📚 Year 13 CIE Physics Formula and Theorem Quick Reference Handbook | Year 13 CIE 物理：公式定理速查手册
This quick reference handbook compiles the essential formulas, theorems and key relationships for CIE A Level Physics (Year 13). Each section presents the core concepts with their mathematical expressions, the conditions under which they apply, and the traps that examiners most often set, making it ideal for last-minute revision and for building a single-page summary before Paper 4 and Paper 5.
本速查手册汇编了 CIE A Level 物理（Year 13）的核心公式、定理和关键关系。每一部分都给出核心概念及其数学表达式、适用条件，以及考官最常设置的陷阱，非常适合考前最后复习，也方便你整理成 Paper 4 与 Paper 5 考前的一页纸总结。
1. Circular Motion | 圆周运动
When an object moves in a circle at constant speed, the magnitude of its velocity is constant but its direction changes continuously. This continuous change of direction produces an acceleration directed towards the centre of the circle, called the centripetal acceleration.
当物体以恒定速率做圆周运动时，速度的大小不变，但方向不断改变。这种方向的持续改变产生指向圆心的加速度，称为向心加速度。
Angular velocity is the rate of change of angular displacement, measured in radians per second (rad s⁻¹). The linear speed of a point on the circle is related to angular velocity through the radius.
角速度是角位移随时间的变化率，单位为弧度每秒（rad s⁻¹）。圆周上某点的线速度通过半径与角速度相联系。
ω = Δθ / Δt v = rω T = 2π / ω = 1 / f
Centripetal acceleration and the resultant centripetal force are given by:
向心加速度与合向心力由下式给出：
a = v² / r = rω² F = mv² / r = mrω²
Centripetal force is not a new kind of force. It is the resultant of the real forces acting towards the centre, such as tension, friction, gravitational attraction or the normal contact force.
向心力不是一种新的力。它是实际作用力（如张力、摩擦力、引力或法向接触力）中指向圆心的合力。
| Situation 情境 | Force providing centripetal force 提供向心力的力 | Key equation 关键方程 |
| Car on a flat bend 平直弯道上的汽车 | Friction 摩擦力 | μmg = mv² / r |
| Car on a banked track 倾斜弯道 | Horizontal component of the normal force 法向力的水平分量 | mg tan θ = mv² / r |
| Conical pendulum 圆锥摆 | Horizontal component of tension 张力的水平分量 | T sin θ = mrω² |
| Satellite in orbit 轨道上的卫星 | Gravitational force 引力 | GMm / r² = mv² / r |
Two exam habits are essential: convert revolutions per minute into rad s⁻¹ by multiplying by 2π/60, and remember that in a vertical circle the tension is greatest at the lowest point because it must support the weight as well as provide the centripetal force.
两个考试习惯必须养成：把每分钟转数换算为 rad s⁻¹ 时要乘以 2π/60；在竖直圆周运动中，最低点张力最大，因为它既要平衡重力，又要提供向心力。
2. Gravitational Fields | 引力场
Newton's law of gravitation describes the attractive force between two point masses. The force is always attractive, acts along the line joining the masses, and follows an inverse square law.
牛顿万有引力定律描述了两个质点之间的吸引力。该力总是吸引力，沿两质点的连线方向，并遵循平方反比定律。
F = G m₁ m₂ / r² G = 6.67 × 10⁻¹¹ N m² kg⁻²
Gravitational field strength is the force per unit mass, and outside a spherical body of mass M it behaves as if all the mass were concentrated at the centre.
引力场强度是单位质量所受的引力，在质量为 M 的球体外部，其效果等价于全部质量集中在球心。
g = F / m = GM / r²
Gravitational potential is the work done per unit mass in bringing a small test mass from infinity to that point, so it is negative and tends to zero at infinity.
引力势是把单位质量的小试探质量从无穷远处移到该点所做的功，因此它为负值，并在无穷远处趋近于零。
V = - GM / r U = mV = - GMm / r
ar orbit, gravity supplies the centripetal force, which leads to the orbital speed and to Kepler's third law.
真正有物理意义的是电势差，把质量 m 在引力场中从一点移到另一点所做的功为 W = mΔV。对于做圆周运动的卫星，引力提供向心力，由此可推出轨道速度和开普勒第三定律。
v = √GM / r v = √gR T² = (4π² / GM) r³
The near-surface form v = √gR applies only when the orbital radius is approximately equal to the planetary radius R. A geostationary orbit must have a period of exactly 24 hours, lie in the equatorial plane and rotate in the same direction as the Earth.
近地形式 v = √gR 仅适用于轨道半径近似等于行星半径 R 的情形。地球同步轨道必须满足周期恰为 24 小时、位于赤道平面内、且转动方向与地球自转相同。
3. Simple Harmonic Motion | 简谐运动
Simple harmonic motion (SHM) occurs when the restoring force is proportional to the displacement from equilibrium and directed towards the equilibrium position. Acceleration is always in antiphase with displacement.
简谐运动发生在回复力与位移成正比、且方向指向平衡位置时。加速度与位移始终反相。
a = - ω² x
x₀, where the choice of sine or cosine simply depends on the initial conditions at t = 0.
定义方程的解是振幅为 x₀ 的正弦式位移函数，选用正弦还是余弦只取决于 t = 0 时的初始条件。
x = x₀ sin(ωt) or x = x₀ cos(ωt)
Velocity and the maximum values follow directly, and these are the relationships most often needed in structured questions.
