📚 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）。
... (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' ten denoted d) act as coherent sources. The screen is at distance D from the slits, with D ≫ rence 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。
(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。
refore, 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
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 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。
ight 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。
/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
(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°。
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
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。
uals the work function plus the maximum kinetic energy of the emitted electron: hf = φ + Kmax. Since Kmax = 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 + Kmax。由于 Kmax = ½mv²_max，得出 hf = φ + ½mv²_max。
The stopping potential Vs is related by eVs = Kmax, providing experimental verification. The threshold frequency f₀ satisfies hf₀ = φ.
遏止电势 Vs 满足 eVs = Kmax，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
ely 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。
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 ∝ e 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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