📚 Year 13 AQA Physics: Quick-Reference Formula & Theorem Handbook | AQA 物理 Year 13：公式定理速查手册
This handbook compiles essential formulas and theorems from the Year 13 AQA Physics syllabus, providing a quick reference for revision. Each formula is accompanied by a concise explanation in both English and Chinese.
本手册汇编了 Year 13 AQA 物理教学大纲中的核心公式与定理，为复习提供快速参考。每个公式均配有中英双语解释。
1. Circular Motion | 圆周运动
ω = 2πf = 2π / T
Angular speed ω (rad s−1) is the rate of change of angle; it links frequency f and period T of a rotating object.
角速度 ω（单位为 rad s−1）是角度的变化率；它将旋转物体的频率 f 和周期 T 关联起来。
v = ωr
Linear speed v of a point at radius r is directly proportional to the angular speed.
半径为 r 处的线速度 v 与角速度成正比。
a = v² / r = ω²r
Centripetal acceleration a always points towards the centre of the circle, with magnitude v²/r.
向心加速度 a 始终指向圆心，大小为 v²/r。
F = mv² / r = mω²r
Centripetal force F is the resultant force causing circular motion, provided by tension, gravity, friction, etc.
向心力 F 是导致圆周运动的合力，可由张力、重力、摩擦力等提供。
2. Simple Harmonic Motion (SHM) | 简谐运动
a = −ω²x
Defining equation of SHM: acceleration is proportional to displacement from equilibrium and directed opposite to it.
简谐运动定义式：加速度与偏离平衡位置的位移成正比，方向相反。
x = A cos(ωt) or x = A sin(ωt)
Displacement x varies sinusoidally with time, where A is amplitude and ω is angular frequency.
位移 x 随时间按正弦（或余弦）变化，A 为振幅，ω 为角频率。
v = ± ω √A² − x²
Velocity at a given displacement: maximum speed vmax = ωA occurs at the equilibrium position.
给定位移下的速度：最大速度 vmax = ωA 出现在平衡位置。
T = 2π √m / k (mass-spring system)
Period of a horizontal mass-spring oscillator depends only on mass m and spring constant k.
水平弹簧振子的周期仅取决于质量 m 和弹簧劲度系数 k。
T = 2π √l / g (simple pendulum, small angles)
For a simple pendulum of length l, the period is independent of mass and amplitude (for small swings).
对于摆长为 l 的单摆，周期与质量和振幅无关（小角度摆动时）。
Etotal = ½ k A² = ½ m ω² A²
Total energy in SHM is constant and proportional to the square of the amplitude.
简谐运动的总能量守恒，与振幅的平方成正比。
3. Gravitational Fields | 重力场
F = Gm₁m₂ / r²
Newton's law of universal gravitation: force between two point masses is proportional to the product of masses and inversely proportional to the square of their separation.
牛顿万有引力定律：两质点间的引力与质量乘积成正比，与距离平方成反比。
g = F / m = GM / r²
Gravitational field strength g at a distance r from a point mass M is the force per unit mass.
距离点质量 M 为 r 处的重力场强 g，等于单位质量所受的力。
Vg = −GM / r
Gravitational potential Vg at a point is the work done per unit mass to bring a test mass from infinity to that point.
重力势 Vg：将单位质量从无穷远处移至该点所做的功。
U = −GMm / r
Gravitational potential energy of two masses separated by distance r.
两质量相距 r 时的重力势能。
vesc = √2GM / r
Escape velocity is the minimum speed needed for an object to escape a planet's gravitational field from its surface.
逃逸速度是物体从行星表面脱离其重力场所需的最小速率。
T² ∝ r³ (Kepler's third law)
For a satellite in a circular orbit, the square of the period is proportional to the cube of the orbital radius.
