📚 Pre-U Cambridge Physics: Formulas and Theorems Quick Reference Handbook | Pre-U Cambridge 物理：公式定理速查手册
This handbook collects the essential equations, constants, and theorems that every Pre-U Cambridge Physics student must have at their fingertips. It is designed for rapid revision and to reinforce the precise language required in examinations, covering classical mechanics, fields, oscillations, thermal physics, and modern physics.
本手册汇集了每位 Pre-U Cambridge 物理学生必须熟记的核心方程、常数和定理。它旨在帮助快速复习并强化考试中要求的精确表述，涵盖经典力学、场、振动、热物理和近代物理。
1. Kinematics and Dynamics | 运动学与动力学
Kinematics describes motion without reference to its causes. The key vector quantities are displacement, velocity, and acceleration. For constant acceleration in one dimension, the SUVAT equations link initial velocity u, final velocity v, acceleration a, displacement s and time t.
运动学描述运动而不考虑其原因。关键的矢量量是位移、速度和加速度。对于一维匀加速运动，SUVAT 方程将初速度 u、末速度 v、加速度 a、位移 s 和时间 t 联系起来。
- v = u + at
- s = ut + ½ at²
- v² = u² + 2as
- s = ½ (u + v) t
Newton’s laws provide the foundation of dynamics. The second law states that the rate of change of momentum of a body is directly proportional to the resultant force and takes place in the direction of the force.
牛顿定律奠定了动力学的基础。第二定律指出，物体的动量变化率与合外力成正比，并发生在力的方向上。
F = dp/dt = ma (constant mass)
Linear momentum is conserved in a closed system with no external forces: Σpbefore = Σpafter. Impulse is the change in momentum, J = FΔt = Δp. Work done by a force is W = Fs cos θ, where θ is the angle between force and displacement. Kinetic energy is Ek = ½mv², and gravitational potential energy near Earth’s surface is Ep = mgh. Power is the rate of doing work, P = W/t = Fv.
线性动量在没有外力的封闭系统中守恒：Σp前 = Σp后。冲量是动量的变化，J = FΔt = Δp。力做的功为 W = Fs cos θ，其中 θ 是力与位移的夹角。动能为 Ek = ½mv²，地球表面附近的引力势能为 Ep = mgh。功率是做功的速率，P = W/t = Fv。
2. Circular Motion and Gravitation | 圆周运动与万有引力
For uniform circular motion, an object moves in a circle of radius r with constant speed v. The velocity is ever-changing in direction, giving a centripetal acceleration directed toward the centre.
对于匀速圆周运动，物体以恒定速率 v 在半径为 r 的圆上运动。速度方向不断改变，产生指向圆心的向心加速度。
ac = v²/r = ω²r
Here ω is the angular velocity in rad s⁻¹. The centripetal force required is Fc = mac = mv²/r = mω²r. Newton’s law of universal gravitation states that any two point masses attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
其中 ω 是角速度，单位为 rad s⁻¹。所需的向心力为 Fc = mac = mv²/r = mω²r。牛顿万有引力定律指出，任何两个质点之间相互吸引的力与它们质量的乘积成正比，与它们之间距离的平方成反比。
F = Gm₁m₂ / r²
G = 6.67 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant. Gravitational field strength g at a distance r from a point mass M is g = F/m = GM/r². Satellite motion combines circular motion and gravitation; for a satellite in a stable orbit, centripetal force is provided by gravity, leading to Kepler’s third law: T² ∝ r³.
G = 6.67 × 10⁻¹¹ N m² kg⁻² 是万有引力常数。距点质量 M 距离 r 处的引力场强 g 为 g = F/m = GM/r²。卫星运动结合圆周运动与引力；对于稳定轨道上的卫星，向心力由引力提供，从而导出开普勒第三定律：T² ∝ r³。
3. Simple Harmonic Motion (SHM) | 简谐运动
SHM is oscillatory motion in which the acceleration is directly proportional to the displacement from equilibrium and always directed toward that equilibrium point. The defining equation is a = -ω²x, where x is displacement and ω is the angular frequency.
