📚 AS Physics Unit 2 Formula Derivations (Jan 2020 Insert) | AS物理单元2公式推导（2020年1月公式表）
In the AS Physics Unit 2 examination (January 2020), students are provided with a formula insert containing key equations. Knowing how these formulas are derived not only deepens conceptual understanding but also reduces reliance on rote memorisation. This article walks through the derivations of essential formulas from the insert, covering waves, optics, electricity, and quantum phenomena.
在AS物理单元2考试（2020年1月）中，考生会得到一张包含关键方程的公式表。了解这些公式的推导过程不仅能加深概念理解，还能减少对死记硬背的依赖。本文将逐一推导公式表中的重要公式，涵盖波、光学、电学和量子现象。
1. Wave Speed: v = fλ | 波速：v = fλ
The fundamental wave equation v = fλ links speed (v), frequency (f), and wavelength (λ). Derivation: Consider a wave travelling at speed v. In one period T, the wave advances by one wavelength λ. Therefore, distance = speed × time gives λ = v T. Since frequency f = 1/T, substituting v = λ / T = λ f yields v = fλ.
基本波动方程 v = fλ 将波速(v)、频率(f)和波长(λ)联系起来。推导：考虑一列以速度 v 传播的波。在一个周期 T 内，波前进一个波长 λ。因此，距离 = 速度 × 时间，得 λ = v T。因为频率 f = 1/T，代入得 v = λ / T = λ f，即 v = fλ。
v = fλ
Thus, v = fλ is derived directly from the definitions of period and wavelength.
因此，v = fλ 直接从周期和波长的定义推导而出。
2. Snell’s Law from Huygens’ Principle | 由惠更斯原理推导斯涅尔定律
Huygens’ principle states that every point on a wavefront acts as a source of secondary wavelets. When a plane wave passes from medium 1 (speed v₁) to medium 2 (speed v₂), the wavelets propagate with different speeds. Consider a wavefront incident at angle θ₁ to the normal. In time t, the wavefront travels distance v₁t in medium 1 and v₂t in medium 2. Geometrically, the distances along the interface satisfy: v₁t / sinθ₁ = v₂t / sinθ₂. Rearranging gives sinθ₁ / v₁ = sinθ₂ / v₂. Using refractive indices n₁ = c/v₁ and n₂ = c/v₂, we obtain n₁ sinθ₁ = n₂ sinθ₂.
惠更斯原理指出，波前上的每一点都可以看作子波的波源。当平面波从介质1（速度为v₁）进入介质2（速度为v₂）时，子波以不同速度传播。考虑一列以入射角 θ₁ 射向法线的波前。在时间 t 内，波前在介质1中传播距离 v₁t，在介质2中传播距离 v₂t。从几何关系可知，沿界面方向满足：v₁t / sinθ₁ = v₂t / sinθ₂。整理得 sinθ₁ / v₁ = sinθ₂ / v₂。利用折射率 n₁ = c/v₁ 和 n₂ = c/v₂，即得 n₁ sinθ₁ = n₂ sinθ₂。
n₁ sinθ₁ = n₂ sinθ₂
3. Critical Angle Formula: sin c = 1/n | 临界角公式：sin c = 1/n
Total internal reflection occurs when light passes from a denser medium (refractive index n) to a rarer medium (air, n ≈ 1). The critical angle c is the angle of incidence for which the angle of refraction is 90°. Applying Snell’s law: n sin c = 1 × sin 90°. Since sin 90° = 1, we get sin c = 1/n.
当光从光密介质（折射率为 n）射向光疏介质（空气，n ≈ 1）时，会发生全反射。临界角 c 是折射角为 90° 时的入射角。应用斯涅尔定律：n sin c = 1 × sin 90°。由于 sin 90° = 1，得到 sin c = 1/n。
sin c = 1 / n
4. Young’s Double‑Slit Fringe Spacing: Δy = λD / d | 杨氏双缝干涉条纹间距：Δy = λD / d
In Young’s double‑slit experiment, monochromatic light of wavelength λ passes through two slits separated by distance d and forms an interference pattern on a screen at distance D. For a bright fringe at a distance y from the centre, the path difference between the two waves is d sinθ. For small angles, sinθ ≈ tanθ = y/D. Constructive interference occurs when path difference = nλ. For the first‑order maximum (n=1), d (y/D) = λ, so y = λD/d. The fringe spacing Δy between adjacent bright fringes (n → n+1) is constant: Δy = λD/d.
在杨氏双缝实验中，波长为 λ 的单色光通过相距为 d 的两条狭缝，在距离为 D 的屏上形成干涉图样。对于距中心 y 处的明纹，两列波的光程差为 d sinθ。当角度很小时，sinθ ≈ tanθ = y/D。相长干涉条件为光程差 = nλ。对第一级明纹（n=1），d (y/D) = λ，故 y = λD/d。相邻明纹（n → n+1）的间距 Δy 恒定：Δy = λD/d。
Δy = λD / d
5. Diffraction Grating Equation: d sinθ = nλ | 衍射光栅方程：d sinθ = nλ
A diffraction grating consists of many equally spaced slits with separation d (the grating spacing). When monochromatic light is incident normally, each slit acts as a coherent source. Constructive interference at angle θ occurs when the path difference between adjacent slits equals an integer multiple of the wavelength: d sinθ = nλ, where n = 0, ±1, ±2… is the order number. This equation is directly obtained from geometry; no additional approximation is needed.
