📚 A2 Physics: Formula Compendium | A2 物理：公式汇总手册
This comprehensive formula compendium covers the essential equations for A2 Physics. Each section presents key relationships in the order they appear in typical syllabuses, with clear explanations to reinforce understanding and aid revision.
这份全面的公式手册涵盖了 A2 物理的基本方程。每个小节按照典型考纲的顺序列出关键关系式，并提供清晰的解释，以加深理解并辅助复习。
1. Circular Motion | 圆周运动
Angular displacement θ is measured in radians, where one complete revolution equals 2π rad. The angular velocity ω is the rate of change of angular displacement.
角位移 θ 以弧度为单位，一整圈等于 2π 弧度。角速度 ω 是角位移的变化率。
ω = Δθ / Δt = 2π / T = 2πf
For an object moving in a circle of radius r at constant speed v, the linear velocity is related to angular velocity by v = ωr. The centripetal acceleration always points towards the centre and has magnitude:
对于以恒定速率 v 在半径为 r 的圆上运动的物体，线速度与角速度的关系为 v = ωr。向心加速度始终指向圆心，大小为：
a = v²/r = ω²r
According to Newton’s second law, a resultant centripetal force is required to maintain circular motion:
根据牛顿第二定律，维持圆周运动需要一个指向圆心的合力：
F = mv²/r = mω²r
2. Gravitational Fields | 引力场
A gravitational field is a region where a mass experiences a force. The gravitational field strength g at a point is the force per unit mass on a small test mass placed there.
引力场是质量受到力的区域。重力场强度 g 定义为在该点放置的小测试质量所受的力与质量之比。
g = F/m
Newton’s law of universal gravitation gives the force between two point masses M and m separated by distance r:
牛顿万有引力定律给出了两个点质量 M 和 m 相距 r 时的力：
F = −GMm/r²
The gravitational field strength at a distance r from the centre of a spherical mass M is:
距离球形质量 M 中心 r 处的重力场强度为：
g = GM/r²
The gravitational potential V at a point is the work done per unit mass in bringing a test mass from infinity to that point:
引力势 V 是将单位质量从无穷远移至该点所做的功：
V = −GM/r
For satellite motion, the centripetal force is provided by gravity. Equating forces yields the orbital period T and escape velocity vₑ:
对于卫星运动，引力提供向心力。令两力相等可以导出轨道周期 T 和逃逸速度 vₑ：
T² = 4π²r³/(GM) vₑ = √(2GM/r)
3. Thermal Physics & Ideal Gases | 热物理与理想气体
The internal energy of a system is the sum of the random kinetic and potential energies of its particles. The first law of thermodynamics relates the increase in internal energy ΔU, heat supplied Q, and work done on the system W.
系统的内能是其粒子无规运动的动能和势能之和。热力学第一定律将内能增量 ΔU、供热量 Q 和对系统做功 W 联系起来。
ΔU = Q + W (work done on the system)
The ideal gas equation in terms of the amount n (in moles) and the universal gas constant R is:
理想气体状态方程，以摩尔数 n 和普适气体常数 R 表示：
pV = nRT
Expressed in terms of the number of molecules N and the Boltzmann constant k, it becomes:
用分子数 N 和玻尔兹曼常数 k 表示为：
pV = NkT
The kinetic theory gives the pressure of an ideal gas in terms of the mean-square speed ⟨c²⟩ of its molecules of mass m:
分子动理论用分子质量 m 和方均速率 ⟨c²⟩ 给出理想气体的压强：
pV = ⅓ N m ⟨c²⟩
The average translational kinetic energy of a molecule is related to absolute temperature:
分子的平均平动动能与绝对温度的关系为：
½m⟨c²⟩ = (3/2) kT
4. Simple Harmonic Motion | 简谐运动
Simple harmonic motion (SHM) occurs when the acceleration a is proportional to the displacement x from the equilibrium position and directed towards it.
简谐运动 (SHM) 发生时，加速度 a 与相对平衡位置的位移 x 成正比，且方向指向平衡位置。
a = −ω²x
The displacement as a function of time can be written using sine or cosine, depending on initial conditions:
位移随时间变化的函数可用正弦或余弦表示，取决于初始条件：
x = A sin(ωt) or x = A cos(ωt)
The velocity v and acceleration a are given by the time derivatives:
速度 v 和加速度 a 由时间导数给出：
v = ± ω √(A² − x²) a = −ω²x
The period T for a mass-spring system (mass m, spring constant k) and for a simple pendulum (length L) are:
质量弹簧系统（质量 m，劲度系数 k）和单摆（摆长 L）的周期 T 分别为：
T = 2π √(m/k) T = 2π √(L/g)
The total energy of an undamped SHM system is constant and is the sum of kinetic and potential energies at any point:
无阻尼简谐运动系统的总能量恒定，等于任一点动能与势能之和：
E_total = ½ k A² (for mass-spring)
5. Electric Fields | 电场
An electric field exists around a charged object, exerting a force on other charges. The electric field strength E is the force per unit positive charge.
