📚 Key Formula Derivations for International AS Physics (PH02) | 国际AS物理(PH02)核心公式推导
This article provides step-by-step derivations of essential formulas found in the Edexcel International AS Physics Unit 2 (PH02) specification, using the Insert sheet from the May 2023 exam as a reference. Understanding these derivations deepens conceptual grasp and aids in solving application problems.
本文以2023年5月国际AS物理单元2(PH02)考试的数据页为参照，逐步推导演绎核心公式。掌握这些推导能加深概念理解，并有助于解决应用题。
1. Diffraction Grating Formula d sinθ = nλ | 衍射光栅公式 d sinθ = nλ
Consider a transmission grating with slit separation d. Coherent monochromatic light of wavelength λ strikes the grating normally. Each slit acts as a source of wavelets. For a distant screen, rays from adjacent slits are nearly parallel.
考虑缝间距为d的透射光栅。波长为λ的相干单色光垂直照射到光栅上。每个狭缝作为子波源。对于远处的屏幕，相邻狭缝的光线近乎平行。
The path difference between waves from two adjacent slits reaching a point at angle θ is d sinθ. Constructive interference (bright fringe) occurs when this path difference equals an integer multiple of the wavelength.
相邻两缝的光波到达与法线成θ角的方向时，光程差为d sinθ。当该光程差等于波长的整数倍时，发生相长干涉（亮纹）。
Thus, d sinθ = nλ, where n = 0, 1, 2, … (order of maximum). This equation accurately predicts the angles of principal maxima.
因此，d sinθ = nλ，其中n = 0, 1, 2, …（极大的级数）。该方程精确地给出了主极大的角度。
2. Young’s Double-Slit Fringe Spacing Δx = λD/d | 杨氏双缝条纹间距 Δx = λD/d
In Young’s experiment, two slits separated by distance a (often denoted d) act as coherent sources. The screen is at distance D from the slits, with D ≫ a. At a point P on the screen, the path difference is approximately a sinθ ≈ a (y/D), where y is the distance from the central maximum.
在杨氏实验中，间距为a（常记作d）的两条缝作为相干光源。屏幕到双缝的距离为D，且D ≫ a。屏幕上一点P，其光程差近似为 a sinθ ≈ a (y/D)，其中y为到中央极大的距离。
Constructive interference occurs when path difference = nλ. For the first-order bright fringe (n=1), a (y/D) = λ, so y₁ = λD/a. The fringe spacing Δx between adjacent bright fringes is y₁ – y₀.
当光程差 = nλ 时发生相长干涉。对于一级亮纹（n=1），a (y/D) = λ，因此 y₁ = λD/a。相邻亮纹的间距 Δx 即 y₁ – y₀。
Since y₀ = 0 for central maximum, Δx = λD/a. Therefore, Δx = λD/d, where d is the slit separation. This formula is valid for small angles.
中央极大处 y₀ = 0，所以 Δx = λD/a。因此 Δx = λD/d，其中d为缝间距。该公式在小角度下成立。
3. Standing Wave Frequency on a Stretched String f = (1/2L)√(T/μ) | 弦上驻波频率 f = (1/2L)√(T/μ)
A string fixed at both ends can sustain standing waves when the length L is an integer multiple of half-wavelengths: L = n (λ/2). For the fundamental mode (n=1), λ = 2L.
两端固定的弦可形成驻波，此时弦长L为半波长的整数倍：L = n (λ/2)。对于基频模式（n=1），λ = 2L。
The speed of a transverse wave on a string is given by v = √(T/μ), where T is tension and μ is mass per unit length. Using the wave equation v = fλ, we obtain f = v/λ.
弦上横波的波速为 v = √(T/μ)，其中T为张力，μ为线密度。由波动方程 v = fλ，可得 f = v/λ。
Substituting λ = 2L for the fundamental gives f = (1/2L)√(T/μ). Harmonics have frequencies n times the fundamental.
将基频的 λ = 2L 代入，得到 f = (1/2L)√(T/μ)。泛音频率为基频的整数倍。
4. Snell’s Law Derivation n₁ sinθ₁ = n₂ sinθ₂ | 斯涅尔定律推导 n₁ sinθ₁ = n₂ sinθ₂
Using Huygens’ principle, consider a plane wavefront incident on a boundary from medium 1 (speed v₁) to medium 2 (speed v₂). The wavefront makes an angle θ₁ with the interface.
利用惠更斯原理，考虑一平面波前从介质1（波速v₁）入射到介质2（v₂）的界面。波前与界面夹角为 θ₁。
In time Δt, the wavefront advances a distance v₁Δt in medium 1 and v₂Δt in medium 2. Geometry of right triangles yields sinθ₁ = (v₁Δt) / x and sinθ₂ = (v₂Δt) / x, where x is a common side.
