📚 Year 13 AQA Physics: Formula & Theorem Quick Reference Handbook | AQA 物理 Year 13：公式定理速查手册
This revision handbook distils the core formulas and theorems from the Year 13 AQA Physics syllabus into a bilingual quick-reference format. Each section covers a major topic—further mechanics, thermal physics, fields, capacitors, and nuclear physics—providing key equations alongside concise paired explanations in English and Chinese. Use this as your go-to summary for active recall and last-minute exam preparation.
本复习手册将 AQA 物理 Year 13 课程的核心公式与定理提炼为双语速查格式。每一节涵盖一个主要专题——进一步力学、热物理、场、电容与核物理——给出关键方程，并配以简明的英中双语解释。请将此手册作为你主动回忆和考前冲刺的首选摘要。
1. Circular Motion | 圆周运动
Angular displacement θ is measured in radians. The angular velocity ω is the rate of change of angle: ω = Δθ/Δt = 2π/T = 2πf, where T is the period and f the frequency. For an object moving in a circle of radius r, the linear speed v and angular velocity are linked by v = ωr. The direction of velocity is tangential at every instant.
角位移 θ 以弧度计量。角速度 ω 是角度的变化率：ω = Δθ/Δt = 2π/T = 2πf，其中 T 为周期，f 为频率。对于沿半径为 r 的圆运动的物体，线速度 v 与角速度的关系为 v = ωr。速度方向在每一瞬时均沿切线方向。
An object moving in a circle experiences a centripetal acceleration directed toward the centre, given by a = v²/r = ω²r. The resultant centripetal force required to maintain circular motion is F = mv²/r = mω²r. This net force always points to the centre of the circle and does no work because it is perpendicular to the velocity.
做圆周运动的物体具有指向圆心的向心加速度 a = v²/r = ω²r。维持圆周运动所需的向心力为 F = mv²/r = mω²r。这一合力始终指向圆心，由于与速度方向垂直，故不做功。
v = ωr | a = v²/r = ω²r | F = mv²/r = mω²r
2. Simple Harmonic Motion (SHM) | 简谐运动
SHM is defined by a restoring force (or acceleration) proportional to displacement and directed towards equilibrium: a = −ω²x. The constant ω is the angular frequency. The displacement as a function of time can be written as x = A cos(ωt) or x = A sin(ωt), where A is the amplitude. Starting conditions determine whether sine or cosine is appropriate.
简谐运动的特征是回复力（或加速度）与位移成正比且指向平衡位置：a = −ω²x。常数 ω 为角频率。位移随时间的变化可写为 x = A cos(ωt) 或 x = A sin(ωt)，其中 A 为振幅。采用正弦还是余弦由初始条件决定。
The velocity in SHM varies with displacement: v = ±ω√(A² − x²). The maximum speed v_max = ωA occurs at the equilibrium position. The period of oscillation for a mass-spring system is T = 2π√(m/k), and for a simple pendulum T = 2π√(l/g), provided the amplitude is small.
简谐运动的速度随位移变化：v = ±ω√(A² − x²)。最大速率 v_max = ωA 出现在平衡位置。弹簧振子的周期为 T = 2π√(m/k)；单摆在小振幅下的周期为 T = 2π√(l/g)。
The total mechanical energy of an undamped SHM system is constant: E_total = ½ mω²A². The kinetic energy is Eₖ = ½ mω²(A² − x²) and the potential energy is Eₚ = ½ mω²x². Damping can be light (oscillations with gradually decreasing amplitude), critical (quickest return to equilibrium without oscillating), or heavy (exponential decay without oscillation).
无阻尼简谐运动系统的总机械能守恒：E_total = ½ mω²A²。动能 Eₖ = ½ mω²(A² − x²)，势能 Eₚ = ½ mω²x²。阻尼可分为欠阻尼（振幅逐渐减小的振荡）、临界阻尼（无振荡而最快回到平衡）和过阻尼（无振荡指数衰减）。
a = −ω²x | T = 2π/ω | E_total = ½ mω²A²
3. Thermal Physics | 热物理
The ideal gas equation links pressure p, volume V, amount of substance n and thermodynamic temperature T: pV = nRT, where R = 8.31 J mol⁻¹ K⁻¹. Equivalently, pV = NkT, with N the number of molecules and k = 1.38×10⁻²³ J K⁻¹. One mole contains Avogadro’s number N_A = 6.02×10²³ particles.
