motion without considering its causes, and mastering it is essential for understanding mechanics, energy, and even fields. This article covers the core definitions, equations, and graph interpretation skills you need for your IB exams.
In kinematics, every physical quantity is either a scalar or a vector. A scalar has magnitude only, while a vector has both magnitude and direction. Distance and speed are scalars; displacement and velocity are vectors.
Distance / 路程: total length of path travelled, always positive. / 物体运动轨迹的总长度，始终为正。
Displacement / 位移: straight-line distance from start to finish with direction. / 从起点到终点的直线距离，并带有方向。
Speed / 速率: distance per unit time, scalar. / 单位时间内通过的路程，标量。
Velocity / 速度: displacement per unit time, vector. / 单位时间内的位移，矢量。
Average speed = total distance ÷ total time
平均速率 = 总路程 ÷ 总时间
Average velocity = displacement ÷ time = Δs / Δt
平均速度 = 位移 ÷ 时间 = Δs / Δt
A common exam trap is confusing distance and displacement when an object changes direction. For example, if a ball rolls 3 m east then 4 m west, the distance is 7 m but the displacement is only 1 m west.
2. Instantaneous Velocity and Acceleration | 瞬时速度与加速度
Instantaneous velocity is the velocity of an object at a specific moment in time. It is found by taking the limit of average velocity as the time interval approaches zero. Acceleration is the rate of change of velocity.
Here, u is initial velocity and v is final velocity. Acceleration is a vector. Negative acceleration does not always mean slowing down; it depends on the direction of the velocity. If velocity and acceleration point in opposite directions, the object slows down. If they point in the same direction, the object speeds up.
Deceleration occurs when v and a have opposite signs.
当v与a符号相反时，物体做减速运动。
3. The SUVAT Equations | 运动学四大公式（SUVAT）
For motion with constant acceleration in a straight line, IB Physics requires you to use the four SUVAT equations. The variables are: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time.
These equations only apply when acceleration is constant. In free-fall near Earth’s surface, a = g = 9.8 m s⁻² downward. Always choose a positive direction and keep signs consistent.
这些公式仅适用于加速度恒定的情况。在地球表面附近的自由落体中，a = g = 9.8 m s⁻²，方向向下。务必选定正方向并保持符号一致。
Suvat variable
Meaning
SI unit
s
displacement / 位移
m
u
initial velocity / 初速度
m s⁻¹
v
final velocity / 末速度
m s⁻¹
a
acceleration / 加速度
m s⁻²
t
time / 时间
s
When using SUVAT equations, always list what you know and what you need. Choose the equation that contains all known variables and only one unknown.
使用SUVAT公式时，先列出已知量和待求量，然后选择包含所有已知量且只有一个未知量的公式。
4. Free Fall and Projectile Motion Basics | 自由落体与抛体运动基础
Free fall is motion under the influence of gravity only. Air resistance is neglected in IB standard-level analysis. The acceleration is always 9.8 m s⁻² downward, regardless of the object’s mass.
自由落体是仅受重力作用的运动。在IB标准分析中忽略空气阻力。无论物体质量如何，加速度始终为9.8 m s⁻²向下。
For projectile motion, treat horizontal and vertical motion independently. The horizontal velocity is constant because there is no horizontal acceleration. The vertical motion follows the SUVAT equations with a = −g.
The time of flight is determined by the vertical motion only. The range is the horizontal distance travelled during that time.
飞行时间仅由竖直运动决定。射程是在这段时间内水平方向运动的距离。
5. Reading Displacement-Time Graphs | 位移-时间图像解读
A displacement-time graph shows how position changes over time. The slope of an s-t graph gives the velocity. A straight line means constant velocity; a curved line means changing velocity, i.e., acceleration.
Positive slope → moving in positive direction / 斜率为正 → 向正方向运动
Negative slope → moving in negative direction / 斜率为负 → 向负方向运动
Increasing slope → speeding up / 斜率增大 → 加速
Decreasing slope → slowing down / 斜率减小 → 减速
To find instantaneous velocity from an s-t graph, draw a tangent line at the point of interest and calculate its gradient. The steepness of the tangent tells you the speed at that instant.
要从s-t图像中求瞬时速度，应在感兴趣的点处画切线并计算其斜率。切线的陡峭程度表示该时刻的速率。
6. Reading Velocity-Time Graphs | 速度-时间图像解读
Velocity-time graphs are among the most important in IB Physics. The slope gives acceleration, and the area under the graph gives displacement.
速度-时间图像是IB物理中最重要的图像之一。斜率表示加速度，图像下方的面积表示位移。
Slope of v-t graph = acceleration
v-t图像的斜率 = 加速度
Area under v-t graph = displacement
v-t图像下方的面积 = 位移
Area above the time axis represents positive displacement. Area below the time axis represents negative displacement. The net displacement is the difference between the two areas.
时间轴上方的面积表示正位移，时间轴下方的面积表示负位移。净位移等于两者之差。
Feature of v-t graph
Physical meaning
Horizontal line / 水平线
Constant velocity, a = 0 / 匀速运动，a = 0
Straight diagonal line / 倾斜直线
Constant acceleration / 匀变速运动
Curved line / 曲线
Changing acceleration / 变加速运动
Line crosses time axis / 穿过时间轴
Direction reversal / 运动方向反转
To find displacement from a curved v-t graph, you may need to count squares or use integration. In IB exams, you will often be given simple shapes such as triangles, rectangles, and trapezoids.
An acceleration-time graph shows how acceleration changes over time. The area under an a-t graph gives the change in velocity, Δv.
加速度-时间图像显示加速度随时间的变化。a-t图像下方的面积表示速度的变化量Δv。
Area under a-t graph = change in velocity = v − u
a-t图像下方的面积 = 速度变化量 = v − u
A horizontal line at a positive value means constant positive acceleration. A horizontal line at zero means constant velocity. The slope of an a-t graph has no direct physical meaning in standard IB kinematics, so do not calculate it unless asked.
IB exam questions often ask you to sketch one graph given another. For example, given a v-t graph, you can determine the a-t graph by taking the slope at every point, and determine the s-t graph by considering the area.
When converting, pay attention to the initial conditions. The initial displacement and initial velocity are needed to determine the integration constants.
转换时注意初始条件。初始位移和初始速度用于确定积分常数。
9. Common Exam Errors and Tips | 常见考试错误与技巧
Many students lose marks in kinematics questions because of simple sign errors or unit mistakes. Always define the positive direction before solving a problem, and always check whether the acceleration is constant before using SUVAT equations.
Tip 1 / 技巧1: Read whether the question asks for distance or displacement. They are different quantities. / 仔细阅读题目问的是路程还是位移，两者不同。
Tip 2 / 技巧2: At the highest point of projectile motion, vertical velocity is 0, but vertical acceleration is still g. / 在抛体运动的最高点，竖直速度为0，但竖直加速度仍为g。
Tip 3 / 技巧3: On a velocity-time graph, a line below the time axis means the object is moving in the negative direction, not necessarily slowing down. / 在速度-时间图像中，时间轴下方的线表示物体向负方向运动，并不一定在减速。
Tip 4 / 技巧4: When using v = u + at, make sure all quantities are in SI units: metres, seconds, metres per second, metres per second squared. / 使用v = u + at时，确保所有量均为SI单位：米、秒、米每秒、米每二次方秒。
Tip 5 / 技巧5: Always sketch a small diagram for motion problems. It clarifies directions and helps avoid confusion. / 运动学问题一定要画示意图，它有助于明确方向，避免混淆。
10. Worked Example | 例题精讲
A ball is thrown vertically upward with an initial speed of 20 m s⁻¹ from the ground. Take g = 10 m s⁻². Find: (a) the maximum height reached; (b) the total time of flight; (c) the velocity just before hitting the ground.
一个球从地面以20 m s⁻¹的初速度竖直上抛。取g = 10 m s⁻²。求：(a)能达到的最大高度；(b)总飞行时间；(c)落地前瞬间的速度。
Solution / 解答:
Choose upward as positive. At the maximum height, v = 0.
取向上为正方向。在最大高度处，v = 0。
(a) Using v² = u² + 2as:
(a) 使用v² = u² + 2as：
0² = 20² + 2(−10)s → s = 20 m
0² = 20² + 2(−10)s → s = 20 m
(b) Using v = u + at for the upward journey:
(b) 上升阶段使用v = u + at：
0 = 20 + (−10)t → t = 2 s
0 = 20 + (−10)t → t = 2 s
Total time of flight = 2 × 2 = 4 s. The downward journey takes the same time as the upward journey because the motion is symmetric.
总飞行时间 = 2 × 2 = 4 s。由于运动是对称的，下落阶段与上升阶段所用时间相同。
(c) Using v = u + at for the whole journey:
(c) 全程使用v = u + at：
v = 20 + (−10)(4) = −20 m s⁻¹
v = 20 + (−10)(4) = −20 m s⁻¹
The negative sign shows the ball is moving downward just before impact, with the same speed as the initial speed.