简谐运动是一种振动运动，其加速度与离开平衡位置的位移成正比并总是指向平衡点。定义方程为 a = -ω²x，其中 x 是位移，ω 是角频率。
The solutions for displacement are
x = A sin(ωt) or x = A cos(ωt)
where A is amplitude. The period T = 2π/ω. For a mass-spring system, ω = √(k/m), so T = 2π√(m/k). For a simple pendulum of length L, ω = √(g/L), giving T = 2π√(L/g). Energy in SHM continually exchanges between kinetic and potential forms, with total energy E = ½kA² = ½mω²A².
其中 A 是振幅。周期 T = 2π/ω。对于弹簧振子，ω = √(k/m)，故 T = 2π√(m/k)。对于摆长为 L 的单摆，ω = √(g/L)，有 T = 2π√(L/g)。简谐运动中的能量在动能和势能之间不断转换，总能量 E = ½kA² = ½mω²A²。
Damping arises when resistive forces remove energy from the system. Light damping gradually reduces amplitude; critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping results in a slow return. Forced oscillations and resonance occur when a periodic driving force matches the natural frequency, causing dramatic amplitude increase.
当阻力使系统能量耗散时产生阻尼。轻阻尼使振幅逐渐减小；临界阻尼使系统在最短时间内返回平衡位置而不发生振动；过阻尼则导致缓慢返回。当周期性驱动力频率与固有频率匹配时发生受迫振动和共振，引起振幅急剧增大。
4. Electric Fields and Capacitance | 电场与电容
Coulomb’s law describes the force between two point charges Q₁ and Q₂ separated by distance r in a vacuum:
F = (1/(4πε₀)) × Q₁Q₂ / r²
where ε₀ = 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space. The electric field strength E is defined as force per unit positive charge, E = F/q. For a point charge, E = Q/(4πε₀ r²). The field lines point away from positive charges.
其中 ε₀ = 8.85 × 10⁻¹² F m⁻¹ 是真空介电常数。电场强度 E 定义为每单位正电荷所受的力，E = F/q。对于点电荷，E = Q/(4πε₀ r²)。电场线从正电荷发出。
For a uniform electric field between parallel plates with potential difference V and separation d, E = V/d. The work done in moving a charge q through a potential difference ΔV is W = qΔV. Electric potential V at a distance r from a point charge Q is V = Q/(4πε₀ r); potential is a scalar. Equipotential surfaces are perpendicular to field lines.
对于间距为 d、电势差为 V 的平行板间的匀强电场，E = V/d。将电荷 q 移动通过电势差 ΔV 所做功为 W = qΔV。距离点电荷 Q 为 r 处的电势 V 为 V = Q/(4πε₀ r)；电势是标量。等势面与电场线垂直。
| Capacitance C | C = Q/V | 单位：法拉 (F) |
| 平行板电容器 | C = ε₀A/d | A 为板面积，d 为间距 |
| 储存的能量 | W = ½QV = ½CV² | |
| 时间常数 τ | τ = RC | 放电时 Q = Q₀e-t/RC |
Capacitors store energy in the electric field between the plates. In series, 1/Ctotal = Σ 1/Ci; in parallel, Ctotal = Σ Ci.
电容器将能量储存在极板间的电场中。串联时，1/C总 = Σ 1/Ci；并联时，C总 = Σ Ci。
5. Magnetic Fields and Electromagnetic Induction | 磁场与电磁感应
A magnetic field is a region where a moving charge experiences a force. Magnetic flux density B is measured in tesla (T). The force F on a charge q moving with velocity v in a magnetic field is given by the Lorentz force:
F = q(v × B) or F = qvB sinθ
For a current-carrying conductor of length L, the motor force is F = BIL sinθ. The direction is given by Fleming’s left-hand rule. For a coil of N turns carrying current I in a magnetic field, the torque is τ = BANI sinθ, where A is the area of the coil.
