📚 Formulae and Relationships | 物理公式与关系
Understanding the fundamental formulae and relationships in A-level Physics is essential for solving problems and grasping the underlying principles of the universe. These equations serve as the language of physics, enabling us to quantify motion, fields, waves, and the behaviour of matter at every scale. This article brings together the most important relationships you need to master for the CIE A-level Physics course, presented with clear explanations and correct usage conditions.
理解A-level物理中的基本公式和关系是解决问题和掌握宇宙基本原理的关键。这些方程构成了物理学的语言，让我们能够量化运动、场、波动以及物质在各个尺度上的行为。本文汇集了CIE A-level物理课程必须掌握的最重要关系，配以清晰的解释和正确的使用条件。
1. Kinematic Equations for Constant Acceleration | 匀加速运动的运动学方程
These equations apply when acceleration is uniform and motion is along a straight line. Always define a positive direction and note that displacement, velocity and acceleration are vector quantities.
这些方程适用于加速度恒定且沿直线运动的情形。务必设定正方向，注意位移、速度和加速度均为矢量。
v = u + at
v = u + at relates final velocity v, initial velocity u, acceleration a and time t. It is derived from the definition of constant acceleration.
v = u + at 描述了末速度v、初速度u、加速度a与时间t的关系，源自匀加速运动的定义。
s = ut + ½ at²
s = ut + ½ at² gives the displacement s after time t. The term ut represents the distance covered without acceleration, and ½at² is the additional displacement due to acceleration.
s = ut + ½ at² 给出t时间内的位移s。ut项代表无加速时的移动距离，½at²是加速度带来的额外位移。
v² = u² + 2as
v² = u² + 2as eliminates time and links velocities directly to displacement. It is useful when t is not given.
v² = u² + 2as 消去了时间，直接关联速度与位移，在未给出时间t时非常实用。
s = ½(u + v)t
s = ½(u + v)t uses the average velocity (u+v)/2, which is valid only under constant acceleration.
s = ½(u + v)t 使用了平均速度(u+v)/2，仅当加速度恒定时成立。
2. Newton's Laws and Momentum | 牛顿定律与动量
Newton's laws form the foundation of classical mechanics. Momentum provides a powerful approach to collisions and explosions.
牛顿定律构成经典力学的基础。动量是处理碰撞与爆炸问题的有力工具。
F = ma
F = ma is the net force causing acceleration a on mass m. The acceleration is always in the direction of the net force.
F = ma 表示净力F使质量m产生加速度a，加速度方向始终与净力方向相同。
p = mv
p = mv defines linear momentum. Momentum is a vector, so direction matters in conservation calculations.
p = mv 定义了线动量。动量是矢量，应用守恒时必须考虑方向。
F = Δp / Δt
F = Δp / Δt ate of change of momentum. For constant mass, this reduces to F = ma. Impulse is FΔt = Δp.
F = Δp / Δt 将净力与动量变化率联系起来。对于恒定质量，即简化为F = ma。冲量FΔt等于Δp。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Conservation of momentum in an isolated system: total momentum before equals total momentum after. Applies to all collisions provided no external resultant force.
动量守恒针对孤立系统：碰撞前总动量等于碰撞后总动量。只要不受外合力，任何碰撞都适用。
3. Work, Energy and Power | 功、能量与功率
Energy can be transformed between forms, but the total remains constant. These relationships allow us to quantify work done and power developed.
能量可在不同形式之间转化，但总量守恒。这些关系让我们能定量计算做功和功率。
W = F d cos θ
Work done the angle between force and displacement. When force is parallel to displacement, W = Fd.
做功 由恒力F使物体移动距离d，θ为力与位移的夹角。当力与位移平行时，W = Fd。
Eₖ = ½ m v²
Kinetic energy of an object with mass m and speed v. The work-energy theorem states that net work done equals change in kinetic energy.
