📚 Year 13 CIE Physics: Formula & Theorem Quick-Reference | CIE A2 物理公式定理速查手册
This master reference gathers the essential formulae, definitions, and key theorems from the Year 13 CIE A Level Physics syllabus (9702). It is designed as a rapid revision tool – every equation is paired with a concise English explanation immediately followed by its Chinese equivalent, alongside the conditions under which it applies. Use it to consolidate your understanding before exams.
本速查手册汇集了 CIE A2 物理课程（9702）的核心公式、定义与关键定理，适合快速复习。每个方程均附有简洁的英文说明和紧接的中文解释，并标明了适用条件，助你考前高效巩固。
1. Circular Motion & Gravitation | 圆周运动与引力
ω = Δθ / Δt
Angular velocity ω (rad s⁻¹) is the rate of change of angular displacement θ. For uniform circular motion, ω = 2π / T = 2πf, where T is the period and f the frequency.
角速度 ω (单位 rad s⁻¹) 是角位移 θ 的变化率。匀速圆周运动中 ω = 2π / T = 2πf，T 为周期，f 为频率。
a = v² / r = rω²
Centripetal acceleration a always points towards the centre of the circle. v is the linear speed along the tangent, r the radius. Since v = rω, the two forms are equivalent.
向心加速度 a 始终指向圆心。v 是切线方向的线速度，r 为半径。由 v = rω 可知两式等价。
F = mv² / r = mrω²
Centripetal force F is the net force required to keep an object of mass m moving in a circle; it does no work because it is perpendicular to velocity.
向心力 F 是使质量为 m 的物体做圆周运动所需的合力；由于始终与速度垂直，向心力不做功。
F = GMm / r²
Newton’s law of gravitation: every point mass attracts every other point mass with a force proportional to the product of their masses and inversely proportional to the square of their separation r. G = 6.67×10⁻¹¹ N m² kg⁻².
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 r 的平方成反比。引力常量 G = 6.67×10⁻¹¹ N m² kg⁻²。
g = GM / r²
Gravitational field strength g at a distance r from a point (or spherical) mass M. Near Earth’s surface it is approximately 9.81 N kg⁻¹.
距离点质量（或球对称质量）M 为 r 处的引力场强 g。地球表面附近约 9.81 N kg⁻¹。
2. Simple Harmonic Motion | 简谐运动
a = − ω²x
Defining equation of SHM: acceleration a is directly proportional to displacement x from the equilibrium position and always directed towards it. ω is the angular frequency.
简谐运动的定义式：加速度 a 与相对平衡位置的位移 x 成正比且方向相反。ω 为角频率。
x = x₀ sin(ωt) or x = x₀ cos(ωt)
Displacement–time solutions. x₀ is the amplitude. The choice of sine or cosine depends on the starting condition (t=0).
位移–时间解。x₀ 为振幅。选择正弦还是余弦取决于初始条件（t=0 时刻）。
vₘₐₓ = ωx₀
Maximum speed occurs as the oscillator passes through the equilibrium position. v = ±ω √(x₀² − x²) at any displacement.
最大速率出现在振子通过平衡位置时。任意位移处的速率 v = ±ω √(x₀² − x²)。
Eₜₒₜₐₗ = ½ mω²x₀²
Total mechanical energy of an undamped simple harmonic oscillator is constant. Kinetic energy and potential energy interchange, each averaging half the total.
无阻尼简谐振子的总机械能守恒。动能与势能相互转换，时间平均值各占总能量的一半。
T = 2π √(m / k) for mass–spring; T = 2π √(l / g) for simple pendulum
Period T is independent of amplitude for small oscillations. k is spring constant; l is pendulum length.
在小角度摆动下周期 T 与振幅无关。k 为弹簧劲度系数；l 为摆长。
3. Thermal Physics & Ideal Gases | 热力学与理想气体
pV = nRT
Ideal gas equation: pressure p (Pa), volume V (m³), amount n (mol), universal gas constant R = 8.31 J mol⁻¹ K⁻¹, absolute temperature T (K). It combines Boyle’s, Charles’s and Avogadro’s laws.
理想气体状态方程：压强 p (Pa)、体积 V (m³)、物质的量 n (mol)、摩尔气体常量 R = 8.31 J mol⁻¹ K⁻¹、热力学温度 T (K)。它综合了玻意耳定律、查理定律和阿伏伽德罗定律。
pV = NkT
Alternative form: N is the number of molecules, k = 1.38×10⁻²³ J K⁻¹ is the Boltzmann constant.
另一种形式：N 为分子数，玻尔兹曼常量 k = 1.38×10⁻²³ J K⁻¹。
½ m⟨c²⟩ = (3/2) kT
Average translational kinetic energy of a gas molecule is proportional to absolute temperature. m is the molecular mass and ⟨c²⟩ the mean square speed.
气体分子的平均平动动能与热力学温度成正比。m 为分子质量，⟨c²⟩ 为方均速率。
ΔU = Q + W
First law of thermodynamics: increase in internal energy ΔU equals heat supplied to the system Q plus work done on the system W. (Sign convention: work done on gas is positive.)
