📚 Formulae and Relationships | 公式与关系
This revision guide summarises the key formulae and relationships required for CIE A Level Physics. Each equation is paired with the conditions under which it applies, because using a formula outside its range of validity is one of the most common errors in calculations.
本复习指南总结了 CIE A Level 物理中需要掌握的关键公式与关系。每个公式都附带其适用条件，因为在计算中超出公式的有效范围是最常见的错误之一。
The emphasis is on understanding what the symbols mean, how the equation is derived, and how to identify the correct relationship in written, graphical and experimental forms.
重点是理解各符号的含义、公式的推导方式，并能在文字、图像和实验三种形式中识别正确的关系。
1. Physical Quantities and SI Units | 物理量与 SI 单位
Every equation in CIE A Level Physics is a relationship between measurable quantities. A formula is only useful when all quantities are expressed in consistent SI units: metre, kilogram, second, ampere, kelvin, mole and candela.
CIE A Level 物理中的每个公式都是可测量物理量之间的关系。只有当所有物理量都采用一致的 SI 单位（米、千克、秒、安培、开尔文、摩尔和坎德拉）时，公式才有意义。
Derived units are constructed from base units. For example, the newton for force is 1 N = 1 kg m s⁻², and the joule for energy is 1 J = 1 N m = 1 kg m² s⁻². Checking unit homogeneity is a powerful way to catch algebraic mistakes.
导出单位由基本单位组合而成。例如，力的单位牛顿为 1 N = 1 kg m s⁻²，能量的单位焦耳为 1 J = 1 N m = 1 kg m² s⁻²。检验单位齐次性是发现代数错误的有效方法。
ρ = m / V ; p = F / A
Density is mass per unit volume, and pressure is normal force per unit area. Both relationships are definitions, so they apply in all situations.
密度是单位体积的质量，压强是单位面积上的法向力。这两种关系都是定义，因此适用于所有情况。
2. Kinematics: Motion in a Straight Line | 直线运动学
For motion with constant acceleration along a straight line, the three principal equations of motion relate initial velocity u, final velocity v, acceleration a, displacement s and time t.
对于沿直线且加速度恒定的运动，三个主要运动学方程联系了初速度 u、末速度 v、加速度 a、位移 s 和时间 t。
v = u + at
s = ut + ½ at²
v² = u² + 2as
These equations are only valid when acceleration is constant. For free fall near Earth’s surface, acceleration a is replaced by g ≈ 9.81 m s⁻², with the direction chosen as positive or negative depending on the coordinate system.
这些方程仅在加速度恒定时有效。对于地球表面附近的自由落体，加速度 a 用 g ≈ 9.81 m s⁻² 替换，并根据所选坐标系取正方向或负方向。
Projectile motion is treated by resolving into independent horizontal and vertical components. Horizontally there is no acceleration, while vertically there is constant acceleration due to gravity.
抛体运动通过分解为独立的水平分量和竖直分量来处理。水平方向没有加速度，竖直方向则存在由重力引起的恒定加速度。
3. Dynamics: Newton’s Laws and Forces | 动力学：牛顿定律与力
Newton’s second law states that the resultant force on an object is equal to the rate of change of its momentum. For constant mass, this simplifies to the familiar equation:
牛顿第二定律指出，作用在物体上的合力等于其动量的变化率。在质量恒定的情况下，该定律简化为熟悉的公式：
ΣF = ma
Weight is the gravitational force on a mass near a planet’s surface, given by W = mg. Friction between two solid surfaces is often modelled by F = μR, where μ is the coefficient of friction and R is the normal contact force.
重量是行星表面附近作用在质量上的引力，由 W = mg 给出。两个固体表面之间的摩擦通常用 F = μR 建模，其中 μ 是摩擦因数，R 是法向接触力。
An object is in equilibrium when the vector sum of all forces is zero and the sum of all moments about any point is zero. This gives the two conditions that must be satisfied simultaneously.
当所有力的矢量和为零，且对任意点的力矩和为零时，物体处于平衡状态。这两项条件必须同时满足。
4. Work, Energy and Power | 功、能量与功率
Work is done when a force moves its point of application. The work done by a constant force at angle θ to the displacement is:
当力的作用点发生移动时，力就对物体做功。恒力与位移成 θ 角时所做的功为：
W = Fs cos θ
Kinetic energy and gravitational potential energy are the main mechanical energy stores. In a uniform gravitational field, the change in GPE is mgh, where h is the vertical height change.
动能和重力势能是主要的机械能储存形式。在均匀引力场中，重力势能的变化为 mgh，其中 h 是竖直高度的变化。
KE = ½ mv² ; ΔGPE = mgh
Power is the rate of doing work. When a force F moves an object at constant velocity v in the same direction, the power delivered is P = Fv.
功率是做功的速率。当力 F 使物体以恒定速度 v 沿同一方向运动时，传递的功率为 P = Fv。
P = W / t ; P = Fv
The principle of conservation of energy states that energy cannot be created or destroyed, only transferred between stores. Efficiency is therefore useful output energy divided by total input energy.