磁场是运动电荷会受到力的区域。磁通量密度 B 的单位是特斯拉 (T)。电荷 q 以速度 v 在磁场中运动所受的力由洛伦兹力给出。对于长度为 L 的载流导体，安培力为 F = BIL sinθ。方向由左手定则确定。对于在磁场中的 N 匝线圈，力矩为 τ = BANI sinθ，其中 A 为线圈面积。
Magnetic flux Φ through a surface is Φ = BA cosθ, where θ is the angle between B and the normal to the surface. Faraday’s law of electromagnetic induction states that the induced emf is equal to the negative rate of change of magnetic flux linkage:
ε = -N dΦ/dt
Lenz’s law determines the direction: the induced current opposes the change producing it. For a conductor of length L moving perpendicularly in a B field with velocity v, the motional emf is ε = BLv. An inductor stores energy in the magnetic field; self-inductance L has induced emf ε = -L dI/dt, and energy stored is ½LI².
穿过某表面的磁通量 Φ 为 Φ = BA cosθ，其中 θ 是 B 与表面法线的夹角。法拉第电磁感应定律指出，感应电动势等于磁通链变化率的负值。楞次定律确定方向：感应电流总是阻碍引起它的变化。对于长度为 L 的导体在磁场 B 中以垂直速度 v 运动，动生电动势为 ε = BLv。电感储存磁场中的能量；自感 L 的感应电动势为 ε = -L dI/dt，储存能量为 ½LI²。
6. Alternating Current and Transformers | 交流电与变压器
AC voltage varies sinusoidally: V = V₀ sin(ωt). Root-mean-square values relate to peak values as Vrms = V₀/√2 and Irms = I₀/√2. For a resistor, VI is in phase; for an inductor, current lags voltage by 90°; for a capacitor, current leads voltage by 90°.
交流电压随时间正弦变化：V = V₀ sin(ωt)。有效值与峰值的关系为 Vrms = V₀/√2，Irms = I₀/√2。对于电阻，电压与电流同相；对于电感，电流落后电压 90°；对于电容，电流超前电压 90°。
Reactance: inductive reactance XL = ωL = 2πfL; capacitive reactance XC = 1/(ωC) = 1/(2πfC). Impedance Z for an LCR series circuit is Z = √(R² + (XL – XC)²). The phase angle φ is given by tan φ = (XL – XC)/R. Resonance occurs when XL = XC, with resonant frequency f₀ = 1/(2π√(LC)). The Q-factor measures the sharpness of resonance.
电抗：感抗 XL = ωL = 2πfL；容抗 XC = 1/(ωC) = 1/(2πfC)。LCR 串联电路的阻抗为 Z = √(R² + (XL – XC)²)。相角 φ 满足 tan φ = (XL – XC)/R。当 XL = XC 时产生谐振，谐振频率 f₀ = 1/(2π√(LC))。Q 因子衡量谐振的尖锐程度。
An ideal transformer has no power loss. The turns ratio determines voltage and current transformation: Vs/Vp = Ns/Np and Is/Ip = Np/Ns. Real transformers have losses due to eddy currents, hysteresis, and resistance.
理想变压器无功率损耗。匝数比决定电压和电流变化：Vs/Vp = Ns/Np，Is/Ip = Np/Ns。实际变压器因涡流、磁滞和电阻而产生损耗。
7. Waves and Optics | 波动与光学
A progressive wave transfers energy without transferring matter. The wave equation links speed v, frequency f and wavelength λ: v = fλ. Transverse waves have oscillations perpendicular to propagation direction; longitudinal waves have oscillations parallel. The phase difference φ between two points separated by distance Δx is φ = (2π/λ) Δx. Intensity is proportional to amplitude squared: I ∝ A².
