📚 AS Physics Unit 2 Past Paper Jan 2019: Formula Derivations | AS 物理 Unit 2 2019年1月真题公式推导
In the Edexcel AS Physics Unit 2 examination, candidates are frequently asked to derive key relationships from fundamental principles. The January 2019 paper featured several such items – from wave optics and electricity to quantum phenomena. This revision guide walks you through the step-by-step derivations that are most likely to appear, ensuring you understand the logical flow behind each equation rather than just memorising it. Use these worked examples to sharpen your problem-solving skills.
在爱德思 AS 物理单元 2 考试中，考生常被要求从基本原理推导关键关系式。2019 年 1 月的试卷就包含了多个此类题目——涵盖波动光学、电学和量子现象。这份复习指南将带你一步步走过最可能出现的推导过程，确保你理解每个方程背后的逻辑流程，而不只是死记硬背。利用这些范例来提升你的解题能力吧。
1. Deriving Snell’s Law and the Critical Angle | 斯涅尔定律与临界角的推导
When light passes from one medium into another, its speed changes, causing refraction. The ratio of the speed of light in a vacuum c to its speed v in the medium defines the absolute refractive index: n = c/v. For two media, Snell’s law arises from the constancy of the wave’s frequency. At a boundary, the incident and refracted wavefronts must match, leading to (sin θ₁)/v₁ = (sin θ₂)/v₂. Multiplying by c gives n₁ sin θ₁ = n₂ sin θ₂, the familiar form. For a light ray travelling from glass to air, n₁ = n (glass) and n₂ = 1 (air). The critical angle C occurs when the refracted angle reaches 90°. Substituting gives n sin C = 1 × sin 90°, so sin C = 1/n.
当光从一种介质进入另一种介质时，速度发生变化，产生折射。介质中光速 v 与真空中光速 c 的比值定义了绝对折射率：n = c/v。对于两种介质，斯涅尔定律源于波频率的恒定。在界面处，入射波前与折射波前必须匹配，由此得出 (sin θ₁)/v₁ = (sin θ₂)/v₂。乘以光速 c 即得熟悉的形式 n₁ sin θ₁ = n₂ sin θ₂。若光线从玻璃射向空气，n₁ = n（玻璃），n₂ = 1（空气）。当折射角恰好达到 90° 时，对应的入射角称为临界角 C。代入公式得 n sin C = 1 × sin 90°，因此 sin C = 1/n。
2. Derivation of the Diffraction Grating Equation d sin θ = nλ | 衍射光栅方程 d sin θ = nλ 的推导
Consider a transmission grating with slit separation d (the grating spacing). When a plane wavefront strikes the grating, each slit acts as a source of secondary wavelets. For light leaving at an angle θ to the normal, the path difference between waves from adjacent slits is d sin θ. Constructive interference – producing a bright fringe (maximum) – occurs when this path difference is an integer multiple of the wavelength: path difference = nλ, where n = 0, ±1, ±2, … Therefore d sin θ = nλ. This equation allows the wavelength of light to be calculated by measuring the angle of the nth order maximum.
考虑缝间距为 d（光栅常数）的透射光栅。当平面波前照射光栅时，每条狭缝成为次级子波波源。对于与法线成角 θ 的出射光，两相邻狭缝发出的波之间的光程差为 d sin θ。当光程差等于波长的整数倍时发生相长干涉，产生亮纹（极大），即光程差 = nλ，其中 n = 0, ±1, ±2, …。故有 d sin θ = nλ。利用该方程，通过测量第 n 级极大的衍射角即可算出入射光的波长。
3. Young’s Double-Slit Fringe Spacing Formula Δx = λD/a | 杨氏双缝条纹间距公式 Δx = λD/a
In Young’s experiment, coherent light illuminates two narrow slits separated by a distance a. On a screen at distance D (where D ≫ a), an interference pattern of bright and dark fringes is observed. At a point on the screen a vertical distance y from the central maximum, the path difference from the two slits is approximately a(y/D) for small angles. Bright fringes satisfy a(y/D) = nλ. The separation Δx between adjacent bright fringes (or dark fringes) is found by considering two successive orders n and n+1: a(yₙ₊₁/D) − a(yₙ/D) = λ, which simplifies to a(Δx/D) = λ. Hence Δx = λD/a. The formula shows that fringe spacing increases with wavelength and screen distance, and decreases with slit separation.
