📚 Deriving Key Formulas from AS Physics Unit 2 June 2019 Insert | AS 物理 Unit 2（2019年6月）核心公式推导
The AS Physics Unit 2 Insert for June 2019 provides a reference list of key formulas covering waves, electricity, and quantum physics. Understanding how these equations are derived deepens your grasp of the underlying principles and prepares you for solving complex problems. This article walks through the derivations of the most important formulas from that insert, showing the logical steps from fundamental concepts.
2019年6月 AS 物理 Unit 2 公式表涵盖了波动、电学和量子物理的核心公式。理解这些方程的推导过程，能够加深对基本原理的掌握，并为解决复杂问题做好准备。本文将逐一推导该公式表中最重要的公式，展示从基本概念出发的逻辑步骤。
1. Wave Speed: v = fλ | 波速公式 v = fλ 的推导
A wave travels exactly one full wavelength λ in the time it takes to complete one oscillation, the period T. Speed is distance over time, so the wave speed v is given by the wavelength divided by the period: v = λ / T. Since frequency f is the reciprocal of the period (f = 1/T), we substitute to obtain the familiar form v = fλ.
波在一个完整的振动周期 T 内恰好传播一个波长 λ。速度等于距离除以时间，因此波速 v = λ / T。又因为频率 f 是周期的倒数（f = 1/T），代入后便得到熟知的公式 v = fλ。
v = fλ
2. Snell’s Law: n₁ sinθ₁ = n₂ sinθ₂ | 斯涅尔定律 n₁ sinθ₁ = n₂ sinθ₂ 的推导
When a wave crosses a boundary between two media, the wavefront must remain continuous. This geometric condition leads to the relationship sinθ₁ / v₁ = sinθ₂ / v₂, where v₁ and v₂ are the wave speeds in the two media. The absolute refractive index of a medium is defined as n = c / v, so v₁ = c / n₁ and v₂ = c / n₂. Substituting these into the wavefront condition gives sinθ₁ / (c/n₁) = sinθ₂ / (c/n₂). Cancelling c from both sides yields Snell’s law: n₁ sinθ₁ = n₂ sinθ₂.
当波穿过两种介质的交界面时，波前必须保持连续。这一几何条件给出关系式 sinθ₁ / v₁ = sinθ₂ / v₂，其中 v₁ 和 v₂ 分别是波在两种介质中的传播速度。介质的绝对折射率定义为 n = c / v，因此 v₁ = c / n₁，v₂ = c / n₂。代入波前条件得 sinθ₁ / (c/n₁) = sinθ₂ / (c/n₂)，消去 c 即得斯涅尔定律：n₁ sinθ₁ = n₂ sinθ₂。
n₁ sinθ₁ = n₂ sinθ₂
3. Critical Angle: sin θc = 1 / n | 临界角公式 sin θc = 1 / n 的推导
Total internal reflection occurs when light travels from an optically denser medium of refractive index n into a less dense medium with n = 1 (such as air) and the angle of refraction reaches 90°. Applying Snell’s law with n₁ = n, n₂ = 1, θ₁ = θc (critical angle), and θ₂ = 90°, we get n sinθc = 1 × sin90° = 1. Hence sinθc = 1 / n. This formula is valid only when the second medium is air or vacuum.
当光从折射率为 n 的光密介质射向折射率为 1 的光疏介质（如空气），且折射角达到 90° 时，发生全内反射。对斯涅尔定律取 n₁ = n，n₂ = 1，θ₁ = θc（临界角），θ₂ = 90°，得到 n sinθc = 1 × sin90° = 1，因此 sinθc = 1 / n。该公式仅在第二介质为空气或真空时成立。
sin θc = 1 / n
4. Diffraction Grating Equation: d sinθ = nλ | 衍射光栅方程 d sinθ = nλ 的推导
A diffraction grating consists of many parallel slits with equal spacing d. When monochromatic light passes through, waves from adjacent slits have a path difference of d sinθ to a distant point at angle θ. Constructive interference occurs when this path difference equals an integer multiple of the wavelength, nλ. Therefore, the condition for a bright fringe is d sinθ = nλ, where n = 0, ±1, ±2, … is the order number.
衍射光栅由许多间距为 d 的平行狭缝组成。单色光通过时，来自相邻狭缝的波到达远处某点、与入射方向成 θ 角时的光程差为 d sinθ。当该光程差等于波长的整数倍 nλ 时，发生相长干涉，因此亮纹条件为 d sinθ = nλ，其中 n = 0, ±1, ±2, … 为级数。
d sinθ = nλ
5. Ohm’s Law and Resistance: V = IR | 欧姆定律 V = IR 的推导
Resistance R is defined as the ratio of the potential difference V across a conductor to the current I flowing through it: R = V / I. For an ohmic conductor at constant temperature, this ratio is constant. Rearranging gives V = IR, which states that the voltage drop is directly proportional to the current. This is the foundation of DC circuit analysis.
