PH02 Physics AS Formula Derivations | PH02 物理AS公式推导

📚 PH02 Physics AS Formula Derivations | PH02 物理AS公式推导

The May 2023 PH02 International AS Physics paper tests a range of key concepts, many of which require students to derive fundamental equations from first principles. Mastering these derivations not only strengthens exam performance but also deepens understanding of the underlying physics. This article walks through the essential formula derivations most relevant to the PH02 syllabus, with clear step‑by‑step reasoning in both English and Chinese.

2023年5月的PH02国际AS物理试卷考查了许多核心概念，其中不少题目要求学生从基本原理出发推导关键公式。掌握这些推导不仅能提升考试成绩，还能加深对物理本质的理解。本文带你逐一梳理与PH02考纲高度相关的核心公式推导，每一步都配有中英文的双语解析。

1. Deriving Resistance from Resistivity | 电阻率公式 R = ρL/A 推导

For a uniform conductor at constant temperature, experiment shows that the resistance R is directly proportional to its length L and inversely proportional to its cross‑sectional area A. Introducing the constant of proportionality, the resistivity ρ, gives R ∝ L/A. Hence the complete relation is:

对于温度恒定的均匀导体，实验表明其电阻 R 与长度 L 成正比，与横截面积 A 成反比。引入比例常数——电阻率 ρ，可写成 R ∝ L/A，因此完整关系式为：

R = ρL / A

The resistivity ρ is a material property and has units of Ω·m. To derive the formula formally, one starts from the definition of resistivity in a rectangular block: if a potential difference V is applied across length L, the electric field E = V/L, and current density J = I/A. Ohm’s law in microscopic form states J = σE, where conductivity σ = 1/ρ. Substituting gives I/A = (1/ρ)(V/L), which rearranges to V/I = ρL/A, yielding R = ρL/A.

电阻率 ρ 是材料属性，单位为 Ω·m。要正式推导这一公式，可从矩形块模型出发：在长度 L 方向施加电势差 V，电场 E = V/L，电流密度 J = I/A。欧姆定律的微观形式为 J = σE，其中电导率 σ = 1/ρ。代入后得 I/A = (1/ρ)(V/L)，整理得到 V/I = ρL/A，即定义 R = ρL/A。

2. Deriving Resistors in Parallel | 并联电阻公式推导

When resistors are connected in parallel, the potential difference V across each branch is the same. The total current I from the source splits: I = I₁ + I₂ + I₃ + … For each resistor, Ohm’s law gives I₁ = V/R₁, I₂ = V/R₂, etc. Substituting into the current sum yields V/R_total = V/R₁ + V/R₂ + V/R₃ + … . Cancelling the common factor V gives the well‑known reciprocal formula:

当电阻并联时，各支路两端的电势差 V 相等。从电源流出的总电流 I 分为各支路电流之和：I = I₁ + I₂ + I₃ + … 。对每个电阻应用欧姆定律可得 I₁ = V/R₁，I₂ = V/R₂ 等等。代入总电流表达式得到 V/R_total = V/R₁ + V/R₂ + V/R₃ + … 。约去公因子 V，就得到大家熟悉的倒数公式：

1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

For two resistors in parallel, this simplifies to R_total = (R₁R₂)/(R₁ + R₂), a result often quoted in exam derivations.

对于两个电阻并联的情况，公式可简化为 R_total = (R₁R₂)/(R₁ + R₂)，这是考试推导中经常引用的结果。

3. Deriving the EMF and Internal Resistance Formula | 电动势与内阻公式 E = I(R + r) 推导

A real cell has an electromotive force (EMF) E and an internal resistance r. When the cell delivers a current I to an external load R, the terminal potential difference V is less than E because some energy is lost across the internal resistance. Energy conservation requires E = V + Ir. Using V = IR gives:

一个实际电池具有电动势 E 和内阻 r。当电池向外部负载 R 输出电流 I 时，端电压 V 会低于电动势，因为部分能量在内阻上消耗。由能量守恒可得 E = V + Ir。利用 V = IR 得到：

E = I(R + r)

The lost volts are Ir, and the terminal voltage follows V = E – Ir. A graph of V against I yields a straight line with gradient –r and y‑intercept E, which is a classic PH02 experiment.