负号表示球在落地前瞬间向下运动，速度大小与初速度相同。
11. Graphical Analysis for a Bouncing Ball | 弹跳球的图像分析
Bouncing ball questions are classic IB kinematics problems. When a ball bounces, its velocity changes sign suddenly at impact. On a v-t graph, this appears as a sharp vertical jump. On an s-t graph, the slope changes sign continuously because the position curve is smooth.
During free flight between bounces, the acceleration is constant at −g, so the v-t graph has the same slope for all segments. The maximum height decreases after each bounce if the ball is not perfectly elastic.
Kinematics is a highly visual topic in IB Physics. You must be able to interpret displacement-time, velocity-time, and acceleration-time graphs fluently, and switch between them. Mastering the SUVAT equations, understanding vector direction, and avoiding common sign errors will boost your exam performance significantly.
Remember that every graph tells a story. Read the story carefully, and the equations will follow naturally. Practice converting between graphs and solving multi-stage motion problems; these skills appear frequently in both paper 1 and paper 2 of IB Physics.
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📚 Ideal Gas Model & Gas Laws Explained | IB物理：理想气体模型与气体定律详解
The ideal gas model is one of the most powerful simplifications in physics. It allows us to describe the behaviour of gases using just a few macroscopic variables: pressure, volume, temperature, and the number of particles. For IB Physics students, mastering this model is essential not only for exam success but also for building a foundation in thermodynamics and statistical mechanics.
The ideal gas model is built on five key assumptions about the microscopic behaviour of gas molecules. These assumptions simplify the mathematics while capturing the most important features of real gases at low pressure and high temperature.
Gas contains a very large number of identical particles (molecules or atoms) in constant random motion.
气体包含大量相同的粒子（分子或原子），它们处于持续的无规则运动中。
The volume of the individual gas molecules is negligible compared to the volume of the container.
与容器的体积相比，单个气体分子的体积可以忽略不计。
Intermolecular forces are negligible except during brief collisions.
除短暂碰撞外，分子间的作用力可以忽略不计。
Collisions between molecules and with container walls are perfectly elastic.
分子之间及分子与容器壁之间的碰撞是完全弹性的。
The average kinetic energy of the molecules is proportional to the absolute temperature.
分子的平均动能与绝对温度成正比。
2. Pressure and the Root-Mean-Square Speed | 压强与根均方速率
Pressure arises from the countless collisions of gas molecules with the walls of the container. Consider a molecule of mass m moving in the x-direction with speed vₓ. When it collides elastically with a wall, its momentum changes from +mvₓ to -mvₓ, giving a change of 2mvₓ.
Here, N is the total number of molecules, m is the mass of each molecule, and ⟨v²⟩ is the mean square speed. The square root of ⟨v²⟩ is called the root-mean-square speed, v_rms. This equation connects the macroscopic pressure with the microscopic motion of molecules.
Boyle’s law states that for a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume. This means that if you compress a gas to half its volume, the pressure doubles, provided the temperature does not change.
A p-V graph for an isothermal process yields a hyperbola. On a p-V diagram, each curve corresponds to a different fixed temperature, with higher temperatures producing curves that lie further from the origin.
4. Charles’s Law: Volume and Temperature | 查理定律：体积与温度
Charles’s law states that for a fixed mass of gas at constant pressure, the volume is directly proportional to the absolute temperature. This relationship explains why a hot-air balloon expands when heated — the gas molecules move faster and push the walls outward.
It is crucial to use the Kelvin scale in all gas law calculations. The Celsius scale cannot be used directly because it is offset relative to absolute zero. A temperature of 0 °C corresponds to 273 K, and 0 K (-273 °C) is the absolute minimum possible temperature.
5. Gay-Lussac’s Law: Pressure and Temperature | 盖-吕萨克定律：压强与温度
Gay-Lussac’s law, also called the pressure law, states that for a fixed mass of gas at constant volume, the pressure is directly proportional to the absolute temperature. When a sealed container of gas is heated, the molecules gain kinetic energy, strike the walls more frequently and harder, and the pressure rises.
If the p-T graph is extrapolated backwards, it passes through the origin of the Kelvin scale. This provides strong experimental evidence for the existence of absolute zero.
如果将p-T图像反向延长，它会通过开尔文温标的原点。这为绝对零度的存在提供了有力的实验证据。
6. The Combined Gas Law | 综合气体定律
The three individual gas laws can be combined into a single equation that relates all three variables simultaneously. This is particularly useful for problems where a gas changes from an initial state (p₁, V₁, T₁) to a final state (p₂, V₂, T₂).
This equation allows you to solve problems where all three variables change simultaneously. Always check the units: pressure must be in a consistent unit on both sides, and temperature must always be in kelvin.
The ideal gas equation combines Boyle’s law, Charles’s law, and Avogadro’s principle into one elegant relationship. It connects pressure p, volume V, the amount of gas n (in moles), and absolute temperature T.
Here, R is the molar gas constant with a value of 8.31 J mol⁻¹ K⁻¹. This equation only applies to ideal gases, which means real gases that are at low pressure and high temperature relative to their critical point.
Typical exam questions involve finding the number of moles from a known volume, pressure, and temperature. For example, calculate the number of moles in 0.025 m³ of gas at 100 kPa and 300 K. Using pV = nRT, we get n = pV/(RT) = (100 × 10³ × 0.025)/(8.31 × 300) ≈ 1.00 mol.
The ideal gas equation can also be written in terms of the total number of molecules N, rather than the number of moles n. This form is particularly useful in explaining the microscopic meaning of temperature.
理想气体方程也可以用分子总数N来表示，而不是物质的量n。这种形式在解释温度的微观意义时特别有用。
pV = NkT
Here, k is Boltzmann’s constant, equal to 1.38 × 10⁻²³ J K⁻¹. The relationship between R and k is R = N_A × k, where N_A = 6.02 × 10²³ mol⁻¹ is Avogadro’s number. Since n = N/N_A, substituting gives pV = (N/N_A) × (N_A k) × T = NkT.
9. Temperature and Mean Molecular Kinetic Energy | 温度与分子平均动能
Combining the kinetic theory result pV = (1/3)Nm⟨v²⟩ with the ideal gas equation pV = NkT, we can derive a profound connection between temperature and molecular motion.
This equation reveals that the average translational kinetic energy of a gas molecule depends only on the absolute temperature, not on the type of gas. At the same temperature, a light molecule like helium moves faster than a heavy molecule like oxygen, but both have the same average kinetic energy.
From this, we can also derive an expression for the root-mean-square speed: v_rms = √(3kT/m). For oxygen at 300 K, with molecular mass m = 32 × 1.66 × 10⁻²⁷ kg, the RMS speed is approximately 483 m/s.
10. Isothermal and Adiabatic Processes | 等温过程与绝热过程
In IB Physics, you are expected to distinguish between two important types of processes on a p-V diagram. An isothermal process occurs at constant temperature, so the ideal gas equation becomes pV = constant, giving a hyperbola on the p-V diagram.
An adiabatic process, by contrast, occurs without any heat exchange with the surroundings (Q = 0). In this case, all the work done on the gas changes its internal energy. For an adiabatic process, pV^γ = constant, where γ = C_p/C_v is the ratio of specific heat capacities. For a monatomic ideal gas, γ = 5/3.
On a p-V diagram, the adiabatic curve is steeper than the isothermal curve passing through the same point. A common IB exam question asks you to identify which curve represents which process.
When a gas expands, it does work on its surroundings. The work done is equal to the area under the curve on a p-V diagram. This is because incremental work is given by dW = p dV.
当气体膨胀时，它对外界做功。所做的功等于p-V图上曲线下方的面积。这是因为微元功的表达式为dW = p dV。
W = ∫ p dV
For an isothermal expansion of an ideal gas from volume V₁ to V₂, the work done is W = nRT ln(V₂/V₁). For a constant-pressure (isobaric) process, the work simply becomes W = pΔV. Understanding this area interpretation is crucial for solving IB problems that ask you to calculate work from a p-V graph.
12. Deviations from the Ideal Gas Model | 理想气体模型的偏差
Real gases deviate from ideal behaviour under certain conditions. At high pressures, the volume of the molecules themselves becomes significant compared to the container volume, and the ideal gas equation underestimates the pressure. At low temperatures, intermolecular attractive forces become important, causing the gas to compress more easily than predicted.
As a general rule, the ideal gas model works best when the gas is at low pressure and high temperature — far from condensation. IB questions may ask you to explain these deviations using the kinetic theory assumptions, so be ready to link the breakdown of assumptions to the breakdown of the model.
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📚 IB Physics: Core Concepts and Knowledge Framework of Thermal Physics | IB物理：热物理核心概念与知识框架
Thermal physics in the IB Physics syllabus (both SL and HL) forms a compact but highly examinable topic. It connects macroscopic measurements such as temperature and pressure with microscopic ideas about molecular motion and energy transfer. Mastering this framework requires a clear understanding of definitions, sign conventions, energy conservation, and the kinetic model of gases.