衍射光栅由许多间距为 d（光栅常数）的等距狭缝组成。当单色光垂直入射时，每条狭缝相当于一个相干光源。在角度 θ 处发生相长干涉的条件是相邻狭缝的光程差等于波长的整数倍：d sinθ = nλ，其中 n = 0, ±1, ±2… 为级次。该方程直接由几何关系得出，无需额外近似。
d sinθ = nλ
6. Standing Waves on a String: λₙ = 2L / n | 弦上驻波：λₙ = 2L / n
When a string of length L is fixed at both ends, standing waves can form only at certain wavelengths. The boundary conditions require nodes at both ends. The simplest standing wave has one antinode in the middle, corresponding to half a wavelength: L = λ₁/2. The next harmonic has two antinodes, fitting a full wavelength: L = λ₂. In general, for the nth harmonic (n = 1,2,3…), the length L equals n half‑wavelengths: L = n (λₙ/2). Hence, the wavelength of the nth harmonic is λₙ = 2L / n. The corresponding frequencies are fₙ = v / λₙ = nv/(2L).
一根长度为 L 两端固定的弦，只能在特定波长下形成驻波。边界条件要求两端为波节。最简单的驻波中间有一个波腹，对应半个波长：L = λ₁/2。下一个谐波有两个波腹，刚好容纳一个完整波长：L = λ₂。一般来说，第 n 次谐波（n = 1,2,3…），长度 L 等于 n 个半波长：L = n (λₙ/2)。因此，第 n 次谐波的波长为 λₙ = 2L / n，相应的频率为 fₙ = v / λₙ = nv/(2L)。
λₙ = 2L / n （fₙ = nv / (2L)）
7. Resistivity and Resistance: R = ρL / A | 电阻率与电阻：R = ρL / A
The resistance R of a uniform conductor depends on its length L, cross‑sectional area A, and a material property called resistivity ρ. For a conductor, resistance is proportional to length and inversely proportional to cross‑sectional area. By definition, resistivity is the constant of proportionality: R = ρL / A. This formula can be understood by considering the drift of electrons; doubling the length doubles the number of collisions, hence doubles resistance, while doubling the area halves the resistance because more charge carriers can flow simultaneously.
均匀导体的电阻 R 取决于其长度 L、横截面积 A 和材料属性——电阻率 ρ。对于导体，电阻与长度成正比，与横截面积成反比。根据定义，电阻率就是比例常数：R = ρL / A。可从电子漂移的角度理解：长度加倍使碰撞次数加倍，因而电阻加倍；而面积加倍使可同时流动的载流子增多，电阻减半。
R = ρL / A
8. Electrical Power: P = IV = I²R = V²/R | 电功率：P = IV = I²R = V²/R
Power is the rate of energy transfer. In an electrical circuit, when a charge Q moves through a potential difference V, the energy transferred is QV. Current I is the rate of charge flow: I = Q/t. Thus, power P = (QV)/t = (Q/t)V = I V. Using Ohm’s law V = IR (for ohmic components), substituting gives P = I (IR) = I²R; substituting I = V/R yields P = V²/R. These three forms are equivalent for resistive loads.
功率是能量转移的速率。在电路中，电荷 Q 通过电势差 V 时所转移的能量为 QV。电流 I 是电荷流动的速率：I = Q/t。因此，功率 P = (QV)/t = (Q/t)V = I V。利用欧姆定律 V = IR（对于欧姆元件），代入得 P = I(IR) = I²R；再代入 I = V/R 得 P = V²/R。对于纯电阻负载，这三种形式等价。
P = IV = I²R = V² / R
9. Photon Energy & Photoelectric Equation: E = hf, hf = φ + K_max | 光子能量与光电效应方程：E = hf, hf = φ + K_max
Quantum theory states that light consists of photons, each carrying energy E = hf, where h is Planck’s constant. In the photoelectric effect, a photon is absorbed by an electron. The electron needs a minimum energy φ (work function) to escape the metal surface. Any excess photon energy becomes the electron’s kinetic energy: hf = φ + ½mv²_max. This is Einstein’s photoelectric equation. The stopping potential V_s relates to maximum kinetic energy: eV_s = K_max. Thus, hf = φ + eV_s.
量子理论指出光由光子组成，每个光子携带能量 E = hf（h 为普朗克常数）。在光电效应中，一个光子被电子吸收。电子需要最小能量 φ（功函数）才能逸出金属表面。多余的光子能量转化为电子的动能：hf = φ + ½mv²_max。这就是爱因斯坦光电方程。遏止电势 V_s 与最大动能的关系为 eV_s = K_max。因此，hf = φ + eV_s。
E = hf, hf = φ + ½mv²_max
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