理想气体状态方程将压强 p、体积 V、物质的量 n 与热力学温度 T 关联：pV = nRT，其中 R = 8.31 J mol⁻¹ K⁻¹。也可表示为 pV = NkT，N 为分子数，k = 1.38×10⁻²³ J K⁻¹。1 mol 物质包含阿伏伽德罗常量 N_A = 6.02×10²³ 个粒子。
Kinetic theory models gas pressure as due to molecular collisions: pV = ⅓ N m ⟨c²⟩, where ⟨c²⟩ is the mean square speed. Equating with pV = NkT yields the average translational kinetic energy per molecule: ½ m ⟨c²⟩ = (3/2) kT. Temperature is thus a measure of average random kinetic energy.
分子动理论将气体压强解释为分子碰撞的结果：pV = ⅓ N m ⟨c²⟩，⟨c²⟩ 为方均速率。与 pV = NkT 联立可得每个分子的平均平移动能：½ m ⟨c²⟩ = (3/2) kT。因此温度是平均无规动能的量度。
When a substance is heated, its internal energy (sum of random kinetic and potential energies) increases. The heat supplied without phase change is Q = mcΔθ, where c is specific heat capacity. During a phase change at constant temperature, Q = ml, where l is specific latent heat of fusion or vaporisation.
物质受热时，其内能（无规则动能与势能之和）增大。无相变时供给的热量 Q = mcΔθ，c 为比热容。在恒温相变过程中，热量 Q = ml，l 为熔化或汽化比潜热。
pV = nRT | ½ m ⟨c²⟩ = (3/2) kT | Q = mcΔθ
4. Gravitational Fields | 引力场
Newton’s law of universal gravitation states that the force between two point masses is F = Gm₁m₂/r², where G = 6.67×10⁻¹¹ N m² kg⁻². The gravitational field strength g at a distance r from a mass M is the force per unit mass: g = F/m = GM/r², directed radially inward. Close to Earth’s surface, g is approximately uniform at 9.81 N kg⁻¹.
牛顿万有引力定律指出两点质量间的力为 F = Gm₁m₂/r²，G = 6.67×10⁻¹¹ N m² kg⁻²。距质量 M 为 r 处的引力场强度 g 定义为单位质量所受的力：g = F/m = GM/r²，方向沿径向向内。在地球表面附近，g 近似为均匀的 9.81 N kg⁻¹。
Gravitational potential V_g at a point is the work done per unit mass in bringing a test mass from infinity to that point: V_g = −GM/r. The field strength is the negative gradient of potential: g = −dV_g/dr. Escape velocity from a planet of radius R is v_esc = √(2GM/R) = √(2gR). For a satellite in a circular orbit, v = √(GM/r) and Kepler’s third law gives T² ∝ r³.
引力势 V_g 是将单位检验质量从无穷远移至该点所做的功：V_g = −GM/r。场强为势的负梯度：g = −dV_g/dr。从半径为 R 的行星逃逸所需的速度为 v_esc = √(2GM/R) = √(2gR)。对于圆轨道卫星，v = √(GM/r)，开普勒第三定律给出 T² ∝ r³。
F = Gm₁m₂/r² | g = GM/r² | V_g = −GM/r | v_esc = √(2GM/R)
5. Electric Fields | 电场
Coulomb’s law describes the force between two point charges: F = k Q₁Q₂/r², where k = 1/(4πε₀) and ε₀ = 8.85×10⁻¹² F m⁻¹ is the permittivity of free space. The electric field strength E is the force per unit positive charge: E = F/q. For a point charge, E = kQ/r² radially outward if Q is positive. In a uniform field between parallel plates, E = V/d, where d is plate separation.