1. Temperature, Heat and Internal Energy | 温度、热量与内能
Temperature is a macroscopic measure of the average random kinetic energy of particles in a substance. It is measured in Kelvin (K) in the SI system, and the Kelvin scale is an absolute scale starting at absolute zero (0 K). A change of 1 K is identical to a change of 1 °C, but the zero points of the two scales differ: T(K) = T(°C) + 273.15.
Heat is defined as thermal energy transferred between two systems because of a temperature difference. It is a process-dependent quantity, not a property of a system. Internal energy U is the sum of the total potential energy and the total random kinetic energy of all particles within a system. For an ideal gas, the potential energy component is assumed to be zero, so the internal energy depends only on temperature.
Specific heat capacity c is the energy required to raise the temperature of 1 kg of a substance by 1 K. The defining equation is:
比热容c是指使1 kg物质温度升高1 K所需的热量。其定义方程为：
E = mcΔT
where E is the thermal energy supplied or removed, m is the mass, c is the specific heat capacity, and ΔT is the temperature change. Common units are J kg⁻¹ K⁻¹. In IB questions, you must identify whether energy is being added to or removed from the system, and the equation works with the sign of ΔT.
When two substances at different temperatures are mixed, the principle of conservation of energy implies that the energy lost by the hotter substance equals the energy gained by the cooler substance, assuming no heat loss to the surroundings. This forms the basis of calorimetry problems.
During a phase change, temperature remains constant while energy is transferred. The energy required to change the phase of 1 kg of a substance without changing its temperature is called the specific latent heat L. The equation is:
There are two important values: the specific latent heat of fusion Lf, for solid-liquid transitions, and the specific latent heat of vaporisation Lv, for liquid-gas transitions. For a given substance, Lv is usually much larger than Lf because the separation of particles against intermolecular forces during vaporisation requires far more energy than breaking the rigid lattice structure during melting.
When a heating curve for ice from below 0 °C to steam above 100 °C is plotted, the graph shows sloping sections (temperature rises) and horizontal plateaus (phase changes). IB students should be able to calculate the energy for each section separately and sum them to find the total energy.
The kinetic model explains the macroscopic properties of solids, liquids and gases in terms of the motion and arrangement of particles. In a solid, particles vibrate about fixed positions in a regular lattice. In a liquid, particles are close together but can move past each other. In a gas, particles are far apart, move rapidly and randomly, and collisions are elastic.
Temperature is related to the average kinetic energy of particles: a higher temperature means a greater average random kinetic energy. Evaporation occurs when the most energetic particles at the liquid surface escape into the gas phase, which reduces the average kinetic energy of the remaining liquid, so evaporation causes cooling.
The ideal gas model is a simplified theoretical model used to describe the behaviour of gases under most ordinary conditions. The assumptions of the model include: the gas consists of a very large number of identical particles; the volume of the particles is negligible compared to the volume of the container; there are no intermolecular forces except during collisions; all collisions are perfectly elastic; and the duration of a collision is negligible compared to the time between collisions.
For an ideal gas, the internal energy is solely the total kinetic energy of the particles. Therefore, if the temperature of an ideal gas is unchanged, its internal energy is unchanged. This is a crucial idea in the first law of thermodynamics applied to isothermal processes.
6. The Equation of State for an Ideal Gas | 理想气体状态方程
The macroscopic behaviour of an ideal gas is described by the equation of state:
理想气体的宏观行为由状态方程描述：
PV = nRT = NkBT
Here P is pressure in pascals, V is volume in cubic metres, n is the amount of gas in moles, R = 8.31 J mol⁻¹ K⁻¹ is the molar gas constant, N is the number of particles, kB = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant, and T is the absolute temperature in kelvin.
In IB problems, a common approach is to compare two states of the same gas using the relation P₁V₁/T₁ = P₂V₂/T₂, provided the amount of gas is fixed. Always convert temperatures to kelvin before substitution, and ensure consistent units for pressure and volume on both sides.
From the kinetic model, the average translational kinetic energy of a single gas molecule is directly proportional to the absolute temperature. The IB equation is:
根据分子动理论，单个气体分子的平均平动动能与绝对温度成正比。IB使用的方程为：
⟨Ek⟩ = (3/2)kBT
For one mole of gas, the total kinetic energy is (3/2)RT. This result shows that at the same temperature, all ideal gases have the same average molecular kinetic energy, regardless of the mass of the molecules. However, lighter molecules move faster on average than heavier molecules at the same temperature.
Pressure arises from the impact of gas molecules on the container walls. Faster molecules collide more frequently and with greater momentum change, producing higher pressure. This microscopic explanation links the macroscopic quantity P with molecular speed and number density.
The first law of thermodynamics is essentially the law of conservation of energy applied to a thermodynamic system. The IB form is:
热力学第一定律本质上是能量守恒定律在热力学系统中的应用。IB使用的形式为：
ΔU = Q + W
where ΔU is the change in internal energy, Q is the thermal energy transferred to the system, and W is the work done on the system. The sign convention is crucial: Q is positive when heat enters the system, and W is positive when work is done on the system by the surroundings. If the system does work on the surroundings, W is negative.
For an expanding gas, the gas does positive work on the surroundings, so W in the equation is negative. For a compressed gas, the surroundings do work on the gas, so W is positive. In an isothermal process, ΔU = 0 for an ideal gas, so Q = -W. In an adiabatic process, Q = 0, so ΔU = W.
9. Thermodynamic Processes and the p-V Diagram | 热力学过程与p-V图
Thermodynamic processes are often represented on a pressure-volume (p-V) diagram. Four special processes are emphasised in IB Physics: isochoric (constant volume), isobaric (constant pressure), isothermal (constant temperature) and adiabatic (no heat transfer).
The work done by a gas during a volume change is equal to the area under the curve on a p-V diagram. The sign convention is important: if the volume increases, the gas does work on the surroundings; if the volume decreases, work is done on the gas. A cyclic process forms a closed loop on the p-V diagram, and the net work done is the area enclosed by the loop.
10. Internal Energy and the First Law in Problem Solving | 内能与第一定律的解题应用
When solving IB thermal physics problems, a systematic approach reduces errors. First, identify the system and write down the known quantities. Then determine which process is involved and which quantities remain constant. Apply the ideal gas law to find missing state variables, and only then apply the first law to calculate ΔU, Q or W.
Common mistakes include forgetting to convert °C to K, using the wrong sign for W, and calculating work as zero for isothermal processes. Remember that in an isothermal expansion of an ideal gas, the temperature and therefore ΔU are zero, so the heat absorbed equals the work done by the gas. In a free expansion into a vacuum, no work is done and no heat is exchanged, so ΔU = 0 and temperature remains constant.
11. Thermal Radiation and Energy Balance | 热辐射与能量平衡
Thermal radiation is energy transferred by electromagnetic waves and does not require a medium. All objects emit radiation, and the rate of emission increases rapidly with temperature. In IB Physics, the Stefan-Boltzmann law is given as:
where P is the power radiated, e is the emissivity (0 to 1), σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature. A black body has emissivity e = 1 and absorbs all incident radiation.
In the energy balance of a planet or a body in space, the power absorbed from a star must equal the power radiated away when the temperature is steady. This leads to equations that can be used to estimate surface temperatures. The greenhouse effect can be modelled by considering the absorption and re-radiation of infrared radiation by atmospheric gases.
12. Common Exam Traps and Revision Strategy | 常见考试陷阱与复习策略
Many IB students lose marks on thermal physics because of small but repeated errors. The most frequent traps are listed below.
许多IB学生在热物理上丢分，原因往往是微小但反复出现的错误。最常见的陷阱如下。
Using degrees Celsius instead of kelvin in ideal gas calculations and in radiation equations: always convert to kelvin.
应用气体方程和辐射方程时使用摄氏度而非开尔文：务必转换为开尔文。
Confusing heat and temperature: heat is energy in transit; temperature measures average kinetic energy.
混淆热量与温度：热量是传递中的能量；温度是平均动能的量度。
Applying E = mcΔT during a phase change: use E = mL instead.
在物态变化过程中使用E = mcΔT：此时应使用E = mL。
Mixing up the sign convention of W in the first law: W is work done on the system in the IB equation.
混淆第一定律中W的符号约定：在IB方程中，W是外界对系统做的功。
Assuming temperature always increases when heat is added: during boiling or melting it remains constant.
假设加入热量时温度一定升高：在沸腾或熔化过程中温度保持不变。
For revision, draw summary tables of the four thermodynamic processes, practise converting between pressure units (Pa, kPa, atm) and volume units (m³, L, cm³), and review past paper questions that combine the ideal gas law with the first law. Understanding the physical meaning behind each equation is more valuable than memorising formulas.