损失电压为 Ir，端电压遵循 V = E – Ir。V‑I 图为一条斜率为 –r、截距为 E 的直线，这是 PH02 经典实验之一。

4. Deriving Snell’s Law Using Huygens’ Principle | 利用惠更斯原理推导斯涅尔定律

Huygens’ principle treats every point on a wavefront as a source of secondary wavelets. Consider a plane wave entering a medium where its speed changes from v₁ to v₂. In time t, the wavefront in medium 1 travels a distance v₁t, while the edge entering medium 2 travels v₂t. The geometry of right triangles gives sinθ₁ = v₁t / x and sinθ₂ = v₂t / x, where x is the common hypotenuse along the boundary. Eliminating x/t yields:

惠更斯原理将波前上的每一点视为子波源。考虑平面波进入波速从 v₁ 变为 v₂ 的介质，在时间 t 内，介质1中的波前传播距离 v₁t，而刚进入介质2的边界点传播距离 v₂t。根据直角三角形几何关系，sinθ₁ = v₁t / x，sinθ₂ = v₂t / x，其中 x 为沿边界的公共斜边。消去 x/t 可得：

sinθ₁ / v₁ = sinθ₂ / v₂

Introducing refractive indices, n₁ = c/v₁ and n₂ = c/v₂, the relation becomes n₁ sinθ₁ = n₂ sinθ₂, which is Snell’s law. This derivation is fundamental for understanding refraction and total internal reflection.

引入折射率 n₁ = c/v₁，n₂ = c/v₂，关系式即变为 n₁ sinθ₁ = n₂ sinθ₂，这就是斯涅尔定律。这一推导对理解折射与全内反射至关重要。

5. Deriving the Critical Angle Formula | 临界角公式 sinC = 1/n 推导

Total internal reflection occurs when light travels from an optically denser medium (refractive index n₁) to a less dense medium (n₂) and the angle of incidence exceeds the critical angle C. At the critical angle, the angle of refraction is exactly 90°. Applying Snell’s law: n₁ sinC = n₂ sin90° = n₂. Hence sinC = n₂ / n₁. For light going from glass or water into air (n₂ = 1), the formula reduces to:

当光从光密介质（折射率 n₁）射向光疏介质（n₂），且入射角超过临界角 C 时，会发生全内反射。在临界角处，折射角恰好为 90°。应用斯涅尔定律：n₁ sinC = n₂ sin90° = n₂，因此 sinC = n₂ / n₁。对于从玻璃或水射入空气的情形（n₂ = 1），公式简化为：

sinC = 1 / n

where n is the refractive index of the denser medium. This expression is frequently used in PH02 questions on optical fibres and prisms.

其中 n 为光密介质的折射率。这个表达式在 PH02 关于光纤和棱镜的考题中经常用到。

6. Deriving the Photoelectric Effect Equation | 光电效应方程 Eₖ(max) = hf – Φ 推导

Einstein’s photon model assumes that each single photon of frequency f carries energy E = hf, where h is the Planck constant. When a photon strikes a metal surface, this energy is transferred to a single electron. The electron must use a minimum energy Φ, the work function, to escape the metal. Any remaining energy appears as the electron’s maximum kinetic energy Eₖ(max). Conservation of energy dictates:

爱因斯坦的光子模型假设每个频率为 f 的光子携带能量 E = h f，其中 h 为普朗克常数。当一个光子撞击金属表面时，这份能量传递给单个电子。电子必须消耗至少为功函数 Φ 的最低能量来脱离金属，剩余能量则转化为电子的最大动能 Eₖ(max)。能量守恒要求：

Eₖ(max) = hf – Φ

Photoelectrons are emitted only when hf > Φ, giving a threshold frequency f₀ = Φ/h. The PH02 paper often asks students to derive this equation from an energy balance argument and to use the stopping potential Vs to find Eₖ(max) via eVs = Eₖ(max).

只有当 hf > Φ 时才会有光电子逸出，由此可得截止频率 f₀ = Φ/h。PH02 试卷常要求学生根据能量平衡推导该方程，并利用遏止电压 Vs 通过 eVs = Eₖ(max) 来求最大动能。

7. Deriving the Thin Lens Equation | 薄透镜公式 1/f = 1/u + 1/v 推导

Consider a thin converging lens with focal length f, object distance u and image distance v. Using similar triangles in the ray diagram: the triangle formed by the object and lens is similar to the triangle formed by the image and lens for a ray through the centre. Another pair of similar triangles involves a ray parallel to the principal axis that passes through the focal point. By equating ratios, one obtains:

考虑焦距为 f 的薄会聚透镜，物距为 u，像距为 v。利用光线图中的相似三角形：过光心的光线构成的物方三角形与像方三角形相似。另一组相似三角形涉及平行于主光轴、通过焦点的光线。通过比例式整理可得到：

1/f = 1/u + 1/v

The sign convention used in the International AS syllabus (real‑is‑positive) must be applied carefully: u is positive for real objects, v positive for real images, f positive for a converging lens. This derivation appears frequently in PH02 structured questions on geometrical optics.