对于圆形轨道上的卫星，周期的平方与轨道半径的立方成正比。
4. Electric Fields | 电场
F = (1 / 4πε₀) × (q₁q₂ / r²)
Coulomb's law: force between two point charges is directly proportional to the product of the charges and inversely proportional to r².
库仑定律：两点电荷之间的作用力与电量乘积成正比，与距离平方成反比。
E = F / q = (1 / 4πε₀) × (Q / r²)
Electric field strength E due to a point charge Q: force per unit positive charge at a point.
点电荷 Q 产生的电场强度 E：某点处单位正电荷所受的力。
E = V / d (uniform field between parallel plates)
For a uniform electric field, field strength is the potential difference V divided by plate separation d.
在匀强电场中，场强等于电势差 V 除以板间距离 d。
Ve = (1 / 4πε₀) × (Q / r)
Electric potential Ve due to a point charge: work done per unit positive charge moved from infinity to that point.
点电荷产生的电势 Ve：将单位正电荷从无穷远处移至该点所做的功。
F = qE
ric field experiences a force F = qE regardless of whether the field is uniform or radial.
电荷 q 在电场中受到的力 F = qE，无论电场是匀强还是辐射状。
5. Capacitance | 电容
C = Q / V
Capacitance C (farad, F) is defined as the charge stored per unit potential difference across a capacitor.
电容 C（单位法拉 F）定义为电容器两端每单位电势差所储存的电荷量。
C = ε₀ A / d (parallel-plate capacitor)
For a parallel-plate capacitor, capacitance is proportional to plate area A and inversely proportional to separation d.
平行板电容器的电容与板面积 A 成正比，与板间距 d 成反比。
E = ½ QV = ½ CV² = ½ Q²/C
Energy stored in a charged capacitor; three equivalent expressions using charge, capacitance and voltage.
充电电容器中储存的能量；使用电荷、电容和电压的三种等价表达式。
Q = Q₀ e−t/RC , V = V₀ e−t/RC
During discharge, charge and voltage decay exponentially with time constant τ = RC.
放电过程中，电荷和电压按指数规律衰减，时间常数 τ = RC。
t½ = RC ln 2
The half-life of a capacitor-discharge circuit is constant and equal to RC ln 2.
电容放电电路的半衰期恒定，为 RC ln 2。
6. Magnetic Fields | 磁场
F = BQv sinθ
Force on a moving charge in a magnetic field: magnitude depends on flux density B, charge Q, speed v and angle θ between v and B.
运动电荷在磁场中所受的力：大小取决于磁通量密度 B、电荷 Q、速度 v 以及 v 与 B 的夹角 θ。
F = BIL sinθ
Force on a current-carrying conductor: length L, current I, flux density B, and angle θ between current and field.
载流导体在磁场中所受的安培力：长度为 L，电流为 I，磁通量密度为 B，θ 为电流与磁场的夹角。
Fleming's left-hand rule gives the direction of the force: thuMb – Motion, First finger – Field, seCond finger – Current.
弗莱明左手定则确定力的方向：拇指 – 运动，食指 – 磁场，中指 – 电流。
r = mv / (BQ) (for charged particle moving perpendicular to B)
Radius of the circular path of a charged particle in a uniform magnetic field, derived from centripetal force equalling magnetic force.
带电粒子在匀强磁场中做圆周运动的半径，由向心力等于磁力推导得出。
Φ = BA cosθ
Magnetic flux Φ through an area A is the product of the flux density B, area and cosine of the angle between B and the normal to the area.
通过面积 A 的磁通量 Φ 是磁通量密度 B、面积以及 B 与面积法线夹角的余弦的乘积。
flux linkage = NΦ
Flux linkage for a coil of N turns: total flux threading the coil.
N 匝线圈的磁链：穿过线圈的总磁通量。
7. Electromagnetic Induction | 电磁感应
ε = − dΦ/dt (Faraday's law)
The magnitude of the induced emf is equal to the rate of change of flux linkage; the minus sign indicates Lenz's law.