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📚 IB Physics: Core Concepts of Force Analysis and Newton’s Laws of Motion | IB物理：受力分析与牛顿运动定律核心要点
In IB Physics, understanding force analysis and Newton’s laws of motion is foundational for solving mechanics problems. This article consolidates the essential points, including free-body diagrams, the three laws, common force types, and practical problem-solving strategies, to help you master this core topic.
Force analysis is the process of identifying all forces acting on a body and representing them in a vector diagram. It is the first step in applying Newton’s laws to a physical situation.
受力分析是确定作用在物体上的所有力并用矢量图表示的过程。这是将牛顿定律应用于物理情境的第一步。
When performing force analysis, you must consider both contact forces (e.g., normal force, friction, tension) and non-contact forces (e.g., gravitational force, electromagnetic force). The object of interest should be isolated; only forces acting on that object are included, not forces it exerts on others.
A free-body diagram (FBD) is a simplified drawing showing the object as a point particle, with all external forces drawn as arrows originating from the centre. The length of each arrow should be proportional to the magnitude of the force, and the direction must be accurate.
Isolate the object and draw it as a dot or a simple shape. | 隔离物体，将其画成一个点或一个简单的形状。
Identify all forces acting on the object. | 找出所有作用在物体上的力。
Draw arrows for each force, with labels (e.g., Fg, FN, FT, Ff). | 为每个力画箭头并标注（如 Fg、FN、FT、Ff）。
Choose a coordinate system (usually x-y or along-incline-perpendicular-to-incline). | 选择坐标系（通常为 x-y 或沿斜面-垂直斜面）。
Resolve forces into components if they are not aligned with the axes. | 如果力不与坐标轴对齐，将其分解为分量。
Common mistake: Including forces that the object exerts on other objects, or omitting forces like air resistance when it is significant. Always ask: “What is touching the object?” and “What is acting at a distance?”
Newton’s first law states that an object remains at rest or in uniform motion in a straight line unless acted upon by a net external force. This is also known as the law of inertia.
牛顿第一定律指出，除非受到合外力的作用，否则物体保持静止或沿直线做匀速运动。这也被称为惯性定律。
Inertia is the tendency of an object to resist changes in its state of motion. The mass of an object is a measure of its inertia; larger mass means greater resistance to acceleration.
惯性是物体抵抗运动状态变化的趋势。物体的质量是其惯性的量度；质量越大，抵抗加速度的能力越强。
For example, a passenger lurching forward when a bus suddenly stops is due to inertia — the passenger’s body tends to maintain its forward velocity while the bus decelerates.
例如，当公交车突然刹车时，乘客前倾是因为惯性——乘客的身体倾向于保持向前的速度，而公交车在减速。
Mathematically, if the net force is zero, the acceleration is zero:
数学上，若合力为零，则加速度为零：
ΣF = 0 → a = 0
4. Newton’s Second Law of Motion | 牛顿第二定律
Newton’s second law establishes the relationship between net force, mass, and acceleration. The net force acting on an object is equal to the product of its mass and its acceleration, and the acceleration is in the same direction as the net force.
This vector equation can be resolved into components. In two dimensions:
该矢量方程可以分解为分量形式。在二维情况下：
ΣFx = m × ax , ΣFy = m × ay
It is important to remember that F_net is the vector sum of all forces, not any individual force. The unit of force is the newton (N); 1 N = 1 kg·m·s⁻².
必须记住，F_net 是所有力的矢量和，而非某个单独的力。力的单位是牛顿（N）；1 N = 1 kg·m·s⁻²。
Example: A 2.0 kg box is pulled horizontally with a force of 10 N, and friction opposes it with 4 N. The net force is 6 N, so the acceleration is a = 6 N / 2.0 kg = 3.0 m·s⁻².
例如：一个2.0 kg的木箱在水平方向被10 N的力拉动，摩擦力为4 N。合力为6 N，因此加速度 a = 6 N / 2.0 kg = 3.0 m·s⁻²。
5. Newton’s Third Law of Motion | 牛顿第三定律
Newton’s third law states that for every action force, there is an equal and opposite reaction force. These two forces act on different objects, so they never cancel each other out.
For example, when you push against a wall, the wall pushes back on you with the same magnitude but opposite direction. When a rocket expels gas downward, the gas exerts an upward force on the rocket.
例如，当你推墙时，墙以相同大小但相反的方向推你。当火箭向下喷出气体时，气体对火箭施加向上的力。
Critical distinction: In Newton’s third law, action-reaction pairs involve the same type of force (e.g., gravitational-gravitational, normal-normal). They act on different bodies. This is different from forces on the same object that may be equal and opposite in equilibrium (e.g., weight and normal force acting on a book on a table), which are not action-reaction pairs.
In IB mechanics problems, the following forces appear frequently:
在IB力学问题中，以下力经常出现：
Force | 力
Symbol | 符号
Direction | 方向
Key Formula | 关键公式
Weight | 重力
Fg or W
Downward toward Earth’s centre | 竖直向下指向地心
Fg = m × g (g ≈ 9.8 m·s⁻²)
Normal force | 支持力
FN
Perpendicular to the contact surface | 垂直于接触面
Varies; often equals component of weight perpendicular to surface | 可变；通常等于重力垂直于表面的分量
Friction | 摩擦力
Ff
Opposes relative motion or attempted motion | 阻碍相对运动或运动趋势
Ff ≤ μ × FN (kinetic: Ff = μk × FN)
Tension | 张力
FT
Along the rope/string, pulling away from the object | 沿绳/弦方向，远离物体
Same magnitude throughout an ideal (massless, inextensible) rope | 理想（无质量、不可伸长）绳上各处大小相等
Applied force | 施加力
Fapp or F
Whatever direction it is applied | 施加的任意方向
Given or determined from other conditions | 由已知条件或其它条件确定
Remember: The normal force is not always equal to the weight. On an inclined plane, FN = m × g × cos θ. If there is an additional vertical force, the normal force adjusts accordingly.
记住：支持力并不总是等于重力。在斜面上，FN = m × g × cos θ。如果有额外的竖直方向的力，支持力也会相应调整。
7. Applying Newton’s Laws in 1D and 2D | 牛顿定律在一维和二维中的应用
For one-dimensional problems, choose a positive direction and write ΣF = m × a along that axis. All forces in the opposite direction are taken as negative.
对于一维问题，选择一个正方向，并沿该方向写出 ΣF = m × a。相反方向的所有力取负值。
For two-dimensional problems, resolve forces into perpendicular components, usually horizontal and vertical, or parallel and perpendicular to an inclined surface. Then apply Newton’s second law separately along each axis.
Consider a mass m on a frictionless incline of angle θ. The weight
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📚 IB Physics: Core Concepts and Key Exam Points in Mechanics | IB物理：力学核心概念与考点梳理
Mechanics is the foundation of IB Physics, covering approximately 20-25% of the entire syllabus across both Standard Level (SL) and Higher Level (HL). This article systematically reviews the core concepts, essential equations, and common exam traps that every IB Physics student must master before sitting for Paper 1 and Paper 2.
Kinematics deals with the description of motion without considering its causes. The three fundamental quantities are displacement, velocity, and acceleration. Displacement is a vector quantity that measures the change in position, while distance is a scalar that measures the total path length traveled.
Velocity is defined as the rate of change of displacement, and acceleration is the rate of change of velocity. For uniform acceleration, the following four equations (often called the SUVAT equations) apply:
v = u + at s = ut + ½at² v² = u² + 2as s = ½(u + v)t
Here, u is initial velocity, v is final velocity, a is acceleration, s is displacement, and t is time. The sign convention is crucial: choose a positive direction and keep it consistent throughout the problem.
Graphical analysis is heavily tested in IB exams. The gradient of a displacement-time graph gives velocity, and the gradient of a velocity-time graph gives acceleration. The area under a velocity-time graph represents displacement.
For projectile motion, the horizontal and vertical components of motion are independent. The horizontal component has constant velocity, while the vertical component has constant acceleration due to gravity (g = 9.81 m s⁻²). The time of flight, maximum height, and range can all be calculated using the SUVAT equations applied to each direction separately.
对于抛体运动，运动的水平分量和垂直分量是相互独立的。水平分量具有恒定速度，而垂直分量具有重力产生的恒定加速度（g = 9.81 m s⁻²）。飞行时间、最大高度和射程都可以通过将SUVAT方程分别应用于每个方向来计算。
2. Forces and Newton’s Laws | 力与牛顿定律
Newton’s three laws of motion form the cornerstone of classical mechanics. The first law states that an object remains at rest or in uniform motion unless acted upon by a net external force. This law introduces the concept of inertia.
Newton’s second law establishes the quantitative relationship between force, mass, and acceleration:
牛顿第二定律建立了力、质量和加速度之间的定量关系：
F = ma or F = Δp/Δt
The second form, F = Δp/Δt, is the more general statement in terms of momentum change rate and is the preferred form in the IB syllabus. Newton’s third law states that for every action, there is an equal and opposite reaction. A common mistake is confusing action-reaction pairs with balanced forces. Action-reaction forces act on different objects, while balanced forces act on the same object.
