📚 Formula Handbook for IB and OCR Physics | IB OCR 物理公式汇总手册
This handbook compiles essential equations for IB Physics (SL and HL) and OCR A Level Physics. Each formula is presented with its physical meaning and typical applications, helping you build confidence for both examinations. The equations are grouped by topic, with every English explanation immediately followed by its Chinese equivalent for bilingual revision.
本手册汇集了IB物理（标准级别与高级级别）和OCR A Level物理的核心公式。每个公式都配有物理含义说明与常见应用场景，帮助你在两种考试体系中建立信心。公式按主题分组，每一条英文解释后面紧跟着对应的中文解释，便于双语复习。
1. Kinematics | 运动学
v = u + a t
The equation v = u + a t links final velocity v to initial velocity u, constant acceleration a and time t. It is used when three of the variables are known.
方程 v = u + a t 描述了末速度 v 与初速度 u、匀加速度 a 和时间 t 的关系。当四个量中有三个已知时，可用该式求解第四个量。
s = u t + ½ a t²
This gives the displacement s of an object under constant acceleration in time t. The term u t represents the distance due to initial velocity, while ½ a t² accounts for additional displacement caused by acceleration.
此式给出物体在匀加速度下，经过时间 t 的位移 s。其中 u t 是由初速度产生的位移，而 ½ a t² 是由加速度引起的额外位移。
v² = u² + 2 a s
It relates final velocity, initial velocity, acceleration and displacement without involving time. Useful for problems where time is not given.
该式将末速度、初速度、加速度和位移联系起来，不显含时间。常用于未给出时间的运动问题。
s = ½ (u + v) t
Displacement equals the average velocity multiplied by time when acceleration is constant. It provides a quick way to find displacement if initial and final velocities are known.
位移等于平均速度乘以时间，前提是加速度恒定。如果初速度和末速度已知，这是快速求位移的方法。
2. Dynamics | 动力学
F = m a
Newton’s second law states that the net force F acting on a body equals the product of its mass m and acceleration a. It applies in inertial frames and is the foundation of classical mechanics.
牛顿第二定律指出，作用在物体上的合力 F 等于其质量 m 与加速度 a 的乘积。该定律适用于惯性参考系，是经典力学的基础。
Fₙₑₜ = m a
The subscript ‘net’ emphasises that only the unbalanced force causes acceleration. When multiple forces act, vector addition must be used to find the net force.
下标“net”强调只有非平衡力才会产生加速度。当多个力同时作用时，必须通过矢量叠加求出合力。
p = m v
Linear momentum p is the product of mass and velocity. It is a vector quantity, conserved in isolated systems. Impulse equals change in momentum: Δp = F Δt.
线动量 p 是质量与速度的乘积，为矢量。在孤立系统中动量守恒。冲量等于动量的变化量：Δp = F Δt。
F_f = μ R
The maximum static or kinetic friction force is proportional to the normal reaction force R, with coefficient μ. Direction always opposes relative motion or attempted motion.
最大静摩擦力或滑动摩擦力与法向反作用力 R 成正比，比例系数为 μ。摩擦力的方向总是阻碍相对运动或相对运动趋势。
3. Work, Energy and Power | 功、能量与功率
W = F s cos θ
Work done by a constant force F is the product of the force magnitude, displacement s and the cosine of the angle θ between them. Only the force component along displacement does work.
恒力 F 所做的功等于力的大小、位移 s 以及力与位移夹角 θ 的余弦三者的乘积。只有沿位移方向的分力做功。
KE = ½ m v²
Kinetic energy is the energy a body possesses due to its motion. It depends on mass and the square of speed. The work-energy theorem states W_net = ΔKE.
动能是物体因运动而具有的能量，依赖于质量与速度的平方。动能定理指出，合力做功等于动能的变化量。
PE_grav = m g h
Near the Earth’s surface, gravitational potential energy change is m g h, where h is the vertical height change. This formula is valid only for small height variations where g is constant.
在地球表面附近，重力势能的变化为 m g h，h 是竖直高度差。该式仅在高度变化较小、g 可视为常数时使用。
P = W / t = F v
Power P is the rate of doing work. For a constant force moving at speed v, instantaneous power can also be written as F v cos θ.
功率 P 是做功的快慢。对于以速度 v 运动的物体，受到恒力时，瞬时功率也可以写成 F v cos θ。
4. Circular Motion and Gravitation | 圆周运动与引力
a_c = v² / r = ω² r
Centripetal acceleration a_c points towards the centre of the circle. It can be expressed in terms of linear speed v and radius r, or angular speed ω. Direction changes continuously.
向心加速度 a_c 指向圆心。可以用线速度 v 和半径 r，或者用角速度 ω 来表示。其方向时刻改变。
F_c = m v² / r = m ω² r
The centripetal force is the net force causing circular motion, always directed towards the centre. It is not a new type of force but a role played by tension, gravity, etc.