动能 由质量m和速率v决定。功能定理指出，净功等于动能的变化量。
ΔEₚ = m g Δh
Change in gravitational potential energy near Earth's surface, where g is uniform. For large distances, use ΔEₚ = -GMm/r.
重力势能变化 在地表附近g均匀时适用。远距离时使用ΔEₚ = -GMm/r。
P = W / t = F v
Power is the rate of doing work. For an object moving at constant velocity v under force F, instantaneous power is P = Fv.
功率 是做功的速率。物体在力F作用下以速度v匀速移动时，瞬时功率P = Fv。
4. Circular Motion | 圆周运动
Uniform circular motion requires a centripetal force directed towards the centre. The angular quantities describe the rotation consistently.
匀速圆周运动需要指向圆心的向心力。角量可以一致地描述转动。
ω = Δθ/Δt = 2π/T = 2πf
Angular speed ω in rad s⁻¹, related to period T and frequency f. One full revolution corresponds to 2π radians.
角速度ω 单位为rad s⁻¹，与周期T和频率f有关。一整圈对应2π弧度。
v = ω r
Linear speed v equals angular speed multiplied by radius. For a given ω, v increases with r.
线速度v 等于角速度乘以半径。对于给定的ω，v随r增大而增大。
a = v²/r = ω² r
Centripetal acceleration always points towards the centre. It does not change speed but continuously alters direction.
向心加速度 始终指向圆心。它不改变速率，但持续改变速度方向。
F = m v²/r = m ω² r
Centripetal force is the net force needed to maintain circular motion. It is not a new kind of force but is provided by tension, gravity, friction, etc.
向心力 是维持圆周运动所需的净力。它并非一种新的力，而是由张力、重力、摩擦力等提供。
5. Gravitational Fields | 引力场
Newton's law of gravitation governs the attractive force between point masses. Field strength and potential describe the field's influence on a test mass.
牛顿万有引力定律支配点质量之间的吸引力。场强和势描述了场对检验质量的影响。
F = G m₁ m₂ / r²
Newton's law of gravitation: G is the gravitational constant 6.67 × 10⁻¹¹ N m² kg⁻². The force is always attractive.
万有引力定律：G为引力常量6.67×10⁻¹¹ N m² kg⁻²。力始终为吸引力。
g = F/m = G M / r²
Gravitational field strength g for a point mass M is the force per unit mass. Near Earth’s surface, g ≈ 9.81 N kg⁻¹.
引力场强g 对于点质量M指单位质量所受的力。地表附近g≈9.81 N kg⁻¹。
Vg = - G M / r
Gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point. The negative sign indicates bound systems.
引力势 是将单位检验质量从无穷远处移至该点所做的功。负号表示系统处于束缚态。
ΔEₚ = m ΔVg
Change in potential energy equals mass times the change in gravitational potential. For uniform fields, ΔEₚ = m g Δh.
势能变化 等于质量乘以引力势的变化。在均匀场中，ΔEₚ = m g Δh。
6. Electric Fields | 电场
Electric fields originate from charges and exert forces on other charges. The concepts mirror gravitational fields but with both attractive and repulsive forces.
电场由电荷产生，并对其他电荷施加力。这些概念与引力场相似，但有吸引和排斥两种力。
F = Q₁ Q₂ / (4π ε₀ r²)
Coulomb's law: ε₀ is the permittivity of free space. Like charges repel, unlike charges attract.
库仑定律：ε₀是真空中介电常数。同种电荷相斥，异种电荷相吸。
E = F / q = Q / (4π ε₀ r²)
Electric field strength E for a point charge Q. The direction is radially outward for positive Q.
电场强度E 对于点电荷Q。Q为正时方向径向向外。
E = V / d
Uniform electric field between two parallel plates separated by distance d with potential difference V. Field lines are straight and equally spaced.
匀强电场 在两块相距d、电势差为V的平行板之间。电场线为直线且等间距。
V = W / q = Q / (4π ε₀ r)
Electric potential Potential difference is the energy per unit charge. The electronvolt 1 eV = 1.6 × 10⁻¹⁹ J.