第二个形式 F = Δp/Δt 是用动量变化率表示的更一般的陈述，也是IB教学大纲中更偏爱的形式。牛顿第三定律指出，每一个作用力都有一个大小相等、方向相反的反作用力。一个常见错误是混淆作用力-反作用力对与平衡力。作用力和反作用力作用于不同物体，而平衡力作用于同一物体。
Free-body diagrams are essential tools for solving force problems. Students must identify all forces acting on an object, including weight, normal reaction, tension, friction, and applied forces, and then resolve them into components along chosen axes.
The normal reaction force is not always equal to weight. It adjusts based on the situation. For example, on an inclined plane, the normal reaction equals mg cos θ, not mg. Friction can be static or kinetic, and the maximum static friction is given by:
法向反力并不总是等于重力。它根据情况而变化。例如，在斜面上，法向反力等于 mg cos θ，而不是 mg。摩擦力可以是静摩擦力或动摩擦力，最大静摩擦力由下式给出：
f ≤ μₛN (static) | fₖ = μₖN (kinetic)
3. Work, Energy, and Power | 功、能量与功率
Work is defined as the product of force and displacement in the direction of the force. Mathematically:
功定义为力与力的方向上位移的乘积。数学上：
W = Fs cos θ
where θ is the angle between the force and displacement vectors. Work is a scalar quantity measured in joules (J). When the force is perpendicular to displacement (θ = 90°), no work is done — this is why the centripetal force does no work on an object in uniform circular motion.
Kinetic energy is the energy an object possesses due to its motion. The equation is:
动能是物体因运动而具有的能量。其方程为：
Eₖ = ½mv²
Gravitational potential energy near the Earth’s surface is given by:
在地球表面附近，重力势能由下式给出：
Eₚ = mgh
where h is the height relative to a chosen reference level. The choice of reference level is arbitrary but must be consistent throughout a calculation.
其中h是相对于选定参考面的高度。参考面的选择是任意的，但在整个计算过程中必须保持一致。
The principle of conservation of mechanical energy states that in the absence of non-conservative forces (such as friction or air resistance), the total mechanical energy (kinetic plus potential) remains constant. This principle is frequently tested in problems involving pendulums, roller coasters, and falling objects.
Power is the rate at which work is done or energy is transferred:
功率是做功或能量转移的速率：
P = W/t = Fv
The relationship P = Fv is particularly useful for problems involving vehicles moving at constant speed against resistance forces.
关系式 P = Fv 特别适用于涉及车辆以恒定速度克服阻力行驶的问题。
4. Momentum and Impulse | 动量与冲量
Momentum is defined as the product of mass and velocity:
动量定义为质量与速度的乘积：
p = mv
Momentum is a vector quantity. Its direction is the same as the velocity. In IB Physics, the principle of conservation of momentum states that in an isolated system (no external forces), the total momentum before a collision equals the total momentum after the collision.
Impulse is the product of force and the time interval over which it acts:
冲量是力与其作用时间间隔的乘积：
J = FΔt = Δp
Impulse equals the change in momentum. On a force-time graph, the area under the curve represents impulse. This concept explains why safety features like airbags and crumple zones reduce injury: they increase the time over which the momentum change occurs, thereby reducing the average force.
Collisions are classified as elastic or inelastic. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not — some energy is converted to heat, sound, or deformation energy. In a perfectly inelastic collision, the objects stick together and move with a common velocity.
Uniform circular motion involves an object moving at constant speed along a circular path. Although the speed is constant, the velocity changes continuously because direction changes. This means there is always an acceleration directed toward the center of the circle, called centripetal acceleration.
The centripetal force required to maintain circular motion is given by:
维持圆周运动所需的向心力由下式给出：
F = mv²/r = mω²r
It is important to note that centripetal force is not a new type of force. It is the net force causing circular motion, and it can be provided by tension, gravity, friction, or the normal reaction force depending on the situation.
Angular velocity ω is measured in radians per second. The relationship between linear speed and angular velocity is v = ωr. The period T and frequency f of revolution are related by T = 1/f and ω = 2πf = 2π/T.
角速度ω以弧度每秒为单位。线速度与角速度之间的关系是 v = ωr。周期的T和频率f之间的关系为 T = 1/f，ω = 2πf = 2π/T。
Common IB exam scenarios include cars on banked curves, objects on rotating turntables, and vertical circular motion such as a ball on a string at the top and bottom of a loop. In vertical circular motion, the tension varies: it is greatest at the bottom and least at the top.
Newton’s law of universal gravitation states that every point mass attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them:
where G = 6.67 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant. This force is always attractive and acts along the line joining the two masses.
其中 G = 6.67 × 10⁻¹¹ N m² kg⁻² 是万有引力常量。该力始终是引力，沿着连接两个质量的直线方向作用。
The gravitational field strength g at a point is defined as the gravitational force per unit mass:
某点的引力场强度g定义为每单位质量所受的引力：
g = F/m = GM/r²
This equation shows that gravitational field strength decreases with the square of the distance from the center of mass. On the Earth’s surface, g ≈ 9.81 N kg⁻¹. Inside a satellite in orbit, objects appear weightless because they are in free fall — the satellite and its contents are accelerating toward Earth at the same rate.
这个方程表明引力场强度随离质心距离的平方而减小。在地球表面，g ≈ 9.81 N kg⁻¹。在轨道上的卫星内部，物体看起来失重是因为它们处于自由落体状态——卫星及其内部物体以相同的速率向地球加速。
For circular orbits, the gravitational force provides the centripetal force:
对于圆形轨道，引力提供向心力：
GMm/r² = mv²/r → v = √(GM/r)
The orbital speed is independent of the mass of the orbiting object. Geostationary satellites orbit at an altitude of approximately 35,800 km above the equator, with a period of 24 hours, appearing stationary relative to a point on Earth’s surface.
Simple harmonic motion (SHM) is a special type of periodic motion in which the acceleration is proportional to the displacement from equilibrium and directed opposite to it. The defining equation is:
where ω is the angular frequency. The solution to this differential equation gives the displacement as a sinusoidal function of time:
其中ω为角频率。该微分方程的解给出位移作为时间的正弦函数：
x = x₀ sin(ωt) or x = x₀ cos(ωt)
where x₀ is the amplitude. The velocity and acceleration of an SHM system are given by:
其中x₀是振幅。简谐运动系统中速度和加速度由下式给出：
v = ±ω√(x₀² – x²) a = -ω²x
The period of SHM for a mass-spring system is T = 2π√(m/k), and for a simple pendulum it is T = 2π√(l/g). Note that the period of a simple pendulum is independent of the mass and amplitude (for small angles).
弹簧振子系统的简谐运动周期为 T = 2π√(m/k)，单摆的周期为 T = 2π√(l/g)。注意，单摆的周期与质量和振幅无关（小角度条件下）。
In SHM, energy continuously converts between kinetic and potential forms. The total mechanical energy is constant and proportional to the square of the amplitude:
在简谐运动中，能量在动能和势能之间不断转换。总机械能恒定且与振幅的平方成正比：
E = ½mω²x₀²
At equilibrium, kinetic energy is maximum and potential energy is zero; at maximum displacement, potential energy is maximum and kinetic energy is zero. Damping and resonance are important related concepts. Resonance occurs when the driving frequency equals the natural frequency, causing maximum amplitude of oscillation.
Torque (also called moment of force) measures the tendency of a force to rotate an object about a pivot point. It is defined as:
力矩（也称为力的矩）衡量力使物体绕支点旋转的趋势。其定义为：
τ = Fr sin θ
where r is the distance from the pivot to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force vector and the lever arm. The SI unit of torque is newton-metre (N m).
The principle of moments states that for an object in rotational equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments. This principle is fundamental for solving problems involving seesaws, levers, ladders, and beams supported at multiple points.
For an object to be in complete equilibrium, two conditions must be satisfied:
物体要达到完全平衡，必须满足两个条件：
The vector sum of all external forces acting on the object is zero (translational equilibrium).
The sum of all external torques about any point is zero (rotational equilibrium).
作用在物体上的所有外力的矢量和为零（平移平衡）。
绕任意点的所有外力矩之和为零（转动平衡）。
When analyzing such problems, it is crucial to choose a convenient pivot point — often at the location of an unknown force — to simplify the torque equation. Common exam traps involve objects on inclined planes, where the weight must be resolved into components parallel and perpendicular to the plane, and hanging sign problems where tension forces must be resolved into horizontal and vertical components.
Just as linear momentum is a conserved quantity in the absence of external forces, angular momentum is conserved in the absence of external torques. The angular momentum L of a rotating object is defined as:
where I is the moment of inertia and ω is the angular velocity. Moment of inertia measures how mass is distributed relative to the axis of rotation. It plays the same role in rotational motion as mass does in linear motion.
The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum remains constant. This explains why a figure skater spins faster when pulling their arms inward — reducing the moment of inertia increases the angular velocity while the product Iω stays constant.