电场存在于带电体周围，并对其他电荷施加力。电场强度 E 定义为单位正电荷所受的力。
E = F/q
Coulomb’s law gives the force between two point charges Q and q separated by distance r:
库仑定律给出两个点电荷 Q 和 q 相距 r 时的力：
F = (1/(4πε₀)) Qq / r²
The electric field strength at a distance r from a point charge Q is therefore:
因此，距离点电荷 Q 为 r 处的电场强度为：
E = (1/(4πε₀)) Q / r²
For a uniform electric field between parallel plates separated by distance d with potential difference V:
对于相距 d、电势差为 V 的平行板之间的匀强电场：
E = V/d
The electric potential V at a point is the work done per unit charge in bringing a test charge from infinity to that point:
电势 V 是将单位正电荷从无穷远移至该点所做的功：
V = (1/(4πε₀)) Q / r
6. Capacitance | 电容
Capacitance C is defined as the ratio of the charge Q stored on a capacitor to the potential difference V across its plates.
电容 C 定义为电容器储存的电荷 Q 与其两极板间电势差 V 之比。
C = Q/V
For a parallel-plate capacitor with plate area A, separation d, and dielectric permittivity ε, the capacitance is:
对于极板面积为 A、间距为 d、介质介电常数为 ε 的平行板电容器，电容为：
C = εA/d where ε = εᵣ ε₀
The energy stored in a charged capacitor can be expressed in three equivalent forms:
已充电电容器储存的能量可用三种等价形式表示：
W = ½ QV = ½ CV² = Q²/(2C)
During discharge through a resistor R, the charge Q and current I decay exponentially with time constant τ = RC:
通过电阻 R 放电时，电荷 Q 和电流 I 以时间常数 τ = RC 呈指数衰减：
Q = Q₀ e^(−t/RC) I = I₀ e^(−t/RC)
For charging, the voltage across the capacitor rises according to V = V₀(1 − e^(−t/RC)). The time constant is the time for the voltage to reach approximately 63% of its final value.
充电时，电容器两端电压按 V = V₀(1 − e^(−t/RC)) 上升。时间常数是电压达到其最终值约 63% 所需的时间。
7. Magnetic Fields & Forces | 磁场与力
A magnetic field exerts a force on a moving charge. The force F on a charge q moving with velocity v at an angle θ to a uniform magnetic field B is given by:
磁场对运动电荷施加力。电荷 q 以速度 v 与匀强磁场 B 成角度 θ 运动时所受的力为：
F = q v B sinθ
For a current-carrying conductor of length L placed in a uniform magnetic field, the force is:
对于放置在匀强磁场中的长度为 L 的载流导体，力为：
F = B I L sinθ
When a charged particle moves perpendicularly to a uniform magnetic field, it travels in a circular path. Equating magnetic force to centripetal force yields the radius r:
当带电粒子垂直于匀强磁场运动时，它做圆周运动。令磁力等于向心力可得半径 r：
r = mv/(Bq)
A Hall probe measures magnetic flux density using the Hall voltage V_H developed across a conductor of thickness t carrying current I in a field B:
霍尔探头利用厚度为 t 的导体在磁场 B 中通以电流 I 时产生的霍尔电压 V_H 来测量磁通密度：
V_H = B I /(n q t)
(where n is the number density of charge carriers).