在时间Δt内，波前在介质1中行进 v₁Δt，在介质2中行进 v₂Δt。由直角三角形几何关系得 sinθ₁ = (v₁Δt) / x，sinθ₂ = (v₂Δt) / x，其中x为公共边。
Dividing the two equations gives sinθ₁ / sinθ₂ = v₁ / v₂. Since refractive index n = c/v, we have v₁ = c/n₁ and v₂ = c/n₂. Therefore, sinθ₁ / sinθ₂ = (c/n₁) / (c/n₂) = n₂ / n₁, leading to n₁ sinθ₁ = n₂ sinθ₂.
两式相除得 sinθ₁ / sinθ₂ = v₁ / v₂。因为折射率 n = c/v，有 v₁ = c/n₁ 及 v₂ = c/n₂。于是 sinθ₁ / sinθ₂ = (c/n₁)/(c/n₂) = n₂/n₁，从而得到 n₁ sinθ₁ = n₂ sinθ₂。
5. Critical Angle sin C = 1/n | 临界角 sin C = 1/n
When light travels from a denser medium (refractive index n) into a rarer medium (e.g. air, n≈1), the angle of refraction becomes 90° at the critical angle C. Using Snell’s law: n sin C = 1 × sin 90°.
当光从光密介质（折射率n）射向光疏介质（如空气，n≈1）时，折射角在临界角C处达到90°。由斯涅尔定律：n sin C = 1 × sin 90°。
Since sin 90° = 1, we obtain n sin C = 1, hence sin C = 1/n. This is valid only when n > 1. For total internal reflection, the angle of incidence must be greater than C.
因为 sin 90° = 1，得到 n sin C = 1，因此 sin C = 1/n。该式仅在 n > 1 时成立。发生全内反射时，入射角必须大于临界角。
6. Photoelectric Effect Equation hf = φ + ½mv²_max | 光电效应方程 hf = φ + ½mv²_max
Einstein proposed that light consists of photons, each with energy E = hf. When a photon strikes a metal surface, it can transfer its energy to an electron. The minimum energy needed to release an electron is the work function φ.
爱因斯坦提出光由光子组成，每个光子能量 E = hf。光子撞击金属表面时，可将能量转移给电子。使电子逸出所需的最小能量为逸出功 φ。
By energy conservation, the photon energy equals the work function plus the maximum kinetic energy of the emitted electron: hf = φ + K_max. Since K_max = ½mv²_max, we have hf = φ + ½mv²_max.
根据能量守恒，光子能量等于逸出功加上出射电子的最大动能：hf = φ + K_max。由于 K_max = ½mv²_max，得出 hf = φ + ½mv²_max。
The stopping potential V_s is related by eVs = K_max, providing experimental verification. The threshold frequency f₀ satisfies hf₀ = φ.
遏止电势 V_s 满足 eVs = K_max，提供了实验验证。截止频率 f₀ 满足 hf₀ = φ。
7. de Broglie Wavelength λ = h/p | 德布罗意波长 λ = h/p
De Broglie hypothesised that particles have wave-like properties, with a wavelength inversely proportional to momentum. For a photon, momentum p = E/c = hf/c = h/λ, giving λ = h/p.
德布罗意假设粒子具有波动性，波长与动量成反比。对于光子，动量 p = E/c = hf/c = h/λ，得出 λ = h/p。
Extending this to particles with mass, de Broglie proposed λ = h/p = h/(mv) for non-relativistic speeds. This is confirmed by electron diffraction experiments. For electrons accelerated through a potential difference V, kinetic energy = eV = ½mv² = p²/(2m). Hence p = √(2meV) and λ = h/√(2meV).
将此推广到有质量粒子，德布罗意提出非相对论速度下 λ = h/p = h/(mv)。电子衍射实验证实了这一点。对于经电势差V加速的电子，动能 eV = ½mv² = p²/(2m)，因此 p = √(2meV)，λ = h/√(2meV)。
8. Resistivity Formula R = ρL/A | 电阻率公式 R = ρL/A
For a uniform conductor at constant temperature, resistance R is directly proportional to its length L and inversely proportional to its cross-sectional area A. The proportionality constant is resistivity ρ, a material property.
对于恒温下的均匀导体，电阻R与长度L成正比，与横截面积A成反比。比例常数是电阻率ρ，为材料特性。
Experimental evidence supports R ∝ L and R ∝ 1/A. Combining these gives R = ρ (L/A). Resistivity is defined as ρ = RA/L, with SI units of ohm-metre (Ω·m).
实验证据支持 R ∝ L 和 R ∝ 1/A。综合得 R = ρ (L/A)。电阻率定义为 ρ = RA/L，SI单位是欧姆·米（Ω·m）。
Microscopically, resistivity depends on the number density of charge carriers and their mean free time. The macroscopic formula is essential for calculating resistance of wires.
从微观角度看，电阻率取决于载流子数密度和平均自由时间。该宏观公式对计算导线电阻至关重要。
9. EMF and Internal Resistance ε = I(R + r) | 电动势与内阻 ε = I(R + r)
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