While the detailed treatment of rotational dynamics is primarily an HL topic, understanding the concept of angular momentum conservation is essential for both SL and HL students, particularly in the context of topics like orbital motion where it explains why planets move faster at perihelion than at aphelion.
10. Common Exam Traps and Problem-Solving Strategies | 常见考试陷阱与解题策略
Many students lose marks in IB Physics mechanics questions due to avoidable errors. Here are the most frequent traps and how to avoid them:
许多学生在IB物理力学题中因可避免的错误而失分。以下是最常见的陷阱及其避免方法：
Sign conventions: Not being consistent with positive and negative directions when using SUVAT equations or resolving forces.
符号约定：在使用SUVAT方程或力分解时，正方向和负方向不一致。
Scalar-vector confusion: Treating acceleration, velocity, or momentum as scalars, or forgetting that work and energy are scalars.
标量与矢量混淆：把加速度、速度或动量当作标量，或忘记功和能量是标量。
Forgetting units: SI units are mandatory in IB, including converting grams to kilograms and centimetres to metres.
忘记单位：IB要求使用SI单位，包括将克转换为千克、将厘米转换为米。
Normal force ≠ weight: The normal reaction is not always equal to mg, especially on inclined planes or accelerating elevators.
法向力 ≠ 重力：法向反力并不总是等于mg，特别是在斜面或加速电梯中。
Third law pairs: Confusing action-reaction pairs with forces that simply cancel each other out.
第三定律力对：将作用力-反作用力对与仅仅是相互抵消的力混淆。
Effective problem-solving strategies include: drawing a clear diagram of the physical situation; identifying known and unknown quantities; selecting the appropriate equation; checking whether the answer is reasonable in terms of magnitude and direction; and always verifying that the units in the final answer are correct.
Students should also practice reading graph questions carefully. In velocity-time graphs, pay attention to whether the graph changes sign (direction reversal) and what the area under each section represents. In force-time graphs, the area gives impulse, not force. When dealing with energy transformations, always identify whether non-conservative forces are present.
The IB Physics mechanics syllabus requires not only memorising equations but also understanding the underlying concepts and knowing when to apply each equation. Regular practice with past papers, careful attention to definitions and sign conventions, and a systematic approach to problem-solving will help you achieve a top score in this foundational topic.
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📚 Vectors and Scalars: Operations and Applications in IB Physics | IB物理：矢量与标量的运算及应用
In IB Physics, distinguishing between vectors and scalars is not merely a classification exercise — it determines how quantities are combined, how equations are applied, and whether your final answer is physically meaningful. This article provides a systematic review of vector and scalar operations, from fundamental definitions to practical problem-solving strategies aligned with the CIE IB syllabus.
A scalar is a quantity that has magnitude only. Examples include mass (5 kg), temperature (300 K), energy (2.0 × 10³ J), and time (4.5 s). A vector has both magnitude and direction. Examples include displacement (5 m due north), velocity (20 m s⁻¹ at 30° above horizontal), force (12 N downward), and momentum (3 kg m s⁻¹ eastward).
标量是仅具有大小的物理量，例如质量（5 kg）、温度（300 K）、能量（2.0 × 10³ J）和时间（4.5 s）。矢量则兼具大小和方向，例如位移（5 m 向北）、速度（20 m s⁻¹ 与水平方向成30°角）、力（12 N 向下）和动量（3 kg m s⁻¹ 向东）。
In IB examination papers, students are often asked to identify whether a quantity is a vector or a scalar. A reliable test: ask “Does it make sense to say this quantity has a direction?” If yes, it is a vector. For instance, electric current has magnitude and a conventional direction, but it is treated as a scalar in IB Physics because it does not obey the parallelogram law of vector addition (currents at a junction add algebraically).
In IB Physics, vectors are represented graphically by directed arrows and algebraically by bold symbols or an arrow above the letter (e.g., a or a). The length of the arrow represents magnitude, and the arrowhead indicates direction.
在IB物理中，矢量用带箭头的线段图形化表示，代数上则用粗体符号或在字母上方加箭头来表示（例如 a 或 a）。箭头的长度代表大小，箭头指向代表方向。
When drawing a vector to scale, choose an appropriate scale so that the diagram fits the space available. For example, a force of 80 N can be drawn as an arrow 8.0 cm long on a scale of 10 N per centimetre. Precision in scale drawing is critical for accurate vector addition in Paper 2 and Paper 3 questions.
3. Addition of Vectors: Triangle and Parallelogram Methods | 矢量加法：三角形法与平行四边形法
Vectors are added head-to-tail. The resultant vector is drawn from the tail of the first vector to the head of the last vector. This is the triangle method, suitable for two vectors. For two vectors a and b, the resultant R = a + b.
矢量采用首尾相连的方式相加。合矢量从第一个矢量的起点画到最后一个矢量的终点。这就是适用于两个矢量的三角形法。对于矢量a和b，合力R = a + b。
The parallelogram method is an equivalent graphical procedure: place both vectors tail-to-tail, construct a parallelogram, and draw the diagonal from the common tail point. The diagonal represents the resultant in both magnitude and direction.
4. Subtraction of Vectors and Scalar Multiplication | 矢量减法与标量乘法
Subtracting vector b from vector a is equivalent to adding the negative of b: a − b = a + (−b). The negative of a vector has the same magnitude but the opposite direction. This is useful in calculating change in velocity, Δv = v − u, where Δv points in the direction of acceleration.
从矢量a中减去矢量b，等价于加上b的负矢量：a − b = a + (−b)。负矢量与原矢量大小相同、方向相反。这在计算速度变化量时非常有用，Δv = v − u，其中Δv的方向即为加速度方向。
Multiplying a vector by a positive scalar k changes its magnitude by a factor of k but leaves direction unchanged. Multiplying by a negative scalar reverses the direction. For example, if velocity v = 10 m s⁻¹ eastward, then −0.5v = 5 m s⁻¹ westward.
矢量乘以正标量 k 时，大小变为原来的 k 倍，方向不变；乘以负标量时方向反转。例如，若速度v = 10 m s⁻¹ 向东，则 −0.5v = 5 m s⁻¹ 向西。
5. Resolution of Vectors into Components | 矢量的分解
The most powerful tool in vector analysis is resolving a vector into perpendicular components. For a vector F making an angle θ with the x-axis:
矢量分析中最强大的工具是正交分解。对于与x轴成θ角的矢量F：
Fₓ = F cos θ, F_y = F sin θ
The magnitude of the original vector is recovered by F = √(Fₓ² + F_y²), and its direction by θ = tan⁻¹(F_y / Fₓ). This decomposition converts vector addition into two independent algebraic operations along the x- and y-axes.
In IB Physics, resolving is essential for analysing motion on inclined planes, projectile motion, and forces at angles. For an object on a frictionless incline of angle θ, the component of weight parallel to the plane is mg sin θ, and the perpendicular component is mg cos θ. These two components replace the original weight vector in all subsequent calculations.
在IB物理中，分解是分析斜面运动、抛体运动和斜向力的关键。对于光滑斜面上倾角为 θ 的物体，重力沿斜面的分量为 mg sin θ，垂直于斜面的分量为 mg cos θ。这两个分量在所有后续计算中替代原始重力矢量。
6. Equilibrium of Forces | 力的平衡
A body is in translational equilibrium when the vector sum of all forces acting on it is zero: ΣF = 0. This is a vector equation, equivalent to two scalar equations: ΣFₓ = 0 and ΣF_y = 0.
For a body in equilibrium under three non-parallel forces, the three force vectors must form a closed triangle when drawn head-to-tail. This is known as the triangle of forces. This geometric condition allows the calculation of unknown forces using the sine rule or cosine rule, without resolving into components.
Consider a 20 N weight suspended by two strings making angles of 30° and 45° with the horizontal. Resolving horizontally: T₁ cos 30° = T₂ cos 45°. Resolving vertically: T₁ sin 30° + T₂ sin 45° = 20 N. Solving these simultaneous equations yields the two tensions.
考虑一个20 N的重物由两根绳子悬挂，两绳与水平方向分别成30°和45°角。水平分解：T₁ cos 30° = T₂ cos 45°；竖直分解：T₁ sin 30° + T₂ sin 45° = 20 N。联立求解即可得到两个张力。
7. Vectors in Kinematics | 运动学中的矢量
Displacement, velocity, and acceleration are vector quantities. In one-dimensional motion, their direction is indicated by positive or negative signs. In two-dimensional projectile motion, the horizontal and vertical motions are analysed independently: horizontal motion has constant velocity (ignoring air resistance), while vertical motion has constant acceleration g = 9.81 m s⁻² downward.