（其中 n 是载流子的数密度）。
8. Electromagnetic Induction | 电磁感应
When the magnetic flux Φ through a circuit changes, an electromotive force (e.m.f.) is induced. Magnetic flux is defined as:
当穿过回路的磁通量 Φ 发生变化时，会感应出电动势。磁通量定义为：
Φ = B A cosθ
Faraday’s law states that the magnitude of the induced e.m.f. equals the rate of change of flux linkage (NΦ):
法拉第定律指出，感应电动势的大小等于磁链 (NΦ) 的变化率：
ε = −N ΔΦ/Δt
Lenz’s law gives the direction of the induced current: it opposes the change producing it. The motional e.m.f. for a conductor of length L moving at speed v perpendicularly to a magnetic field B is:
楞次定律给出了感应电流的方向：它阻碍引起它的变化。长度为 L 的导体以速度 v 垂直于磁场 B 运动时的动生电动势为：
ε = B L v
Transformers step voltage up or down according to the turns ratio. For an ideal transformer:
变压器根据匝数比升高或降低电压。对于理想变压器：
V_s / V_p = N_s / N_p = I_p / I_s
9. Alternating Currents | 交流电
An alternating current (a.c.) varies sinusoidally with time. The instantaneous voltage V and current I can be written as:
交流电 (a.c.) 随时间按正弦变化。瞬时电压 V 和电流 I 可写为：
V = V₀ sin(ωt) I = I₀ sin(ωt)
The root-mean-square (r.m.s.) values for a sinusoidal waveform relate to the peak values by:
正弦波形的均方根 (r.m.s.) 值与峰值的关系为：
V_rms = V₀/√2 I_rms = I₀/√2
The average power in an a.c. circuit with a purely resistive load is given by:
纯电阻负载交流电路中的平均功率为：
P_avg = V_rms I_rms = I²_rms R
For circuits containing inductors and capacitors, reactance X and impedance Z describe the opposition to current flow. Inductive reactance X_L and capacitive reactance X_C are:
对于包含电感和电容的电路，感抗 X_L 和容抗 X_C 描述对电流的阻碍作用：
X_L = ωL = 2πfL X_C = 1/(ωC) = 1/(2πfC)
The phase angle φ between voltage and current in an LCR series circuit satisfies:
LCR 串联电路中电压和电流之间的相位角 φ 满足：
tanφ = (X_L − X_C) / R
10. Quantum Physics | 量子物理
The energy of a photon is directly proportional to its frequency f and inversely proportional to its wavelength λ. The Planck constant h is the proportionality factor.
光子的能量与其频率 f 成正比，与其波长 λ 成反比。普朗克常数 h 是比例因子。
E = hf = hc/λ
The photoelectric effect is explained by Einstein’s equation, where Φ is the work function of the metal and K_max is the maximum kinetic energy of the emitted electrons:
光电效应由爱因斯坦方程解释，其中 Φ 是金属的逸出功，K_max 是出射电子的最大动能：
K_max = hf − Φ
The threshold frequency f₀ is related to the work function by Φ = hf₀. The stopping potential V_s is given by:
截止频率 f₀ 与逸出功的关系为 Φ = hf₀。遏止电势 V_s 由下式给出：
e V_s = K_max
According to de Broglie, all particles have a wavelength associated with their momentum p:
根据德布罗意，所有粒子都具有与其动量 p 相关的波长：
λ = h/p = h/(mv)
11. Nuclear Physics | 核物理
The nuclear radius R depends on the mass number A raised to the one-third power:
原子核半径 R 依赖于质量数 A 的三分之一次方：
R = r₀ A^(1/3) (r₀ ≈ 1.2 fm)
Radioactive decay follows an exponential law. The number N of undecayed nuclei after time t is:
放射性衰变遵循指数规律。时间 t 后未衰变的原子核数 N 为：
N = N₀ e^(−λt)
The decay constant λ is related to the half-life T₁/₂ by:
衰变常数 λ 与半衰期 T₁/₂ 的关系为：
λ = ln2 / T₁/₂
The activity A of a sample is the rate of decay:
样品的活度 A 是衰变率：
A = λN
In nuclear reactions, the energy released (Q-value) can be calculated from the mass defect Δm using Einstein’s mass-energy relation:
在核反应中，释放的能量（Q 值）可通过质量亏损 Δm 利用爱因斯坦质能方程计算：
E = Δm c²
12. Medical Imaging (Optional A2 Topic) | 医学成像（选学主题）
Ultrasound imaging relies on the pulse-echo principle. The distance d to a reflecting surface is determined from the time delay Δt and the speed of sound c in the medium:
超声成像依赖于脉冲回波原理。到反射面的距离 d 由时间延迟 Δt 和介质中的声速 c 确定：
d = c Δt / 2
The acoustic impedance Z of a material determines the fraction of ultrasound intensity reflected at a boundary. For normal incidence, the intensity reflection coefficient α is:
材料的声阻抗 Z 决定了在边界处反射的超声强度比例。对于垂直入射，强度反射系数 α 为：
α = (Z₂ − Z₁)²/(Z₂ + Z₁)² where Z = ρc
In X-ray imaging, the half-value thickness x₁/₂ characterises the attenuation of a beam in matter according to I = I₀ e^(−μx), where μ is the linear attenuation coefficient:
在 X 射线成像中，半值厚度 x₁/₂ 表征射线在物质中的衰减，遵循 I = I₀ e^(−μx)，其中 μ 是线性衰减系数：
x₁/₂ = ln2 / μ
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