电势 对于点电荷。电势差是单位电荷的能量。电子伏特1 eV = 1.6 × 10⁻¹⁹ J。
7. Capacitance | 电容
Capacitors store energy in an electric field. The capacitance depends on geometry and the dielectric material.
电容器在电场中储存能量。电容取决于几何结构和介电材料。
C = Q / V
Capacitance is the charge stored per unit potential difference. Unit farad (F).
电容 是每单位电势差储存的电荷量，单位法拉(F)。
C = ε₀ εᵣ A / d
Parallel-plate capacitor rea, d is separation, εᵣ is relative permittivity of the dielectric. Air or vacuum εᵣ = 1.
平行板电容器：A为极板面积，d为间距，εᵣ为介质的相对介电常数。空气或真空εᵣ=1。
E = ½ Q V = ½ C V² = ½ Q² / C
Energy stored in a capacitor. The area under a Q–V graph gives the energy.
储存的能量 在电容器中。Q–V图下方的面积表示能量。
Series: 1/Ctotal = 1/C₁ + 1/C₂ , Parallel: Ctotal = C₁ + C₂
Capacitors in series reduce total capacitance; in parallel they add up.
串联电容 总电容减小；并联电容相加。
8. Current, Resistance and DC Circuits | 电流、电阻与直流电路
Electric current is the flow of charge. Resistance determines how much a component opposes current, and circuit rules govern the distribution of current and potential difference.
电流是电荷的流动。电阻决定元件对电流的阻碍程度，电路规则支配电流和电压的分配。
I = ΔQ / Δt
Electric current n is the rate of flow of charge. For electron drift: I = n A v e, where n is number density, A cross-sectional area, v drift velocity.
电流 是电荷流动的速率。对于电子漂移：I = n A v e，其中n为电子数密度，A为截面积，v为漂移速度。
V = I R
Ohm's law holds for ohmic conductors at constant temperature. R is resistance in ohms (Ω).
欧姆定律 对恒温下的欧姆导体成立。R为电阻，单位欧姆(Ω)。
R = ρ L / A
Resistivity ρ links resistance to length L and cross-sectional area A. Resistivity increases with temperature for metals.
电阻率ρ 将电阻与长度L和截面积A联系起来。金属的电阻率随温度升高而增大。
P = I V = I² R = V² / R
Electrical power dissipated in a resistor converts electrical energy to heat. Also applies to power supplied by a source.
电功率 耗散在电阻上，将电能转化为热能。也适用于电源提供的功率。
Σ Iin = Σ Iout (Kirchhoff's current law); Σ Vloop = 0 (Kirchhoff's voltage law)
Kirchhoff's laws are conservation of charge and energy applied to circuits. They are essential for solving complex networks.
基尔霍夫定律 是电荷和能量守恒在电路中的应用，对于求解复杂电路至关重要。
V = E – I r
Terminal potential difference of a source with emf E and internal resistance r when delivering current I.
路端电压 电动势E和内阻r的电源在提供电流I时的关系。
9. Electromagnetic Induction and Alternating Currents | 电磁感应与交流电
Changing magnetic flux induces an emf. Transformers and ac generators rely on this principle. Alternating currents are described by rms and peak values.
变化的磁通量会感应出电动势。变压器和交流发电机依赖这一原理。交流电由均方根值和峰值描述。
ε = - dΦ / dt
Faraday's law n. of induction. The induced emf equals the negative rate of change of magnetic flux Φ = B A cos θ. Lenz's law is represented by the minus sign.
法拉第电磁感应定律：感应电动势等于磁通量Φ = B A cos θ变化率的负值。负号体现了楞次定律。
Vₚ / Vₛ = Nₚ / Nₛ = Iₛ / Iₚ (ideal transformer)
Transformer equation links turns ratio to voltage and current ratios. An ideal transformer has no energy losses.
