📚 Year 13 CAIE Physics: Formula & Theorem Quick Reference Guide | Year 13 CAIE 物理：公式定理速查手册
This is a condensed reference guide covering the essential formulas and theorems you need for your Year 13 CAIE Physics exam. Every key relationship is presented with a short explanation, helping you to review efficiently and avoid common mistakes.
这是一份凝练的速查手册，涵盖了你 Year 13 CAIE 物理考试所需的核心公式与定理。每一个重要关系式都配有简短的解释，帮助你高效复习、避开常见误区。
1. Circular Motion | 圆周运动
Angular velocity ω measures how fast an object rotates. It is defined as ω = Δθ / Δt, and for uniform circular motion ω = 2π / T = 2πf, where T is the period and f the frequency.
角速度 ω 衡量物体转动的快慢，定义为 ω = Δθ / Δt。在匀速圆周运动中 ω = 2π / T = 2πf，其中 T 为周期，f 为频率。
Linear speed and angular speed are linked by v = rω, where r is the radius of the circle.
线速度与角速度的关系是 v = rω，r 为圆周半径。
Centripetal acceleration is always directed towards the centre: a = v² / r = rω². The centripetal force required to keep an object moving in a circle is F = mv² / r = mrω².
向心加速度始终指向圆心：a = v² / r = rω²。维持物体做圆周运动所需的向心力为 F = mv² / r = mrω²。
2. Gravitational Fields | 引力场
Newton’s law of gravitation: F = G m₁ m₂ / r², where G is the gravitational constant. The gravitational field strength at a distance r from a point mass M is g = F / m = G M / r².
牛顿万有引力定律：F = G m₁ m₂ / r²，G 为引力常量。距离点质量 M 为 r 处的引力场强度为 g = F / m = G M / r²。
Gravitational potential φ = – G M / r, measured in J kg⁻¹. The gravitational potential energy of a system is U = – G M m / r.
引力势 φ = – G M / r，单位为 J kg⁻¹。系统的引力势能为 U = – G M m / r。
For a satellite in a circular orbit, the orbital speed is v = √(G M / r) and Kepler’s third law gives T² = (4π² / G M) r³.
对于圆形轨道上的卫星，轨道速度为 v = √(G M / r)，开普勒第三定律给出 T² = (4π² / G M) r³。
3. Temperature & Ideal Gases | 温度与理想气体
Temperature is a measure of the average random kinetic energy of particles. For an ideal gas, the absolute temperature T is proportional to the mean translational kinetic energy: ⟨KE⟩ = ½ m ⟨c²⟩ = (3/2) kT, where k is Boltzmann’s constant.
温度是粒子平均无规动能的量度。对于理想气体，绝对温度 T 与平均平动动能成正比：⟨KE⟩ = ½ m ⟨c²⟩ = (3/2) kT，k 为玻尔兹曼常数。
The equation of state for an ideal gas can be written as pV = nRT or pV = NkT, linking pressure p, volume V and absolute temperature T.
理想气体状态方程可写为 pV = nRT 或 pV = NkT，将压强 p、体积 V 和绝对温度 T 联系起来。
Boyle’s law (pV = constant at constant T), Charles’s law (V ∝ T at constant p) and the pressure law (p ∝ T at constant V) all follow from this equation.
波义耳定律（恒温下 pV = 常数）、查理定律（恒压下 V ∝ T）及压强定律（恒容下 p ∝ T）均可由该方程导出。
4. Thermodynamics | 热力学
The first law of thermodynamics states that the increase in internal energy ΔU equals the heat added Q plus the work done on the system W: ΔU = Q + W. Often it is written as ΔU = Q – W when W is work done by the system; be careful with sign conventions.
热力学第一定律表明，内能的增加 ΔU 等于加入的热量 Q 加上对系统做的功 W：ΔU = Q + W。当 W 表示系统对外做的功时，常写作 ΔU = Q – W；注意符号规定。
For a gas expanding at constant pressure, the work done on the gas is W = – p ΔV, so ΔU = Q – p ΔV.
对于恒压下膨胀的气体，对气体做的功为 W = – p ΔV，因此 ΔU = Q – p ΔV。
For an isothermal process ΔU = 0, so Q = – W. For an adiabatic process Q = 0, so ΔU = W. The relationship pVγ = constant applies for an ideal gas undergoing a reversible adiabatic change, where γ = Cp / Cv.
等温过程中 ΔU = 0，故 Q = – W。绝热过程中 Q = 0，故 ΔU = W。理想气体做可逆绝热变化时满足 pVγ = 常数，其中 γ = Cp / Cv。
5. Simple Harmonic Motion | 简谐运动
Simple harmonic motion (SHM) occurs when the acceleration is directly proportional to the displacement from equilibrium and directed opposite to it: a = – ω² x.
当加速度与偏离平衡位置的位移成正比且方向相反时，物体做简谐运动：a = – ω² x。
The displacement can be described by x = x₀ sin(ωt) or x = x₀ cos(ωt), where x₀ is the amplitude. The velocity is v = ± ω √(x₀² – x²) and the maximum speed is vmax = ω x₀.
位移可描述为 x = x₀ sin(ωt) 或 x = x₀ cos(ωt)，x₀ 为振幅。速度为 v = ± ω √(x₀² – x²)，最大速率为 vmax = ω x₀。
The period of SHM is T = 2π / ω. For a mass–spring system T = 2π √(m / k), and for a simple pendulum T = 2π √(l / g) (small angles).