国际AS考纲中使用的符号规则（实正虚负）需要谨慎应用：实物 u 为正，实像 v 为正，会聚透镜焦距 f 为正。这一推导经常出现在 PH02 几何光学的结构题中。

8. Deriving Wave Speed on a Stretched String | 弦上波速 v = √(T/μ) 推导

Consider a small segment of a stretched string under tension T. When a transverse pulse travels at speed v, the centripetal force on a curved element of length Δl and radius R is approximately T Δθ, where Δθ is the small angle subtended. The mass of the segment is μ Δl, with μ being the linear density. Applying Newton’s second law: T Δθ = (μ Δl) v² / R. For small angles, Δl ≈ R Δθ, giving T = μ v². Hence:

考虑一微小段张紧的弦，张力为 T。当横波脉冲以速度 v 传播时，一个长度为 Δl、对应圆心角为 Δθ 的弧形微元所受向心力约为 T Δθ。该微元的质量为 μ Δl，其中 μ 为线密度。应用牛顿第二定律：T Δθ = (μ Δl) v² / R。对于小角度，Δl ≈ R Δθ，于是得到 T = μ v²，从而有：

v = √(T / μ)

This formula is essential for standing wave experiments on strings, such as Melde’s experiment, which may be referenced in PH02 contexts involving waves and vibrations.

该公式对弦上驻波实验（如梅尔迪实验）至关重要，在 PH02 涉及波动与振动的题目中可能会有所涉及。

9. Deriving Young’s Double‑Slit Fringe Spacing | 杨氏双缝条纹间距 Δy = λD/d 推导

In Young’s double‑slit experiment, coherent light of wavelength λ passes through two slits separated by a distance d, and forms an interference pattern on a screen at a perpendicular distance D (D ≫ d). For the m‑th order bright fringe, the path difference from the two slits to the screen is mλ. Geometry gives path difference ≈ d sinθ ≈ d (y / D) for small angles, where y is the distance from the central maximum. Setting d (y / D) = mλ yields y = mλD/d. The fringe separation Δy between adjacent bright fringes (m=1 and m=0) is therefore:

在杨氏双缝实验中，波长为 λ 的相干光通过间距为 d 的两条狭缝，在垂直距离为 D 远处的屏幕上形成干涉图样（且 D ≫ d）。对于第 m 级亮纹，两缝到达屏幕的光程差为 mλ。几何关系给出光程差 ≈ d sinθ ≈ d (y / D)（小角度近似），其中 y 为距中央极大的距离。令 d (y / D) = mλ，得到 y = mλD/d。因此相邻亮纹（如 m=1 与 m=0）的间距 Δy 为：

Δy = λD / d

This derivation is a favourite in PH02 questions on the wave nature of light, often combined with measurements of the fringe spacing to determine the wavelength of a laser.

该推导是 PH02 光线波动性考题中的常见内容，常结合条纹间距的测量来确定激光波长。

10. Deriving the Condition for Constructive Interference | 相长干涉条件推导

When two coherent waves of the same amplitude and wavelength meet, constructive interference occurs if the path difference is an integer multiple of the wavelength. If the individual displacements are x₁ = A sin(ωt) and x₂ = A sin(ωt + φ), the resultant amplitude is 2A|cos(φ/2)|. Maximum amplitude (2A) occurs when cos(φ/2) = ±1, i.e. φ = 2πn, where n = 0,1,2,… A phase difference of 2πn corresponds to a path difference Δx = nλ, giving the condition:

当两列振幅相同、波长相等的相干波相遇时，若光程差等于波长的整数倍，则发生相长干涉。若两列波的位移分别为 x₁ = A sin(ωt) 和 x₂ = A sin(ωt + φ)，合振幅为 2A|cos(φ/2)|。当 cos(φ/2) = ±1，即 φ = 2πn（n=0,1,2,…）时，振幅最大。相位差 2πn 对应光程差 Δx = nλ，因此条件为：

Δx = nλ

For destructive interference, the path difference is an odd multiple of half‑wavelengths: Δx = (2n+1)λ/2. These conditions underlie the analysis of double‑slit patterns and thin‑film interference included in the PH02 specification.

对于相消干涉，光程差为半波长的奇数倍：Δx = (2n+1)λ/2。这些条件是分析双缝图样和薄膜干涉（均包含在 PH02 考纲中）的基础。

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PH02 Physics AS Formula Derivations | PH02 物理AS公式推导