行波传递能量而不传递物质。波动方程将波速 v、频率 f 和波长 λ 联系起来：v = fλ。横波的振动方向垂直于传播方向；纵波的振动方向平行于传播方向。相距 Δx 的两点之间的相差 φ 为 φ = (2π/λ) Δx。强度与振幅的平方成正比：I ∝ A²。
Principle of superposition: when two or more waves meet, the resultant displacement is the vector sum of individual displacements. Standing waves form when two identical waves travelling in opposite directions superpose, producing nodes (zero amplitude) and antinodes (maximum amplitude). For a string fixed at both ends, L = nλ/2; for a pipe open at both ends, L = nλ/2; open-closed pipe gives L = (2n-1)λ/4.
叠加原理：当两个或多个波相遇时，合位移是各个位移的矢量和。当两列相同的波沿相反方向传播时叠加产生驻波，形成波节（振幅为零）和波腹（振幅最大）。对于两端固定的弦，L = nλ/2；两端开口管，L = nλ/2；一端开口一端封闭管，L = (2n-1)λ/4。
Diffraction is the spreading of waves at a slit or obstacle; noticeable when λ is comparable to the slit width. In the double-slit experiment, the fringe spacing Δy = λD/d, where D is screen distance and d is slit separation. Diffraction grating with N lines per metre has maxima at d sinθ = nλ (d = 1/N).
衍射是波在通过缝隙或绕过障碍物时的扩展现象；当波长 λ 与缝隙宽度可比时尤为明显。双缝实验中，条纹间距 Δy = λD/d，其中 D 为屏幕距离，d 为双缝间距。每米有 N 条刻线的衍射光栅在 d sinθ = nλ (d = 1/N) 处产生主极大。
Refraction is governed by Snell’s law: n₁ sin θ₁ = n₂ sin θ₂. Refractive index n = c/v, where c is speed of light in vacuum. Total internal reflection occurs when light travels from a medium of higher n to lower n at incident angles greater than the critical angle, sin C = 1/n.
折射遵循斯涅尔定律：n₁ sin θ₁ = n₂ sin θ₂。折射率 n = c/v，其中 c 为真空中的光速。当光从高折射率介质射向低折射率介质且入射角大于临界角 (sin C = 1/n) 时发生全内反射。
8. Thermal Physics and Gases | 热物理与气体
Temperature measures average random kinetic energy of particles. On the Kelvin scale, absolute zero T = 0 K corresponds to zero thermal energy; 0°C = 273.15 K. Specific heat capacity c is the energy required to raise the temperature of 1 kg of a substance by 1 K: Q = mcΔT. Specific latent heat L is the energy required to change the state of 1 kg without temperature change: Q = mL.
温度是粒子平均无规则动能的量度。在开尔文温标中，绝对零度 T = 0 K 对应热能为零；0°C = 273.15 K。比热容 c 是使 1 kg 物质温度升高 1 K 所需的能量：Q = mcΔT。比潜热 L 是在温度不变时使 1 kg 物质状态改变所需的能量：Q = mL。
The ideal gas equation is pV = nRT, where n is number of moles and R = 8.31 J mol⁻¹ K⁻¹. Alternatively, pV = NkT, where N is number of molecules and k = 1.38 × 10⁻²³ J K⁻¹ is Boltzmann’s constant. A mole of any substance contains Avogadro’s number NA = 6.02 × 10²³ particles.
理想气体状态方程为 pV = nRT，其中 n 为摩尔数，R = 8.31 J mol⁻¹ K⁻¹。或者 pV = NkT，其中 N 为分子数，k = 1.38 × 10⁻²³ J K⁻¹ 为玻尔兹曼常数。任何物质一摩尔含有阿伏伽德罗常数 NA = 6.02 × 10²³ 个粒子。
The kinetic theory of gases relates pressure to microscopic motion: pV = ⅓ N m ⟨c²⟩, where m is molecular mass and ⟨c²⟩ is mean square speed. Hence average translational kinetic energy of a molecule is ⟨KE⟩ = ³⁄₂ kT. The root mean square speed crms = √(3kT/m) = √(3RT/M).