库仑定律描述两点电荷间的力：F = k Q₁Q₂/r²，其中 k = 1/(4πε₀)，ε₀ = 8.85×10⁻¹² F m⁻¹ 为真空介电常数。电场强度 E 是作用于单位正电荷上的力：E = F/q。点电荷场强 E = kQ/r²，若 Q 为正则沿径向向外。在两平行板间的匀强电场中，E = V/d，d 为板间距。
Electric potential V at a point is work done per unit charge to bring a positive test charge from infinity: V = kQ/r. The potential difference ΔV = V_A − V_B. The work done on a charge q moving through a p.d. is W = qΔV. A useful energy unit in particle physics is the electronvolt: 1 eV = 1.60×10⁻¹⁹ J. The field is the negative potential gradient: E = −dV/dr.
电势 V 是将单位正电荷从无穷远移至该点所做的功：V = kQ/r。电势差 ΔV = V_A − V_B。电荷 q 通过电势差时电场做功 W = qΔV。粒子物理中常用的能量单位是电子伏特：1 eV = 1.60×10⁻¹⁹ J。场强与势的关系为 E = −dV/dr。
F = k Q₁Q₂/r² | E = kQ/r² | E = V/d | V = kQ/r
6. Capacitors | 电容
Capacitance C is the charge stored per unit potential difference: C = Q/V, measured in farads (F). For a parallel-plate capacitor, C = ε₀εᵣ A/d, where A is the plate area, d the separation, and εᵣ the relative permittivity of the dielectric. The addition of a dielectric increases capacitance by reducing the effective electric field.
电容 C 定义为单位电势差下储存的电荷量：C = Q/V，单位为法拉（F）。对于平行板电容器，C = ε₀εᵣ A/d，A 为板面积，d 为间距，εᵣ 为介质的相对介电常数。添加电介质因削弱有效电场而使电容增大。
The energy stored by a capacitor is the work to move charge onto its plates: E = ½QV = ½CV² = ½Q²/C. During charging through a resistor R, charge, voltage and current vary exponentially. For charge: Q = Q₀ (1 − e^{-t/RC}); during discharge: Q = Q₀ e^{-t/RC}, V = V₀ e^{-t/RC}, I = I₀ e^{-t/RC}. The product RC is the time constant τ; after τ, the quantity falls to 37% of its initial value. The half-life is T_half = RC ln 2.
电容器储存的能量等于将电荷移至极板所做的功：E = ½QV = ½CV² = ½Q²/C。通过电阻 R 充电时，电荷、电压与电流按指数规律变化。充电时 Q = Q₀ (1 − e^{-t/RC})；放电时 Q = Q₀ e^{-t/RC}，V = V₀ e^{-t/RC}，I = I₀ e^{-t/RC}。乘积 RC 称为时间常数 τ；经过 τ 后该量衰减至初始值的 37%。半衰期 T_half = RC ln 2。
C = Q/V | C = ε₀εᵣ A/d | E = ½CV² | τ = RC
7. Magnetic Fields | 磁场
A magnetic field exerts a force on a moving charge: the Lorentz force is F = BQv sinθ, where B is magnetic flux density in tesla (T), Q the charge, v its speed, and θ the angle between velocity and field. The direction is given by Fleming’s left-hand rule (or right-hand rule for negative charges). For a current-carrying wire of length l, the force is F = BIl sinθ.
磁场对运动电荷施加洛伦兹力：F = BQv sinθ，B 为磁通量密度（特斯拉 T），Q 为电荷量，v 为速率，θ 为速度与磁场的夹角。方向由弗莱明左手定则确定（负电荷可用右手定则）。对载流直导线，力为 F = BIl sinθ。
Magnetic flux Φ through an area A is Φ = BA cosφ, where φ is the angle between the field and the normal to the area. Flux linkage for a coil of N turns is NΦ. The unit of flux is the weber (Wb). When a conductor moves in a magnetic field or when flux linkage changes, an emf is induced (see next section).
穿过面积 A 的磁通量 Φ = BA cosφ，φ 为磁场与面积法线间的夹角。N 匝线圈的磁链为 NΦ。磁通量的单位是韦伯（Wb）。当导体在磁场中运动或磁链变化时，会感应出电动势（见下一节）。
F = BQv sinθ | F = BIl sinθ | Φ = BA cosφ
8. Electromagnetic Induction | 电磁感应
Faraday’s law states that the magnitude of the induced emf in a circuit is equal to the rate of change of magnetic flux linkage: ε = N |ΔΦ/Δt|. Instantaneously, ε = −N dΦ
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