热力学第一定律：内能增量 ΔU 等于系统吸收的热量 Q 加上外界对系统做的功 W。（符号约定：对气体做功为正。）
4. Electric Fields | 电场
F = kQq / r² (k = 1/(4πε₀))
Coulomb’s law: force between two point charges Q and q separated by r in vacuum. ε₀ = 8.85×10⁻¹² F m⁻¹ is the permittivity of free space.
库仑定律：真空中两个点电荷 Q 与 q 相距 r 时的作用力。真空介电常量 ε₀ = 8.85×10⁻¹² F m⁻¹。
E = F / q
Electric field strength E (N C⁻¹ or V m⁻¹) is the force per unit positive charge.
电场强度 E（单位 N C⁻¹ 或 V m⁻¹）定义为单位正电荷所受的力。
E = kQ / r²
Radial field due to a point charge Q. The field points away from a positive Q.
点电荷 Q 产生的径向电场。若 Q 为正，电场方向沿径向向外。
E = ΔV / Δd
Uniform electric field: field strength equals potential gradient. For parallel plates separated by d with p.d. V, E = V / d.
匀强电场：场强等于电势梯度。对于相距 d 且电势差为 V 的平行板，E = V / d。
W = qΔV
Work done on a charge q moving through a potential difference ΔV. In a uniform field, it equals force × distance.
电荷 q 经过电势差 ΔV 时电场力做功。匀强电场中等于力乘位移。
5. Capacitance | 电容
C = Q / V
Capacitance C (farad, F) is the charge stored per unit potential difference. 1 F = 1 C V⁻¹.
电容 C (单位法拉 F) 定义为电荷量与电势差之比。1 F = 1 C V⁻¹。
E = ½ QV = ½ CV² = ½ Q² / C
Energy stored in a charged capacitor. The energy resides in the electric field between the plates.
电容储存的能量。能量储存在极板间的电场中。
τ = RC
Time constant for a capacitor–resistor circuit. In discharge, Q = Q₀ e^(−t/RC); in charging, Q = Q₀ (1 − e^(−t/RC)). After one time constant the charge falls to ~37 % or rises to ~63 % of the final value.
阻容电路的时间常数。放电时 Q = Q₀ e^(−t/RC)；充电时 Q = Q₀ (1 − e^(−t/RC))。经过一个时间常数，电荷量降至约 37 % 或升至约 63 %。
6. Magnetic Fields & Electromagnetic Induction | 磁场与电磁感应
F = BIl sin θ
Force on a current-carrying conductor of length l in a magnetic field B. θ is the angle between the current direction and the B‑field. When perpendicular, F = BIl.
通电长度为 l 的直导线在磁场 B 中所受的力。θ 为电流方向与磁场的夹角。当两者垂直时 F = BIl。
F = Bqv sin θ
Magnetic force on a moving charge q with speed v. For circular motion in a uniform field, Bqv = mv² / r, giving r = mv/(Bq).
运动电荷在磁场中所受的洛伦兹力。在匀强磁场中做圆周运动时满足 Bqv = mv² / r，得轨道半径 r = mv/(Bq)。
Φ = BA cos θ
Magnetic flux Φ through an area A; B is the magnetic flux density. Flux linkage through a coil of N turns is NΦ.
磁通量 Φ 通过面积 A；B 是磁通密度。通过 N 匝线圈的磁链为 NΦ。
ε = − dΦ/dt (for one turn)
Faraday’s law: induced e.m.f. equals the rate of change of magnetic flux linkage. Lenz’s law is represented by the minus sign – the induced current opposes the change that produced it.
法拉第定律：感应电动势等于磁链的变化率。负号体现楞次定律——感应电流阻碍引起它的磁通变化。
7. Alternating Currents | 交流电
V = V₀ sin(ωt) ; I = I₀ sin(ωt)
Instantaneous a.c. voltage and current. V₀ and I₀ are peak values; ω = 2πf.
交流电压和电流的瞬时值。V₀ 和 I₀ 是峰值；ω = 2πf。
Vᵣₘₛ = V₀ / √2 ; Iᵣₘₛ = I₀ / √2
Root-mean-square (r.m.s.) values for a sinusoidal a.c. supply. The r.m.s. value of an alternating current produces the same heating effect as a direct current of the same value.
正弦交流电的有效值（均方根值）。交流有效值与相同数值的直流电产生相等的热效应。
⟨P⟩ = Iᵣₘₛ Vᵣₘₛ = Iᵣₘₛ² R (for a resistive load)
Average power delivered to a purely resistive load. When reactance is present, the power factor cos φ must be considered.
纯电阻负载的平均功率。若电路中存在电抗，须考虑功率因数 cos φ。
Vₛ / Vₚ = Nₛ / Nₚ (ideal transformer)
An ideal transformer has no power loss: Iₚ Vₚ = Iₛ Vₛ, hence Iₛ / Iₚ = Nₚ / Nₛ. Energy losses occur due to eddy currents, hysteresis and copper losses in real transformers.