向心力是产生圆周运动的合力，始终指向圆心。它不是一种新的力，而是由拉力、引力等扮演的角色。
F_g = G M m / r²
Newton’s law of gravitation gives the attractive force between two point masses. G is the universal gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻².
牛顿的万有引力定律给出两个质点间的吸引力。G 为万有引力常数，6.67 × 10⁻¹¹ N m² kg⁻²。
g = G M / r²
Gravitational field strength g at a distance r from a spherical mass M. On Earth’s surface, g ≈ 9.81 m s⁻².
距离球状质量 M 的 r 处的引力场强度 g。在地球表面，g ≈ 9.81 m s⁻²。
5. Thermal Physics | 热学
Q = m c Δθ
The heat energy Q required to change the temperature of mass m by Δθ depends on specific heat capacity c. No phase change occurs.
使质量为 m 的物体温度变化 Δθ 所需的热量 Q 取决于比热容 c。该式仅适用于不发生相变的情况。
Q = m L
During melting or boiling, the heat energy supplied changes the state without temperature change. L is the specific latent heat (fusion or vaporisation).
在熔化或沸腾过程中，吸收的热量用于改变物态，温度保持不变。L 是比潜热（熔化潜热或汽化潜热）。
p V = n R T
The ideal gas equation links pressure p, volume V, amount n and thermodynamic temperature T. R is the molar gas constant, 8.31 J K⁻¹ mol⁻¹.
理想气体状态方程将压强 p、体积 V、物质的量 n 和热力学温度 T 联系在一起。R 是摩尔气体常数，8.31 J K⁻¹ mol⁻¹。
E_k = ½ m c²
For a single gas molecule, the average translational kinetic energy is proportional to temperature: KE_avg = (3/2) k T, where k is the Boltzmann constant.
对于单个气体分子，平均平移动能与温度成正比：KE_avg = (3/2) k T，其中 k 为玻尔兹曼常数。
6. Simple Harmonic Motion and Waves | 简谐振动与波
a = – ω² x
The defining equation for simple harmonic motion (SHM): acceleration a is proportional to displacement x and directed towards the equilibrium position. ω is the angular frequency.
简谐运动的定义式：加速度 a 与位移 x 成正比且方向指向平衡位置。ω 是角频率。
v = ± ω √(A² – x²)
The speed of an oscillator at displacement x depends on amplitude A. Maximum speed v_max = ω A occurs at equilibrium.
振子在位移 x 处的速度取决于振幅 A。最大速度 v_max = ω A 出现在平衡位置。
T = 1 / f
Period T and frequency f are reciprocals. In SHM, period of a mass-spring system: T = 2π √(m/k); for a simple pendulum: T = 2π √(L/g).
周期 T 和频率 f 互为倒数。在简谐运动中，弹簧振子的周期为 T = 2π √(m/k)；单摆的周期为 T = 2π √(L/g)。
v = f λ
Wave speed v equals frequency multiplied by wavelength. This holds for all progressive waves, including sound and electromagnetic waves.
波速 v 等于频率乘以波长。这适用于所有行波，包括声波与电磁波。
7. Electric Fields and Circuits | 电场与电路
F = k Q q / r²
Coulomb’s law describes the electrostatic force between two point charges. k = 1/(4π ε₀), and ε₀ is the permittivity of free space.
库仑定律描述两点电荷间的静电力。k = 1/(4π ε₀)，ε₀ 为真空介电常数。
E = F / q
Electric field strength E is the force per unit positive charge. For a uniform field between parallel plates, E = V / d.
电场强度 E 是每单位正电荷所受的力。对于平行板间的匀强电场，E = V / d。
V = I R
Ohm’s law states that potential difference V across a conductor is directly proportional to the current I, provided temperature remains constant. Resistance R is measured in ohms.
欧姆定律指出，在温度不变时，导体两端的电势差 V 与通过的电流 I 成正比。电阻 R 的单位为欧姆。
P = I V = I² R = V² / R
Electrical power can be expressed in three equivalent forms. The choice depends on which quantities are known. Energy dissipated: E = P t.
电功率有三种等价表达式，选择取决于已知量。耗散能量：E = P t。
8. Magnetism and Electromagnetic Induction | 磁场与电磁感应
F = B I L sin θ
The magnetic force on a current-carrying conductor in a magnetic field depends on flux density B, current I, length L and the angle θ between the conductor and the field.
载流导体在磁场中所受的磁力取决于磁通密度 B、电流 I、导体长度 L 以及导体与磁场的夹角 θ。
F = q v B sin θ
A moving charge in a magnetic field experiences a force: the foundation of circular motion for charged particles. Direction is given by Fleming’s left-hand rule (or right-hand for positive charges).