感应电动势的大小等于磁链随时间的变化率；负号表示楞次定律。
Lenz's law: the direction of the induced current is such that it opposes the change in magnetic flux that produced it.
楞次定律：感应电流的方向总是阻碍引起感应电流的磁通量变化。
ε = BLv
Emf induced across a straight conductor of length L moving with velocity v perpendicular to a uniform magnetic field B.
长度为 L 的直导体以速度 v 垂直于匀强磁场 B 运动时，产生的感应电动势 ε = BLv。
Vs / Vp = Ns / Np (ideal transformer)
For an ideal transformer, the ratio of secondary to primary voltage equals the turns ratio.
对于理想变压器，次级电压与初级电压之比等于匝数比。
In practice, transformer efficiency = (IsVs / IpVp) × 100%; losses arise from eddy currents, hysteresis and resistive heating.
实际变压器效率 = (IsVs / IpVp) × 100%；能量损耗来自涡流、磁滞和电阻发热。
8. Radioactive Decay & Nuclear Physics | 放射性衰变与核物理
A = λ N
Activity A (becquerel, Bq) is the rate of decay; λ is the decay constant, N is the number of undecayed nuclei.
活度 A（单位贝可 Bq）是衰变速率；λ 为衰变常数，N 为未衰变原子核数目。
N = N₀ e−λt
Exponential decay law: the number of nuclei decreases exponentially with time.
指数衰变规律：原子核数目随时间呈指数减少。
t½ = ln 2 / λ
Half-life is the time taken for half the radioactive nuclei to decay; related to the decay constant.
半衰期是半数放射性原子核发生衰变所需的时间，与衰变常数相关。
ΔE = Δm c²
Mass defect Δm is the difference between the mass of a nucleus and the sum of its individual nucleons; binding energy is Δm c².
质量亏损 Δm 是原子核质量与其核子单独质量总和之差；结合能为 Δm c²。
1 atomic mass unit u = 1.661 × 10−27 kg = 931.5 MeV / c².
1 原子质量单位 u = 1.661 × 10−27 kg = 931.5 MeV / c²。
Nuclear fission: heavy nucleus splits into lighter nuclei, releasing energy; fusion: light nuclei combine, energy released when binding energy per nucleon increases.
核裂变：重核分裂为轻核，释放能量；核聚变：轻核结合，当比结合能增加时释放能量。
9. Thermal Physics & Gas Laws | 热物理与气体定律
pV = nRT = NkT
mperature T (kelvin); n = number of moles, R = molar gas constant, k = Boltzmann constant.
理想气体状态方程：压强 p，体积 V，绝对温度 T（开尔文）；n = 物质的量，R = 摩尔气体常数，k = 玻尔兹曼常数。
k = R / NA
Boltzmann constant k relates the average kinetic energy of particles to temperature; NA is Avogadro's number.
玻尔兹曼常数 k 将粒子平均动能与温度关联起来；NA 为阿伏伽德罗常数。
pV = ⅓ N m (crms)²
Kinetic theory of gases: pressure arises from the collisions of N molecules each of mass m, with root-mean-square speed crms.
气体动理论：压强源于 N 个质量为 m 的分子以方均根速率 crms 碰撞壁面。
⟨Ek⟩ = ⅔ kT
Average translational kinetic energy of a molecule in an ideal gas is directly proportional to absolute temperature.
理想气体中单个分子的平均平动动能与绝对温度成正比。
ΔU = Q + W
First law of thermodynamics (AQA sign convention): increase in internal energy ΔU equals heat added to the system Q plus work done ON the system W. For a gas expanding, W is negative (work done by gas is −W).
热力学第一定律（AQA 符号约定）：内能增加 ΔU 等于系统吸收的热量 Q 加上对系统做的功 W。气体膨胀时，W 为负（气体对外做功为 −W）。
10. Key Constants & Units | 关键常数与单位
| Constant | Symbol & Value | Unit |
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