在杨氏实验中，相干光照射两条相距为 a 的窄缝。在距离 D 远处的屏幕上（且 D ≫ a），可观察到明暗相间的干涉条纹。对于屏幕上距中央极大竖直距离为 y 的一点，在角度很小的情况下，来自两条缝的光程差近似为 a(y/D)。明纹条件为 a(y/D) = nλ。考虑两个相邻级数 n 和 n+1，可得 a(yₙ₊₁/D) − a(yₙ/D) = λ，化简为 a(Δx/D) = λ。因此相邻条纹的间距 Δx = λD/a。该式表明，条纹间距随波长和屏距增大而增大，随缝距增大而减小。
4. Deriving the Resistivity Formula R = ρL/A | 电阻率公式 R = ρL/A 的推导
Experiment shows that the resistance R of a uniform wire is directly proportional to its length L and inversely proportional to its cross-sectional area A. This can be written as R ∝ L/A. Introducing a constant of proportionality – the resistivity ρ, which depends on the material and temperature – gives R = ρL/A. Resistivity has units of ohm-metre (Ω m). The derivation is rooted in the microscopic model: resistance arises from collisions between conduction electrons and the metal lattice; a longer wire provides more scattering centres, while a larger cross-section offers a wider pathway for electron flow, reducing resistance.
实验表明，均匀导线的电阻 R 与其长度 L 成正比，与其横截面积 A 成反比。这一关系可表示为 R ∝ L/A。引入比例常数——电阻率 ρ，它取决于材料与温度——即得 R = ρL/A。电阻率的单位是欧姆·米（Ω m）。该推导基于微观模型：电阻源于传导电子与金属晶格之间的碰撞；更长的导线提供更多的散射中心，而更大的横截面积则为电子流动提供了更宽的通道，从而降低电阻。
5. Deriving Equivalent Resistance for Series and Parallel | 串联与并联等效电阻的推导
For resistors in series, the same current I flows through each. The total potential difference V is the sum of the individual p.d.s: V = V₁ + V₂ + … . Applying Ohm’s law, V = I R_eq and V₁ = I R₁, V₂ = I R₂, so I R_eq = I R₁ + I R₂. Cancelling I yields R_eq = R₁ + R₂ + … . For parallel resistors, the p.d. V across each branch is identical. The total current I splits: I = I₁ + I₂ + … . Using Ohm’s law, I = V/R_eq and I₁ = V/R₁, I₂ = V/R₂. Substituting gives V/R_eq = V/R₁ + V/R₂, so 1/R_eq = 1/R₁ + 1/R₂ + … . These rules are essential for circuit analysis.
对于串联电阻，通过的电流 I 相同。总电压 V 等于各分电压之和：V = V₁ + V₂ + …。应用欧姆定律，有 V = I R_eq 及 V₁ = I R₁, V₂ = I R₂，故 I R_eq = I R₁ + I R₂。约去 I 得 R_eq = R₁ + R₂ + …。对于并联电阻，各支路两端电压 V 相等。总电流 I 分流：I = I₁ + I₂ + …。由欧姆定律，I = V/R_eq，I₁ = V/R₁，I₂ = V/R₂。代入得 V/R_eq = V/R₁ + V/R₂，因此 1/R_eq = 1/R₁ + 1/R₂ + …。这些规则是电路分析的基础。
6. Internal Resistance and Terminal Potential Difference | 内阻与路端电压的推导
A real cell has internal resistance r due to the materials inside it. When a current I is drawn, some of the electromotive force (emf) E is lost across the internal resistance. The terminal p.d. V (the voltage across the cell’s terminals) is therefore less than E. Applying the conservation of energy around the circuit: E = I (R + r), where R is the external load resistance. Rearranging, V = IR = E − Ir. This shows that the terminal p.d. decreases linearly as the current increases. The open-circuit voltage equals E, and the short-circuit current is E/r.
实际电池因内部材料而具有内阻 r。当有电流 I 输出时，一部分电动势（emf）E 在内阻上降落。路端电压 V（电池两端的电压）因此小于 E。根据回路中的能量守恒：E = I (R + r)，其中 R 为外接负载电阻。整理得 V = IR = E − Ir。这表明路端电压随电流增加而线性下降。开路电压等于 E，短路电流为 E/r。
7. The Potential Divider Derivation | 分压器公式的推导
A potential divider consists of two resistors R₁ and R₂ in series across a supply voltage V_s. The current I in the series chain is V_s/(R₁ + R₂). The output voltage V_out is the p.d. across R₂, so V_out = I R₂. Substituting for I gives V_out = V_s × R₂/(R₁ + R₂). This circuit is widely used to obtain a variable voltage from a fixed supply, for example in sensor circuits where R₂ is a thermistor or LDR.
分压器由两个串联电阻 R₁ 和 R₂ 跨接在电源电压 V_s 上构成。串联回路中的电流 I 为 V_s/(R₁ + R₂)。输出电压 V_out 是 R₂ 两端的电压，故 V_out = I R₂。代入 I 即得 V_out = V_s × R₂/(R₁ + R₂)。该电路广泛用于从固定电源获得可变电压，例如在传感器电路中 R₂ 为热敏电阻或光敏电阻。
8. Einstein’s Photoelectric Equation: hf = φ + KEₘₐₓ | 爱因斯坦光电方程：hf = φ + KEₘₐₓ
Einstein proposed that electromagnetic radiation consists of photons, each with energy E = hf, where h is Planck’s constant and f is the frequency. When a photon strikes a metal surface, it can transfer its energy to a single electron. A minimum energy, the work function φ, is required to extract the electron from the metal. Any surplus energy becomes the electron’s maximum kinetic energy: hf = φ + KEₘₐₓ. This equation explains the threshold frequency f₀ = φ/h and the stopping potential V_s such that KEₘₐₓ = eV_s.