运动电荷在磁场中受到力的作用，这是带电粒子做圆周运动的基础。方向通过左手定则（正电荷用右手定则）判断。
Φ = B A cos θ
Magnetic flux Φ is the product of flux density B and the area A perpendicular to the field. θ is the angle between B and the normal to the area.
磁通量 Φ 是磁通密度 B 与垂直于磁场方向的面积 A 的乘积。θ 为 B 与面积法线间的夹角。
ε = – N dΦ / d t
Faraday’s law of induction: the induced emf ε in a coil of N turns equals the negative rate of change of magnetic flux linkage. Lenz’s law gives the direction.
法拉第电磁感应定律：N 匝线圈中产生的感应电动势 ε 等于磁通链变化率的负值。楞次定律给出了电动势的方向。
9. Quantum Physics | 量子物理
E = h f
The energy of a photon is proportional to its frequency f, with Planck’s constant h = 6.63 × 10⁻³⁴ J s. Higher frequency means greater photon energy.
光子的能量与其频率 f 成正比，普朗克常数 h = 6.63 × 10⁻³⁴ J s。频率越高，光子能量越大。
h f = Φ + KE_max
The photoelectric effect equation: photon energy is used to overcome the work function Φ of the metal, with the remainder becoming the electron’s maximum kinetic energy.
光电效应方程：光子的能量一部分用于克服金属的逸出功 Φ，剩余的部分转化为电子的最大动能。
λ = h / p
De Broglie wavelength associates a wavelength λ with a particle of momentum p. It demonstrates wave-particle duality, significant for electrons and other microscopic particles.
德布罗意波长将动量 p 的粒子与波长 λ 联系起来，体现了波粒二象性，对电子等微观粒子意义重大。
Δx Δp ≥ h / 4π
The Heisenberg uncertainty principle: it is impossible to know precisely both the position and momentum of a particle simultaneously. The product of uncertainties has a lower limit.
海森堡不确定性原理：无法同时精确地知道粒子的位置和动量，它们的测量不确定度之积存在下限。
10. Nuclear Physics | 核物理
E = m c²
Mass-energy equivalence: mass m can be converted into energy E, with c the speed of light. In nuclear reactions, mass defect appears as binding energy.
质能方程：质量 m 可以转化为能量 E，c 为光速。在核反应中，质量亏损以结合能的形式展现。
N = N₀ e^(-λ t)
The radioactive decay law gives the number N of undecayed nuclei after time t. λ is the decay constant, related to half-life by λ = ln 2 / T_½.
放射性衰变定律给出了 t 时刻未衰变的原子核数目 N。λ 是衰变常数，与半衰期的关系为 λ = ln 2 / T_½。
A = λ N
Activity A, the number of decays per unit time, is the product of decay constant and the number of radioactive nuclei present. It decreases exponentially with time.
活度 A 是单位时间内发生的衰变次数，等于衰变常数与当前放射性核子数的乘积。活度随时间指数衰减。
Binding energy per nucleon = (Z m_p + N m_n – m_nucleus) c² / A
Average binding energy per nucleon indicates nuclear stability. Iron-56 has one of the highest values. Fusion and fission release energy by moving towards a greater binding energy per nucleon.
平均每个核子的结合能反映原子核的稳定性。铁-56 具有极高的值。聚变和裂变通过使每个核子结合能增大来释放能量。
11. Data, Measurement and Uncertainty | 数据、测量与不确定度
If Q = a ± Δa and R = b ± Δb, then:
For addition: Q + R = (a + b) ± (Δa + Δb). For multiplication: Q × R ≈ a b ± (a b)(Δa/a + Δb/b). Absolute uncertainties add in addition; relative (percentage) uncertainties add in multiplication.
若 Q = a ± Δa，R = b ± Δb，则相加时：Q + R = (a + b) ± (Δa + Δb)。相乘时：Q × R ≈ a b ± (a b)(Δa/a + Δb/b)。加减法用绝对不确定度相加，乘除法用相对（百分）不确定度相加。
mean = Σ xᵢ / n
The mean of n measurements xᵢ is the sum divided by n. Standard deviation s = √[ Σ (xᵢ – mean)² / (n – 1) ] estimates the spread of data.
n 个测量值 xᵢ 的平均值等于总和除以 n。标准差 s = √[ Σ (xᵢ – mean)² / (n – 1) ] 用于估计数据的分散程度。
% uncertainty = (absolute uncertainty / measured value) × 100%
Converting between absolute and percentage uncertainty is crucial for combining errors. Always quote uncertainties to one or two significant figures.
在绝对不确定度和百分不确定度之间转换，对误差的综合计算很关键。不确定度通常保留一到两位有效数字。
12. Formula Manipulation and Exam Tips | 公式运用与应试技巧
Rearranging equations: reverse operations step by step
When solving for an unknown, perform inverse operations systematically. For example
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