简谐运动的周期为 T = 2π / ω。对于弹簧–质量系统有 T = 2π √(m / k)，对于单摆（小角度）有 T = 2π √(l / g)。
The total energy remains constant: E = ½ m ω² x₀² = ½ k x₀². Kinetic and potential energies interchange throughout the cycle.
总能量守恒：E = ½ m ω² x₀² = ½ k x₀²。动能和势能在周期内相互转化。
6. Electric Fields | 电场
Coulomb’s law gives the force between two point charges: F = k Q q / r² = (1 / 4π ε₀) (Q q / r²). Electric field strength is E = F / q, so for a point charge E = k Q / r².
库仑定律给出两点电荷之间的作用力：F = k Q q / r² = (1 / 4π ε₀) (Q q / r²)。电场强度为 E = F / q，因此点电荷的电场大小为 E = k Q / r²。
Electric potential at a distance r from a point charge is V = k Q / r. The work done on a charge q moving through a potential difference ΔV is W = q ΔV.
距离点电荷 r 处的电势为 V = k Q / r。电荷 q 通过电势差 ΔV 时电场力做的功为 W = q ΔV。
In a uniform electric field between parallel plates, the field strength is E = V / d, where V is the potential difference and d the plate separation.
在平行板之间的匀强电场中，场强为 E = V / d，V 为电势差，d 为板间距。
7. Capacitance | 电容
Capacitance is defined as C = Q / V, measured in farads (F). For a parallel-plate capacitor, C = ε₀ A / d, where A is the area of overlap and d the separation.
电容定义为 C = Q / V，单位为法拉 (F)。对于平行板电容器，C = ε₀ A / d，A 为正对面积，d 为板间距。
The energy stored in a capacitor is W = ½ Q V = ½ C V² = ½ Q² / C.
电容器储存的能量为 W = ½ Q V = ½ C V² = ½ Q² / C。
For a discharging RC circuit, the charge, voltage and current decay exponentially: Q = Q₀ e^{-t / RC}, V = V₀ e^{-t / RC}, and the time constant is τ = RC.
在 RC 放电回路中，电荷、电压和电流按指数衰减：Q = Q₀ e^{-t / RC}，V = V₀ e^{-t / RC}，时间常数为 τ = RC。
8. Magnetic Fields & Electromagnetic Induction | 磁场与电磁感应
A current-carrying conductor in a magnetic field experiences a force: F = B I L sin θ, where θ is the angle between the current and the magnetic field direction.
磁场中的载流导体受到力的作用：F = B I L sin θ，θ 为电流方向与磁场方向之间的夹角。
A moving charged particle experiences a magnetic force (Lorentz force): F = B q v sin θ, which provides the centripetal force for circular motion: B q v = m v² / r.
运动电荷受到洛伦兹力：F = B q v sin θ，该力为圆周运动提供向心力：B q v = m v² / r。
Magnetic flux through an area is Φ = B A cos θ. Faraday’s law states that the induced emf is proportional to the rate of change of flux linkage: ε = – N (ΔΦ / Δt), and Lenz’s law gives the direction.
通过某一面积的磁通量 Φ = B A cos θ。法拉第定律指出感应电动势与磁链的变化率成正比：ε = – N (ΔΦ / Δt)，楞次定律给出方向。
9. Alternating Currents | 交流电
An alternating voltage can be described by V = V₀ sin(ωt), and the corresponding current by I = I₀ sin(ωt). The root-mean-square (rms) values are Vrms = V₀ / √2 and Irms = I₀ / √2.
交流电压可表示为 V = V₀ sin(ωt)，相应的电流为 I = I₀ sin(ωt)。方均根值（有效值）为 Vrms = V₀ / √2 和 Irms = I₀ / √2。
The average power in a resistive AC circuit is P = Irms Vrms = Irms² R. For a pure resistor, V and I are in phase.
纯电阻交流电路中的平均功率为 P = Irms Vrms = Irms² R，电压与电流同相。
An ideal transformer satisfies Vs / Vp = Ns / Np = Ip / Is, assuming 100% efficiency.
理想变压器满足 Vs / Vp = Ns / Np = Ip / Is，假定效率为100%。
10. Quantum & Nuclear Physics | 量子与核物理
The energy of a photon is E = h f = h c / λ, where h is Planck’s constant. In the photoelectric effect, h f = Φ + KEmax, where Φ is the work function of the metal.
光子的能量为 E = h f = h c / λ，h 为普朗克常数。在光电效应中，h f = Φ + KEmax，Φ 为金属的逸出功。
The de Broglie wavelength of a particle is λ = h / p, linking wave and particle properties.
粒子的德布罗意波长为 λ = h / p，将波动性与粒子性联系起来。
Nuclear decay follows the exponential law: N = N₀ e^{-λ t}, and the activity is A = λ N. The half-life is T₁/₂ = ln 2 / λ.
核衰变遵循指数规律：N = N₀ e^{-λ t}，活度 A = λ N。半衰期为 T₁/₂ = ln 2 / λ。
Mass and energy are equivalent: E = m c². The binding energy of a nucleus is the energy required to separate it into its individual nucleons.
质量与能量等价：E = m c²。原子核的结合能是将其分裂为单个核子所需的能量。
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