位移、速度和加速度都是矢量。在一维运动中，方向由正负号表示。在二维抛体运动中，水平与竖直运动独立分析：水平方向为匀速运动（忽略空气阻力），竖直方向为加速度 g = 9.81 m s⁻² 向下的匀变速运动。
For a projectile launched at speed u and angle θ above the horizontal:
对于以速度 u、仰角 θ 发射的抛体：
uₓ = u cos θ, u_y = u sin θ
The time of flight, maximum height, and range follow from substituting these components into the kinematic equations. Crucially, the velocity at any instant must be reconstructed as a vector from its components using v = √(vₓ² + v_y²), not simply by adding the magnitudes.
飞行时间、最大高度和射程均由将这些分量代入运动学方程得出。关键在于，任意时刻的速度必须由分量按 v = √(vₓ² + v_y²) 重新合成矢量，而不是简单相加大小。
8. Momentum and Impulse as Vectors | 动量与冲量的矢量性
Linear momentum is defined as p = mv. Since velocity is a vector, momentum is also a vector, pointing in the same direction as the velocity. The impulse delivered by a force is J = FΔt, also a vector. The impulse–momentum theorem states that the impulse equals the change in momentum: FΔt = Δp = mv − mu.
In collision problems, the law of conservation of momentum must be applied separately along perpendicular axes. In a two-dimensional collision, the total momentum before the collision equals the total momentum after the collision for both the x-direction and the y-direction independently.
For a ball bouncing off a wall with speed u at angle θ, the change in momentum is Δp = −2mu sin θ in the normal direction, provided the wall is smooth and the collision is elastic. This vector treatment explains why the impulse is perpendicular to the wall, not along the incoming direction.
对于以速度u、入射角θ撞击光滑墙壁并弹性反弹的球，法线方向的动量变化为 Δp = −2mu sin θ。这种矢量处理解释了为何冲量垂直于墙面，而不是沿入射方向。
9. Electric and Magnetic Fields: Vector Representation | 电场与磁场中的矢量表示
Electric field strength is defined as E = F/q, a vector quantity pointing in the direction of the force experienced by a positive test charge. The electric force on a charge q in a field is F = qE. If q is negative, the force opposes the field direction — a direct consequence of scalar multiplication of a vector by a negative number.
Magnetic force on a moving charge is given by F = qv × B, where × denotes the vector product. The direction of the force is perpendicular to both the velocity and the magnetic field. The right-hand rule is used to determine the force direction for a positive charge; the direction reverses for a negative charge. This is a frequent source of errors in IB Paper 1 multiple-choice questions — candidates often forget the sign of the charge.
运动电荷所受磁场力为F = qv × B，其中×表示矢积。力的方向同时垂直于速度方向和磁场方向。右手定则用于确定正电荷的受力方向；负电荷则方向相反。这是IB Paper 1选择题的常见失分点——考生经常忘记考虑电荷的正负。
10. Common Exam Pitfalls and Strategies | 常见考试陷阱与解题策略
Pitfall 1: Confusing direction with sign. In one-dimensional problems, a negative sign indicates direction, not magnitude. A velocity of −5 m s⁻¹ has a speed of 5 m s⁻¹. Never state “the velocity is bigger because the sign is negative.”
陷阱1：混淆方向与符号。在一维问题中，负号表示方向而非大小。速度为 −5 m s⁻¹ 时，速率是 5 m s⁻¹。切勿写”因为负号所以速度更大”。
Pitfall 2: Adding vector magnitudes directly. This is only valid when the vectors are parallel. For perpendicular vectors, use the Pythagorean theorem. For vectors at arbitrary angles, use the cosine rule or resolve into components.
Pitfall 3: Forgetting the angle convention. When resolving, always state which angle is used — to the horizontal, to the vertical, or between two vectors. A force of magnitude F at angle θ to the horizontal has horizontal component F cos θ; at angle θ to the vertical, the horizontal component is F sin θ.
陷阱3：忘记角度约定。分解时，务必说明所用的角度——是与水平方向、竖直方向的夹角，还是两矢量之间的夹角。大小为F、与水平成θ角的力，水平分量为 F cos θ；若与竖直方向成θ角，则水平分量为 F sin θ。
Strategy: For any vector problem, establish a coordinate system, draw a clear diagram, resolve all vectors into x and y components, write two scalar equations, and solve. Always verify the final answer by checking whether the direction reported is consistent with the diagram.
11. Worked Example: Wind and Aircraft Velocity | 例题精解：风与飞机速度
An aircraft flies due north at an airspeed of 120 m s⁻¹ while a wind blows from the west at 40 m s⁻¹. Determine the resultant velocity relative to the ground.
一架飞机以120 m s⁻¹的空速向正北飞行，同时有从西吹来的40 m s⁻¹的风。求飞机相对地面的合速度。
Let north be the positive y-direction and east the positive x-direction. The aircraft’s velocity is vₐ = 120j m s⁻¹. The wind velocity is vw = 40i m s⁻¹ (blowing from west to east). The ground velocity is the vector sum:
设正北为y轴正方向，正东为x轴正方向。飞机的速度为vₐ = 120j m s⁻¹。风速为vw = 40i m s⁻¹（从西向东吹）。对地速度为矢量和：
v_g = 40i + 120j m s⁻¹
The magnitude is v_g = √(40² + 120²) = √(1600 + 14400) = √16000 ≈ 126 m s⁻¹. The direction is θ = tan⁻¹(120/40) ≈ 71.6° east of north. So the ground velocity is approximately 126 m s⁻¹ at 71.6° east of north.
Mastering vector and scalar operations is foundational for success in IB Physics. Ensure you can: (1) classify quantities correctly; (2) add and subtract vectors graphically and algebraically; (3) resolve vectors into perpendicular components; (4) apply vector conditions for equilibrium; and (5) reconstruct resultant vectors from components with the correct quadrant.
For Paper 2 and Paper 3, show all working clearly: draw the vector diagram, write the component equations explicitly, and state the final answer with direction. In Paper 1, check the units and signs carefully. Consistent practice with past-paper vector questions is the most effective way to build speed and accuracy.
Published by TutorHao | IB Physics Revision Series | aleveler.com
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📚 IB Physics: Measurement Methods and Unit Conversion | IB 物理：物理量测量方法与单位换算
Measurement is the foundation of physics. Every physical quantity must be expressed with a number and a unit, and accurate measurement methods allow us to compare observations and test theories. Unit conversion, meanwhile, is an essential skill for solving problems and communicating results across different systems.
The International System of Units (SI) defines seven base units. All other physical quantities are derived from these base units. In IB Physics, you must know their names, symbols and the quantities they measure.
Notice that the kilogram is the only base unit with a prefix (kilo). Also remember that the kelvin is not written with a degree symbol in SI.
注意，千克是唯一带有前缀（千）的基本单位。另外，开尔文在SI中不使用度符号。
2. Derived Units | 导出单位
Derived units are combinations of base units. Many have special names, such as the newton (N) for force, the joule (J) for energy, the pascal (Pa) for pressure, and the watt (W) for power.
To derive these expressions, substitute the base units of each quantity according to its defining equation. For example, pressure = force / area, so 1 Pa = 1 N / 1 m² = 1 (kg·m·s⁻²) / m² = 1 kg·m⁻¹·s⁻².
要推导这些表达式，需要根据定义方程代入每个量的基本单位。例如，压强 = 力 / 面积，所以 1 Pa = 1 N / 1 m² = 1 (kg·m·s⁻²) / m² = 1 kg·m⁻¹·s⁻²。
3. Metric Prefixes and Decimal Conversion | 十进制前缀与换算
SI uses prefixes to denote powers of ten. The most common in IB physics are pico (p, 10⁻¹²), nano (n, 10⁻⁹), micro (μ, 10⁻⁶), milli (m, 10⁻³), centi (c, 10⁻²), kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹) and tera (T, 10¹²).
When converting, write the prefix as its power of ten and multiply. For squared or cubed units, apply the factor to the whole power. For example, 1 cm² = (10⁻² m)² = 10⁻⁴ m², not 10⁻² m².
Length can be measured using a metre ruler (precision ±0.1 cm), a vernier calliper (precision ±0.01 cm) or a micrometer screw gauge (precision ±0.001 cm). Choose the instrument according to the required significant figures.
Mass is usually measured with a digital balance or an analytical balance. For very small masses, such as individual atoms, other techniques are needed, but in the laboratory a balance is sufficient.
Time is measured with a stopwatch, a light gate or a data logger. Modern atomic clocks define the second using caesium-133 vibrations, reaching precision better than 10⁻⁹ s.
时间用秒表、光电门或数据记录器测量。现代原子钟利用铯-133的振动定义秒，精度可达10⁻⁹秒以上。
Every measurement has uncertainty. The uncertainty of an analogue instrument is usually half the smallest scale division; for digital instruments it is often the smallest displayed digit.
5. Measuring Area, Volume and Density | 面积、体积与密度的测量
Area is derived from length squared and has the unit m². Volume is length cubed with unit m³. In practical work, area can be found from dimensions for regular shapes, or by counting squares on graph paper for irregular shapes.
For liquids, a measuring cylinder measures volume directly in cm³ or mL. Note that 1 cm³ = 1 mL. For irregular solids, the displacement method is used: record the volume of liquid before and after completely submerging the object.