气体动理论将压强与微观运动联系起来：pV = ⅓ N m ⟨c²⟩，其中 m 为分子质量，⟨c²⟩ 为方均速率。因此分子的平均平动动能为 ⟨KE⟩ = ³⁄₂ kT。方均根速率 crms = √(3kT/m) = √(3RT/M)。
The first law of thermodynamics is ΔU = Q – W, where ΔU is change in internal energy, Q is heat added to the system, and W is work done by the system. For an isothermal process (ΔT = 0), ΔU = 0; for adiabatic, Q = 0. The efficiency of a heat engine is η = W/Qh = 1 – Qc/Qh. Maximum Carnot efficiency between temperatures Th and Tc is ηCarnot = 1 – Tc/Th (kelvin).
热力学第一定律为 ΔU = Q – W，其中 ΔU 是内能变化，Q 是系统吸收的热量，W 是系统对外做功。对于等温过程 (ΔT = 0)，ΔU = 0；对于绝热过程，Q = 0。热机效率 η = W/Qh = 1 – Qc/Qh。在温度 Th 和 Tc 之间工作的最大卡诺效率为 η卡诺 = 1 – Tc/Th（开尔文）。
9. Nuclear and Particle Physics | 核物理与粒子物理
The nucleus is characterised by atomic number Z (protons) and mass number A (protons + neutrons). Protons and neutrons are held together by the strong nuclear force. Nuclear binding energy is the energy required to separate a nucleus into its individual nucleons; the mass defect Δm is converted to binding energy via E = Δm c².
原子核的特征是原子序数 Z（质子数）和质量数 A（质子+中子）。质子和中子由强核力结合在一起。原子结合能是将原子核分解成单个核子所需的能量；质量亏损 Δm 通过 E = Δm c² 转化为结合能。
Radioactive decay follows the exponential law: N = N₀ e-λt, where λ is the decay constant. Activity A = λN. Half-life T½ = ln2/λ. The three main types of decay are alpha (⁴₂He nucleus), beta-minus (electron and antineutrino), beta-plus (positron and neutrino), and gamma (high-energy photon). In all nuclear reactions, mass-energy, charge, and nucleon number are conserved.
放射性衰变遵循指数规律：N = N₀ e-λt，其中 λ 是衰变常数。活度 A = λN。半衰期 T½ = ln2/λ。三种主要衰变类型为 α 衰变（⁴₂He 核）、β⁻ 衰变（电子和反中微子）、β⁺ 衰变（正电子和中微子）和 γ 衰变（高能光子）。在所有核反应中，质能、电荷和核子数守恒。
Einstein’s mass-energy equivalence: E = mc². In fission, a heavy nucleus splits into smaller fragments, releasing energy. In fusion, light nuclei combine to form a heavier nucleus, also releasing energy; the binding energy per nucleon curve peaks around iron-56. Particle physics classifies fundamental particles into quarks and leptons. The Standard Model describes forces mediated by gauge bosons: photon (electromagnetic), W and Z (weak), and gluons (strong).
爱因斯坦质能方程：E = mc²。裂变中，重核分裂成较小碎片并释放能量。聚变中，轻核结合形成更重的核，也释放能量；比结合能曲线在铁-56 附近达到峰值。粒子物理将基本粒子分为夸克和轻子。标准模型描述由规范玻色子传递的力：光子（电磁力）、W 和 Z（弱力）以及胶子（强力）。
10. Quantum Physics and Photons | 量子物理与光子
Light exhibits both wave and particle properties. The photon energy is E = hf = hc/λ, where h is Planck’s constant (6.63 × 10⁻³⁴ J s). The photoelectric effect demonstrates the particle nature: electrons are emitted from a metal surface when incident photons have energy greater than the work function Φ. Einstein’s photoelectric equation:
Ek max = hf – Φ
The stopping potential Vs gives e Vs = hf – Φ. Threshold frequency f0 = Φ/h. Intensity affects the number of photoelectrons, not their maximum kinetic energy.