爱因斯坦提出电磁辐射由光子组成，每个光子的能量为 E = hf，其中 h 为普朗克常量，f 为频率。当光子撞击金属表面时，可将能量传递给单个电子。将电子从金属中拉出所需的最小能量称为逸出功 φ。剩余能量转变为电子的最大动能：hf = φ + KEₘₐₓ。此方程解释了极限频率 f₀ = φ/h 以及遏止电压 V_s 满足 KEₘₐₓ = eV_s。
9. Deriving the Electronvolt (eV) as a Unit of Energy | 推导电子伏特（eV）作为能量单位
An electronvolt is the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 volt. From the definition of potential difference V = W/Q, the work done on a charge Q is W = QV. For an electron, Q = e = 1.60 × 10⁻¹⁹ C. Therefore, 1 eV = (1.60 × 10⁻¹⁹ C) × (1 V) = 1.60 × 10⁻¹⁹ J. This conversion is a staple in quantum and particle physics whenever energies are expressed in eV instead of joules.
一个电子伏特是指一个电子经 1 伏特电势差加速后获得的动能。根据电势差定义 V = W/Q，对电荷 Q 做的功为 W = QV。对电子而言，Q = e = 1.60 × 10⁻¹⁹ C。因此，1 eV = (1.60 × 10⁻¹⁹ C) × (1 V) = 1.60 × 10⁻¹⁹ J。这一换算在量子物理和粒子物理中十分常用，能量常以 eV 而非焦耳表示。
10. De Broglie Wavelength λ = h/p | 德布罗意波长 λ = h/p 的推导
Louis de Broglie proposed that moving particles exhibit wave-like behaviour, with a wavelength inversely proportional to their momentum. From Einstein’s energy–momentum relation for photons, E = pc, and the photon energy E = hf = hc/λ, equating yields pc = hc/λ, so λ = h/p. De Broglie generalised this to all matter, assigning a wavelength to a particle of mass m moving at speed v as λ = h/(mv). This bold hypothesis was confirmed by electron diffraction experiments and underpins wave–particle duality.
德布罗意提出运动的粒子表现出波动性，其波长与动量成反比。根据爱因斯坦的光子能量–动量关系 E = pc，以及光子能量 E = hf = hc/λ，令两者相等得 pc = hc/λ，因此 λ = h/p。德布罗意将这一关系推广到一切实物粒子，质量为 m、速度为 v 的粒子其波长为 λ = h/(mv)。这一大胆假说被电子衍射实验证实，奠定了波粒二象性的基础。
11. Deriving the Wave Equation v = fλ | 波速公式 v = fλ 的推导
Any continuous wave can be characterised by its frequency f (the number of complete oscillations per second) and its wavelength λ (the distance between successive points in phase). In one period T = 1/f, the wave advances by exactly one wavelength. Speed is distance divided by time, so v = λ/T = λ × (1/T) = fλ. This simple derivation applies to all mechanical and electromagnetic waves, linking the spatial and temporal aspects of wave motion.
任何连续波都可用频率 f（每秒完整振动的次数）和波长 λ（相邻同相点间的距离）来描述。在一个周期 T = 1/f 内，波形正好前进一个波长。速度等于距离除以时间，因此 v = λ/T = λ × (1/T) = fλ。这一简单推导适用于所有机械波与电磁波，将波动的空间特征与时间特征联系起来。
12. Deriving the Power Dissipation Formula P = I²R = V²/R | 电功率公式 P = I²R = V²/R 的推导
Power is the rate of energy transfer. When a steady current I flows through a component with potential difference V, the charge Q = I t passes in time t, and the energy transferred is W = QV = I t V. Hence power P = W/t = I V. Using Ohm’s law for a resistor (V = IR), we can substitute to obtain two alternative forms: P = I (IR) = I²R, or similarly P = (V/R) V = V²/R. These expressions are vital for calculating energy dissipated as heat in circuit elements.
功率是能量转换的速率。当稳定电流 I 通过电势差为 V 的元件时，在时间 t 内通过的电荷量 Q = I t，传递的能量 W = QV = I t V。因此功率 P = W/t = I V。利用欧姆定律 V = IR，代入可得另两种形式：P = I (IR) = I²R，或 P = (V/R) V = V²/R。这些表达式对于计算电路元件中作为热量耗散的能量至关重要。
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