能量守恒定律指出，能量不能凭空产生或消失，只能在不同的能量储存之间转移。因此效率等于有用的输出能量除以总输入能量。
5. Momentum and Impulse | 动量与冲量
Linear momentum is the product of mass and velocity. Impulse is the change in momentum caused by a resultant force acting over a time interval.
线动量是质量与速度的乘积。冲量是合力在一段时间内作用所产生的动量变化。
p = mv ; Δp = FΔt = mv − mu
In a closed system with no external resultant force, total momentum is always conserved. This principle is used to analyse collisions and explosions.
在没有外力合力的封闭系统中，总动量始终守恒。该原理用于分析碰撞和爆炸。
Momentum conservation is a vector relationship, so directions must be assigned positive and negative signs before solving problems. In elastic collisions kinetic energy is also conserved; in inelastic collisions it is not.
动量守恒是矢量关系，因此解题前必须对方向规定正负号。在弹性碰撞中动能也守恒；在非弹性碰撞中动能不守恒。
6. Circular Motion and Gravitational Fields | 圆周运动与引力场
An object moving in a circle at constant speed has a changing velocity because its direction changes. Its angular velocity, centripetal acceleration and centripetal force are related by:
物体以恒定速率做圆周运动时，由于方向不断改变，速度是变化的。其角速度、向心加速度和向心力之间的关系为：
ω = Δθ / Δt ; v = ωr ; a = v² / r = ω²r
F = mv² / r = mrω²
Newton’s law of gravitation gives the force between two point masses. The gravitational field strength at a distance r from a point mass M is obtained by dividing the force by a small test mass m.
牛顿万有引力定律给出两个质元之间的引力。距质元 M 距离为 r 处的引力场强度可通过力除以检验质量 m 得到。
F = Gm₁m₂ / r² ; g = GM / r²
For a satellite in a circular orbit, gravitational force provides the centripetal force. Equating these two expressions leads to Kepler’s third law for circular orbits: T² ∝ r³.
对于沿圆轨道运行的卫星，引力提供向心力。将两者相等即可得到圆轨道下的开普勒第三定律：T² ∝ r³。
The gravitational potential at a point is the work done per unit mass in bringing a small mass from infinity to that point:
某点的引力势是将单位质量从无穷远移到该点所做的功：
φ = −GM / r
7. Electric Fields, Current and Resistance | 电场、电流与电阻
Electric field strength is the force per unit positive charge. In a uniform electric field between two parallel plates separated by distance d with potential difference V, the field strength is constant.
电场强度是单位正电荷所受的力。在两个平行板之间、间距为 d、电势差为 V 的均匀电场中，电场强度是恒定的。
E = F / q ; E = V / d
Electric current is the rate of flow of charge. Resistance is defined as the ratio of potential difference to current, and resistivity links resistance to the dimensions of a conductor.
电流是电荷的流动速率。电阻定义为电势差与电流之比，而电阻率则将电阻与导体的尺寸联系起来。
I = ΔQ / Δt ; V = IR ; R = ρL / A
Electrical power can be expressed in three equivalent forms using Ohm’s law. In a circuit with emf E and internal resistance r, the terminal potential difference is less than the emf when current flows.
电功率可以利用欧姆定律表示为三种等价形式。在电动势为 E、内阻为 r 的电路中，有电流流过时端电势差小于电动势。
P = VI = I²R = V² / R ; V = E − Ir
A potential divider produces a fraction of the input voltage across one resistor in series. The ratio depends only on the resistance values, not on the current.
分压器在串联的某个电阻上产生输入电压的一部分。该比例只取决于电阻值，与电流无关。
Vout = Vin R₂ / (R₁ + R₂)
8. Capacitance | 电容
Capacitance is the charge stored per unit potential difference across a capacitor. For a parallel-plate capacitor, capacitance is proportional to plate area and inversely proportional to plate separation.
电容是电容器每单位电势差所储存的电荷量。对于平行板电容器，电容与板面积成正比，与板间距成反比。
C = Q / V ; C = ε₀A / d
The energy stored in a charged capacitor can be written in three equivalent ways. It represents the work done in moving charge onto the plates.
充电电容器中储存的能量有三种等价表达式。它表示将电荷搬运到极板上所做的功。
E = ½ QV = ½ CV² = Q² / 2C
For capacitors in parallel, the total capacitance increases, while for capacitors in series the total capacitance is smaller than the smallest individual value.
电容器并联时总电容增大，而串联时总电容小于其中最小的单个电容。
C = C₁ + C₂ + … ; 1/C = 1/C₁ + 1/C₂ + …
During discharge through a resistor, charge, current and potential difference all decrease exponentially. The time constant RC is the time for the charge to fall to about 37% of its initial value.
通过电阻放电时，电荷、电流和电势差都按指数规律减小。时间常数 RC 是电荷下降到初始值约 37% 所需的时间。
Q = Q₀ exp(−t / RC) ; V = V₀ exp(−t / RC)
9. Magnetic Fields and Electromagnetic Induction |
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