理想变压器无功率损耗：Iₚ Vₚ = Iₛ Vₛ，故 Iₛ / Iₚ = Nₚ / Nₛ。实际变压器存在涡流、磁滞和铜损等能量损失。
8. Quantum Physics | 量子物理
E = hf = hc / λ
Photon energy: h is Planck’s constant (6.63×10⁻³⁴ J s). The photoelectric effect shows that light consists of discrete photons.
光子能量：普朗克常量 h = 6.63×10⁻³⁴ J s。光电效应证实光由分立的光子组成。
hf = Φ + KEₘₐₓ
Einstein’s photoelectric equation: the energy of an incident photon equals the work function Φ (minimum energy to eject an electron) plus the maximum kinetic energy of the emitted electron. Below the threshold frequency f₀ = Φ/h, no electrons are emitted.
爱因斯坦光电效应方程：入射光子能量等于逸出功 Φ（逸出电子的最小能量）与光电子最大动能之和。低于截止频率 f₀ = Φ/h 时，无电子逸出。
λ = h / p = h / (mv)
De Broglie wavelength: particles exhibit wave-like behaviour. A diffraction pattern for electrons confirms their wave nature.
德布罗意波长：粒子具有波动性。电子衍射图样证实了其波动本质。
ΔE = hf = E₂ − E₁
Energy-level transitions in atoms: an electron falling from a higher energy level E₂ to a lower level E₁ emits a photon of frequency f = (E₂ − E₁)/h. Absorption spectra are the reverse process.
原子能级跃迁：电子由高能级 E₂ 跃迁至低能级 E₁ 时辐射频率 f = (E₂ − E₁)/h 的光子。吸收光谱是其逆过程。
9. Nuclear Physics | 核物理
E = mc²
Mass–energy equivalence: a mass m can be converted into energy E. In nuclear reactions, the mass defect Δm accounts for the binding energy released.
质能等价：质量 m 可转换为能量 E。核反应中，质量亏损 Δm 对应释放的结合能。
Binding energy per nucleon = Δmc² / A
Total binding energy divided by the nucleon number A. The iron‑56 region has the highest binding energy per nucleon, making it the most stable.
比结合能 = 总结合能除以核子数 A。铁‑56 附近比结合能最大，因此最稳定。
A = λN ; N = N₀ e^(−λt) ; T₁/₂ = ln 2 / λ
Radioactive decay law: activity A is proportional to the number of undecayed nuclei N; λ is the decay constant. Half‑life T₁/₂ is the time for half the nuclei to decay.
放射性衰变规律：活度 A 与未衰变核的数目 N 成正比；λ 为衰变常量。半衰期 T₁/₂ 为半数核发生衰变所需时间。
α, β⁻, β⁺, γ decay notation
In nuclear equations, conservation of nucleon number and proton number applies. α decay reduces A by 4 and Z by 2. β⁻ decay turns a neutron into a proton; β⁺ decay converts a proton into a neutron. γ emission often accompanies α or β decay to release excess energy.
写核反应方程时遵守质量数守恒和电荷数守恒。α 衰变使质量数减 4、质子数减 2；β⁻ 衰变中子变为质子；β⁺ 衰变质子变为中子；γ 辐射常伴随 α 或 β 衰变放出多余能量。
10. Medical Imaging | 医学成像
Z = ρc (acoustic impedance)
Ultrasound imaging relies on the reflection of sound at boundaries between tissues. Acoustic impedance Z equals the product of density ρ and speed of sound c. The intensity reflection coefficient for normal incidence is (Z₂ − Z₁)² / (Z₂ + Z₁)².
超声成像依靠声波在组织界面处的反射。声阻抗 Z 等于组织密度 ρ 与声速 c 的乘积。垂直入射时的强度反射系数为 (Z₂ − Z₁)² / (Z₂ + Z₁)²。
I = I₀ e^(−μx)
Attenuation of X‑rays: the transmitted intensity I after passing through a material of thickness x decreases exponentially. μ is the linear attenuation coefficient; half‑value thickness x₁/₂ = ln 2 / μ.
X 射线衰减规律：穿透厚度为 x 的材料后出射强度 I 按指数衰减。μ 为线性衰减系数；半值厚度 x₁/₂ = ln 2 / μ。
CT number = 1000 × (μ − μw) / μw
In computed tomography, the Hounsfield scale assigns air −1000, water 0 and compact bone +1000. The number depends on the linear attenuation coefficient relative to water.
在计算机断层成像中，亨氏单位规定空气为 −1000，水为 0，密质骨为 +1000。CT 值取决于相对于水的线性衰减系数。
λ = hc / E for X‑ray photons
Characteristic X‑ray spectra show sharp peaks due to electron transitions between atomic shells; the continuous bremsstrahlung spectrum is produced when electrons are decelerated.
标识 X 射线谱的尖锐峰源于原子壳层间的电子跃迁；连续谱（轫致辐射）由电子减速产生。
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