光呈现波粒二象性。光子能量为 E = hf = hc/λ，其中 h 是普朗克常数 (6.63 × 10⁻³⁴ J s)。光电效应展示了粒子性：当入射光子能量大于功函数 Φ 时，电子从金属表面逸出。爱因斯坦光电方程：遏止电势 Vs 满足 e Vs = hf – Φ。阈频率 f0 = Φ/h。光强影响光电子数目，而非其最大动能。
Matter also has wave properties, given by de Broglie wavelength λ = h/p = h/mv. Energy levels in atoms are quantized. The hydrogen spectrum is explained by transitions between energy levels: En = -13.6 eV / n². When an electron falls from level ni to nf, the emitted photon has energy hf = Eni – Enf. The Lyman, Balmer, and Paschen series correspond to transitions to n=1, n=2, and n=3 respectively.
实物粒子也具有波动性质，由德布罗意波长 λ = h/p = h/mv 给出。原子中的能级是量子化的。氢原子光谱由能级间跃迁解释：En = -13.6 eV / n²。当电子从能级 ni 跃迁到 nf 时，发射的光子能量为 hf = Eni – Enf。莱曼系、巴耳末系和帕邢系分别对应于跃迁到 n=1、n=2 和 n=3。
The Heisenberg uncertainty principle places fundamental limits: Δx Δp ≥ h/4π and ΔE Δt ≥ h/4π. Wavefunctions and probability densities replace deterministic trajectories in the quantum world.
海森伯不确定性原理给出了基本限制：Δx Δp ≥ h/4π 和 ΔE Δt ≥ h/4π。在量子世界中，波函数和概率密度取代了确定性的轨迹。
11. Essential Constants and Data | 基本常数与数据
Every physicist must know these values and their units. Use them precisely in calculations and explanations.
每位物理学家都必须牢记这些数值及其单位，并在计算和解释中准确使用。
| Speed of light in vacuum, c | 3.00 × 10⁸ m s⁻¹ |
| Planck constant, h | 6.63 × 10⁻³⁴ J s |
| Electron charge, e | 1.60 × 10⁻¹⁹ C |
| Electron mass, me | 9.11 × 10⁻³¹ kg |
| Proton mass, mp | 1.67 × 10⁻²⁷ kg |
| Unified atomic mass unit, u | 1.66 × 10⁻²⁷ kg = 931.5 MeV/c² |
| Avogadro constant, NA | 6.02 × 10²³ mol⁻¹ |
| Gas constant, R | 8.31 J mol⁻¹ K⁻¹ |
| Boltzmann constant, k | 1.38 × 10⁻²³ J K⁻¹ |
| Gravitational constant, G | 6.67 × 10⁻¹¹ N m² kg⁻² |
| Permittivity of free space, ε₀ | 8.85 × 10⁻¹² F m⁻¹ |
| Permeability of free space, μ₀ | 4π × 10⁻⁷ H m⁻¹ |
| Acceleration of free fall (standard), g | 9.81 m s⁻² |
12. Key Theorems and Principles Summary | 核心定理与原理汇总
Beyond formulas, exam success hinges on fluent recall of fundamental theorems and principles. The conservation laws—energy, momentum, charge, baryon number, lepton number—are universal pillars. The principle of superposition applies to all waves and fields. The laws of thermodynamics govern energy flow, entropy always increases for an isolated system.
除公式外，考试成功还依赖于对基本定理和原理的流畅回忆。守恒定律——能量、动量、电荷、重子数、轻子数——是普遍的支柱。叠加原理适用于所有波和场。热力学定律支配能量流动，孤立系统的熵总是增加。
Maxwell’s equations unify electricity and magnetism, predicting electromagnetic waves travelling at speed c
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