Published by TutorHao | IB Physics Revision Series | aleveler.com
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Radioactive decay is a core topic in the IB Physics syllabus, appearing in both SL and HL. You must be able to write decay equations, interpret decay graphs, and understand the statistical nature of the process.
This article organises the essential definitions, equations, and exam strategies, including the extra HL concepts such as decay constant derivations and the continuous beta spectrum.
本文整理了必要的定义、方程和应试策略，并包含HL的额外概念，如衰变常数的推导和连续β能谱。
1. Nucleus Notation and Types of Decay | 核素符号与衰变类型
A nuclide is written with the mass number as a superscript and the atomic number as a subscript before the chemical symbol, for example ²³⁸₉₂U.
核素的写法是在元素符号前用上标表示质量数、下标表示质子数，例如²³⁸₉₂U。
The mass number A is the total number of protons and neutrons; the atomic number Z is the number of protons.
质量数A是质子数和中子数之和；原子序数Z是质子数。
There are four common types of decay: alpha (α), beta-minus (β⁻), beta-plus (β⁺), and gamma (γ). Naturally occurring radioactive nuclides typically emit α, β⁻, and γ, while proton-rich nuclides can undergo β⁺ decay.
An alpha particle is a helium nucleus, written as ⁴₂He or α²⁺. In alpha decay, the parent nucleus loses two protons and two neutrons, so the mass number decreases by 4 and the atomic number decreases by 2.
Alpha particles have low penetrating power (a few centimetres of air or a sheet of paper can stop them) but very strong ionising power.
α粒子穿透力很弱（几厘米空气或一张纸即可阻挡），但电离能力非常强。
3. Beta-Minus Decay | β⁻衰变
Beta-minus decay occurs when a neutron changes into a proton, an electron, and an antineutrino. The electron is emitted as the β⁻ particle.
β⁻衰变发生时，一个中子转变为质子，同时释放一个电子和一个反中微子。释放的电子即β⁻粒子。
The general equation for the decay of a neutron inside the nucleus is:
核内中子的衰变方程一般写作：
n → p + e⁻ + ṽₑ
For example, carbon-14 decays to nitrogen-14:
例如，碳-14衰变为氮-14：
¹⁴₆C → ¹⁴₇N + ⁰₋₁e + ṽₑ
The atomic number increases by 1, while the mass number stays the same because the total number of nucleons is unchanged.
原子序数增加1，质量数保持不变，因为核子总数没有改变。
4. Beta-Plus Decay | β⁺衰变
Beta-plus decay is the reverse process: a proton changes into a neutron, a positron, and a neutrino. The emitted positron is the antiparticle of the electron.
β⁺衰变是逆过程：一个质子转变为中子，同时释放一个正电子和一个中微子。发射的正电子是电子的反粒子。
The equation is:
方程为：
p → n + e⁺ + νₑ
An example is carbon-11 decaying to boron-11:
例如碳-11衰变为硼-11：
¹¹₆C → ¹¹₅B + e⁺ + νₑ
In beta-plus decay, the atomic number decreases by 1 and the mass number is again unchanged.
在β⁺衰变中，原子序数减少1，质量数同样不变。
5. Gamma Decay | γ衰变
Gamma decay does not change the composition of the nucleus. It only releases excess energy in the form of a high-energy photon.
γ衰变不改变原子核的组成，只以高能光子的形式释放多余能量。
It frequently follows alpha or beta decay when the daughter nucleus is left in an excited state. For example:
γ衰变通常发生在α或β衰变之后，此时子核处于激发态。例如：
⁶⁰₂₇Co* → ⁶⁰₂₇Co + γ
Gamma rays are very penetrating and require thick lead or concrete to absorb them.
γ射线穿透力极强，需要厚厚的铅块或混凝土才能吸收。
6. Conservation Laws in Decay Equations | 衰变方程中的守恒定律
Every decay equation must satisfy conservation of mass number (nucleon number), conservation of charge, and conservation of energy-momentum.
每个衰变方程都必须满足质量数守恒、电荷守恒和能量-动量守恒。
For example, in the beta-minus decay of ¹⁴₆C, the mass number is 14 on both sides and the total charge is +6 on the left and +7 − 1 = +6 on the right.
例如，在¹⁴₆C的β⁻衰变中，两边质量数均为14；左边总电荷为+6，右边为+7−1=+6。
When balancing any equation, always check the superscripts and subscripts explicitly. A common exam question is to find the missing particle.
平衡任何方程时，务必检查上标和下标。常见的考题是求缺失的粒子。
7. Half-Life and Decay Constant | 半衰期与衰变常数
The half-life t½ is the time required for half of the radioactive nuclei in a sample to decay. It is measured in seconds, minutes, years, or any suitable time unit.
半衰期t½是指样品中一半放射性原子核发生衰变所需的时间。可用秒、分钟、年等合适的时间单位来度量。
The decay constant λ is the probability per unit time that a given nucleus will decay. The relationship is:
衰变常数λ是某个原子核在单位时间内发生衰变的概率。两者关系为：
t½ = (ln 2) / λ
Therefore λ = (ln 2) / t½. In IB, this derivation is expected in both SL and HL, but HL often asks you to calculate λ from a decay graph.
The number of radioactive nuclei remaining after time t follows an exponential decay law:
经过时间t后剩余的放射性核数目满足指数衰变定律：
N(t) = N₀ e^(−λt)
Here N₀ is the initial number of nuclei, and λ is the decay constant.
其中N₀为初始核数目，λ为衰变常数。
The activity A is the number of decays per second, measured in becquerels (Bq). It is proportional to N:
活度A是每秒衰变次数，单位为贝克勒尔（Bq）。它与N成正比：
A = λN
The activity also decays exponentially: A = A₀ e^(−λt). On a semi-log graph of N versus t, the gradient is −λ.
活度同样指数衰减：A = A₀ e^(−λt)。在半对数图中，以N对t作图，斜率为−λ。
9. Decay Diagrams and Decay Chains | 衰变图与衰变链
A decay diagram plots neutron number N on the vertical axis and proton number Z on the horizontal axis. Alpha decay moves the nuclide two steps left and two steps down; beta-minus decay moves it one step right and one step down.
Beta-plus decay moves one step left and one step up, while gamma decay is shown as a vertical or nearly vertical transition between energy levels of the same nuclide.
β⁺衰变使核素向左移一步、向上移一步；γ衰变则表现为同一核素能级之间垂直或近似垂直的跃迁。
Some heavy nuclides undergo a long decay chain. For example, ²³⁸₉₂U eventually decays through 14 steps to stable ²⁰⁶₈₂Pb.
某些重核素会发生很长的衰变链。例如²³⁸₉₂U经过14步最终衰变为稳定的²⁰⁶₈₂Pb。
10. Radioactive Dating | 放射性测年
Carbon-14 dating uses the beta-minus decay of ¹⁴₆C with a half-life of about 5730 years. The activity in a living organism is roughly constant because it continuously absorbs carbon-14 from the environment.
After death, no new carbon-14 is absorbed, and the observed activity decreases according to the exponential law. The age t can be found from:
死亡后不再吸收碳-14，观测到的活度按指数规律减少。样品的年龄t可由下式求出：
t = (1/λ) ln(A₀/A)
Because carbon-14 has a relatively short half-life, this method is useful for objects up to about 50,000 years old. Other isotopes such as uranium-238 are used for older rocks.
Common detectors include the Geiger-Müller tube, the cloud chamber, and the scintillation counter. The Geiger-Müller tube produces a pulse for each ionising particle that enters it.
常见的探测器包括盖革-米勒管、云室和闪烁计数器。盖革-米勒管每进入一个电离粒子就产生一个脉冲。
Radiation dose depends on the type of radiation and the tissue involved. In the laboratory, safe practice includes minimising time near a source, maximising distance, and using appropriate shielding.
Shielding rules: α particles are stopped by paper, β⁻ particles by a few millimetres of aluminium, and γ rays by lead or concrete. A beta source should be handled with tongs because the braking radiation (X-rays) from beta in metal can be dangerous.
12. HL: Beta Spectrum and Quantum Tunnelling | HL拓展：β能谱与量子隧穿
In beta-minus decay, the emitted electron has a continuous range of energies, not discrete values. This was initially puzzling because alpha particles have discrete energies.
The explanation is that the decay also produces an antineutrino, which carries away a variable amount of energy. The electron and antineutrino share the total energy release in random proportions.
解释是衰变过程还产生一个反中微子，它带走了可变比例的能量。电子和反中微子随机分配总释放能量。
For alpha decay, HL students should understand that the alpha particle is emitted by quantum tunnelling through the Coulomb barrier around the nucleus. The decay constant λ therefore depends on both the energy of the alpha particle and the height and width of the barrier.
These HL concepts are often tested in Paper 2 or Paper 3 data-analysis questions, especially when interpreting beta spectra or explaining the role of the neutrino.
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