速度及最大值可直接推出，这些关系是结构化作答题中最常用的。
v = ± ω √x₀² - x² vmax = ω x₀ amax = ω² x₀
The period of a mass-spring system and of a simple pendulum demonstrates that the period is independent of amplitude, which is the defining feature of isochronous oscillation.
弹簧振子和单摆的周期表明周期与振幅无关，这正是等时性振荡的本质特征。
T = 2π √m / k T = 2π √l / g
Energy in SHM oscillates between kinetic and potential forms while the total energy remains constant. The total energy is proportional to the square of the amplitude.
简谐运动中的能量在动能与势能之间往复转化，但总能量保持不变。总能量与振幅的平方成正比。
Etotal = ½ m ω² x₀² KE = ½ m ω² (x₀² - x²)
In an exam, sketch energy against displacement as two parabolas that add to a constant total, and remember that damping reduces the amplitude while slightly changing the period.
考试中画能量-位移图像时，应画成两条抛物线，其和等于恒定总能量；同时要记住阻尼会减小振幅，并略微改变周期。
4. Thermal Physics and Ideal Gases | 热物理与理想气体
Macroscopic thermal behaviour is described by heat capacity, latent heat and the ideal gas equation. Temperature must always be converted into kelvin before substitution into the gas laws.
宏观热学行为由热容、潜热和理想气体状态方程描述。代入气体定律前，温度必须换算为开尔文。
T / K = θ / °C + 273.15
The energy required to change temperature or phase is given by the specific heat capacity and the specific latent heat.
改变温度或发生相变所需的能量由比热容和比潜热给出。
Q = mcΔθ Q = mL
Specific heat capacity is the energy needed to raise the temperature of 1 kg by 1 K; specific latent heat is the energy needed to change the phase of 1 kg at constant temperature.
比热容是使 1 kg 物质温度升高 1 K 所需的能量；比潜热是在恒温下使 1 kg 物质发生相变所需的能量。
The ideal gas equation links pressure, volume, amount of substance and temperature, and can also be written in terms of the number of molecules using the Boltzmann constant.
理想气体状态方程把压强、体积、物质的量和温度联系起来，也可用玻尔兹曼常量写成分子数的形式。
pV = nRT pV = NkT
R = 8.31 J mol⁻¹ K⁻¹ k = 1.38 × 10⁻²³ J K⁻¹ NA = 6.02 × 10²³ mol⁻¹ R = NAk
nstant: pV = constant (Boyle), V/T = constant (Charles) and p/T = constant (pressure law).
三大气体定律是理想气体状态方程在固定某一个变量时得到的特例：pV = 常量（玻意耳定律）、V/T = 常量（查理定律）、p/T = 常量（压强定律）。
The first law of thermodynamics states that the increase in internal energy equals the energy supplied by heating plus the work done on the system. For a gas expanding at constant pressure, the work done by the gas is pΔV.
热力学第一定律指出，内能增量等于外界供热加上外界对系统所做的功。对于等压膨胀的气体，气体对外做功为 pΔV。
ΔU = q + W W = - pΔV (work done on the gas 外界对气体做功)
5. Kinetic Theory and Internal Energy | 分子动理论与内能
The kinetic theory model explains the macroscopic gas laws in terms of the random motion of molecules. Its assumptions are the starting point for every derivation question in this topic.
分子动理论用分子的无规则运动解释宏观气体定律。它的假设是本专题所有推导题的出发点。
The standard assumptions are: the gas contains a very large number of identical molecules; the molecules move randomly in all directions; the volume of the molecules is negligible compared with the volume of the container; collisions between molecules and with the walls are perfectly elastic and of negligible duration; and no forces act between molecules except during collisions.
标准假设包括：气体含大量全同分子；分子向各个方向无规则运动；分子自身体积远小于容器体积；分子之间及与器壁的碰撞是完全弹性的且持续时间极短；除碰撞瞬间外分子间无作用力。
Combining Newtonian mechanics with these assumptions gives the pressure of an ideal gas in terms of the mean square speed of the molecules.
把这些假设与牛顿力学结合，可把理想气体的压强用分子的均方速率表示出来。
pV = ⅓ N m <c²>
average translational kinetic energy of a molecule, which depends only on the absolute temperature.
把它与 pV = NkT 比较，可得到分子的平均平动动能，它只与绝对温度有关。
½ m <c²> = (3/2) kT
For an ideal gas, the internal energy is entirely kinetic, because there are no intermolecular potential energy changes, so the internal energy of n moles is:
对理想气体而言，内能完全是动能，因为分子间势能不变，因此 n 摩尔理想气体的内能为：
U = (3/2) NkT = (3/2) nRT
Brownian motion provides direct evidence for the random motion of molecules, since smoke particles are seen to move irregularly after being struck unevenly by gas molecules.
布朗运动为分子的无规则运动提供了直接证据：烟雾颗粒因受到气体分子不均匀的撞击而做无规则运动。
6. Electric Fields | 电场
An electric field is a region in which a charged particle experiences a force. Field strength is defined as the force per unit positive charge, and it is a vector.
电场是带电粒子会受到力的区域。电场强度定义为单位正电荷所受的力，它是一个矢量。
E = F / Q
Between two parallel plates the field is uniform, and the field strength is related to the potential difference and the separation of the plates.
在两块平行板之间电场是匀强的，其场强与板间电势差和板间距有关。
E = V / d
For point charges, Coulomb's law gives the force and the inverse square field of a point charge, with the constant
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