变压器公式 将匝数比与电压比和电流比关联。理想变压器无能量损耗。
Vrms = V₀ / √2, Irms = I₀ / √2
Root mean square values for sinusoidal ac. Rms voltage and current are the effective dc equivalents for power calculations: Pavg = Vrms Irms.
均方根值 适用于正弦交流电。均方根电压和电流是用于功率计算的有效直流等效值：Pavg = Vrms Irms。
10. Simple Harmonic Motion | 简谐运动
SHM occurs when the restoring force is proportional to displacement from equilibrium. It is characterised by sinusoidal displacement, velocity and acceleration.
当回复力与离开平衡位置的位移成正比时，发生简谐运动。其特征是位移、速度和加速度按正弦规律变化。
a = - ω² x
Defining equation for SHM ortional to negative displacement. ω is the angular frequency, related to period T = 2π/ω.
简谐运动的定义方程：加速度与负位移成正比。ω为角频率，与周期T = 2π/ω有关。
x = x₀ sin(ωt) or x = x₀ cos(ωt)
Displacement–time relation with amplitude x₀. The choice of sine or cosine depends on the starting condition.
位移-时间关系，振幅为x₀。选择正弦或余弦取决于初始条件。
v = ± ω √x₀² - x²; vmax = ω x₀
Velocity in SHM is maximum at equilibrium, zero at extremes. The ± sign indicates direction.
速度 在平衡位置最大，在端点为零。±号表示方向。
T = 2π √m/k for mass-spring; T = 2π √l/g for simple pendulum
Period of a mass-spring system depends on mass and spring constant. The simple pendulum period depends on length and gravitational field strength (small angles only).
周期 弹簧振子取决于质量和弹簧常数。单摆周期取决于摆长和重力场强（仅小角度）。
Etotal = ½ m ω² x₀²
Total mechanical energy in SHM remains constant, interchanging between kinetic and potential forms.
总机械能 在简谐运动中保持不变，在动能和势能之间转化。
11. Thermal Physics and Ideal Gases | 热物理与理想气体
Macroscopic properties of gases are linked to microscopic particle behaviour. The kinetic theory model explains pressure, temperature and internal energy.
气体的宏观性质与微观粒子行为相联。分子运动论模型解释了压强、温度和内能。
pV = n R T, pV = N k T
Ideal gas equation sure, V volume, n number of moles, T absolute temperature. R = 8.31 J K⁻¹ mol⁻¹, k = R / NA, NA = 6.02 × 10²³ mol⁻¹.
理想气体状态方程：p压强，V体积，n摩尔数，T绝对温度。R=8.31 J K⁻¹ mol⁻¹，k = R/NA，NA=6.02×10²³ mol⁻¹。
Eₖ_avg = (3/2) k T
Average kinetic energy of a monatomic gas molecule is directly proportional to absolute temperature. Each degree of freedom contributes (1/2)kT.
单原子气体分子的平均动能 与绝对温度成正比。每个自由度贡献(1/2)kT。
U = (f/2) n R T
Internal energy ure and the number of degrees of freedom f. For monatomic f=3, diatomic at moderate T f=5.
内能 对于理想气体仅取决于温度和自由度f。单原子f=3，双原子在适中温度下f=5。
ΔU = q + W
First law of thermodynamics ded to the system plus work done on the system. W = - p ΔV for a gas at constant pressure.
热力学第一定律：内能增加等于系统吸收的热量加上对系统做的功。恒压下气体W = -pΔV。
12. Quantum and Nuclear Physics | 量子物理与核物理
Particles exhibit wave-like properties, and energy is quantised in photons. Nuclear processes involve mass-energy equivalence and exponential decay.
粒子表现出波动性，光子能量是量子化的。核过程涉及质能等价和指数衰变。
E = h f = h c / λ
Photon energy portional to frequency, h = 6.63 × 10⁻³⁴ J s. Higher frequency means larger photon energy.
光子能量 与频率成正比，h = 6.63×10⁻³⁴ J s。频率越高，光子能量越大。