电阻 R 定义为导体两端的电势差 V 与通过导体的电流 I 之比：R = V / I。对于温度不变的欧姆导体，该比值为常数。将定义式变形即得 V = IR，表明电压降与电流成正比。这是直流电路分析的基础。
V = IR
6. Power in DC Circuits: P = IV = I²R = V²/R | 直流电路功率公式推导
Electrical power is the rate at which energy is transferred. When a charge Q moves through a potential difference V, the work done is W = QV. Power P = W / t = (Q / t) V = I V. By substituting Ohm’s law V = IR into P = IV, we get P = I (IR) = I²R. Alternatively, writing I = V / R gives P = (V / R) V = V² / R. These three forms are equivalent for ohmic conductors.
电功率是能量传输的速率。电荷 Q 通过电势差 V 时，所做的功为 W = QV。功率 P = W / t = (Q / t) V = I V。将欧姆定律 V = IR 代入 P = IV，得 P = I (IR) = I²R；或由 I = V / R 代入，得 P = (V / R) V = V² / R。这三个表达式对欧姆导体等价。
P = IV = I²R = V²/R
7. EMF and Internal Resistance: ε = I(R + r) | 电动势与内阻公式 ε = I(R + r) 的推导
A real source of emf ε has an internal resistance r. According to energy conservation, the total power supplied by the source equals the power dissipated in the external load R plus the power wasted inside the source: ε I = I²R + I²r. Dividing through by the current I yields ε = IR + Ir = I(R + r). The terminal potential difference V across the source is then V = ε – Ir.
实际电源具有电动势 ε 和内阻 r。根据能量守恒，电源提供的总电功率等于外电路负载 R 消耗的功率与内阻上消耗的功率之和：ε I = I²R + I²r。两边同除以电流 I 得 ε = IR + Ir = I(R + r)。此时电源的端电压 V 为 V = ε – Ir。
ε = I(R + r)
8. Energy Transferred: W = VIt | 电功公式 W = VIt 的推导
The energy dissipated in a circuit component can be derived from the definitions of power and current. Since power P = VI and time t is the duration, the total energy transferred W equals power multiplied by time: W = Pt = VIt. Alternatively, from the definition of current I = Q / t, the charge Q = It, and work done W = QV, we again obtain W = VIt.
电路中元件消耗的能量可从功率和电流的定义导出。因功率 P = VI，持续时间为 t，故总能量 W = Pt = VIt。也可从电流定义 I = Q / t 得电荷 Q = It，再结合做功 W = QV，同样得到 W = VIt。
W = VIt
9. Electronvolt: 1 eV = e × 1 V | 电子伏特与焦耳的换算推导
An electronvolt is defined as the kinetic energy gained by a single electron when it is accelerated through a potential difference of exactly 1 volt. The work done by the electric field is W = qΔV, where q is the elementary charge e = 1.60 × 10⁻¹⁹ C. Thus, 1 eV = e × 1 V = 1.60 × 10⁻¹⁹ J. This unit is enormously helpful when dealing with energies on the atomic scale.
电子伏特定义为一个电子经过恰好 1 伏电势差加速后所获得的动能。电场力做功 W = qΔV，其中 q 为元电荷 e = 1.60 × 10⁻¹⁹ C。因此 1 eV = e × 1 V = 1.60 × 10⁻¹⁹ J。在处理原子尺度的能量时，这一单位极其便利。
1 eV = 1.60 × 10⁻¹⁹ J
10. Photoelectric Effect Stopping Potential: eVs = hf – φ | 光电效应截止电压公式推导
Einstein’s photoelectric equation states that the maximum kinetic energy Kmax of emitted photoelectrons is Kmax = hf – φ, where hf is the photon energy and φ is the work function of the metal. In the stopping potential experiment, a reverse voltage Vs is applied until even the most energetic electrons are brought to rest. At that point, the loss of kinetic energy equals the gain in electrical potential energy: eVs = Kmax. Combining the two gives eVs = hf – φ.
爱因斯坦光电方程指出，逸出光电子的最大动能 Kmax = hf – φ，其中 hf 是光子能量，φ 为金属的逸出功。在截止电势差的实验中，施加反向电压 Vs，直到动能最大的电子也被遏止。此时动能损失等于电势能的增加：eVs = Kmax。联立即得 eVs = hf – φ。
eVs = hf – φ
11. de Broglie Wavelength: λ = h / p | 德布罗意波长公式 λ = h / p 的推导
The de Broglie relation emerges from the wave-particle duality first applied to light. A photon has energy E = hf and travels at speed c, so its momentum p = E / c = hf / c. Using the wave equation c = fλ, we replace c to find p = hf / (fλ) = h / λ. Rearranging gives λ = h / p for photons. De Broglie proposed that the same equation holds for all particles, linking their wavelength to momentum.
德布罗意关系来源于最初应用于光的波粒二象性。光子能量 E = hf，以光速 c 传播，其动量 p = E / c = hf / c。利用波速公式 c = fλ 消去 c，得 p = hf / (fλ) = h / λ。变形后对光子有 λ = h / p。德布罗意提出，所有实物粒子均满足同一方程，将其波长与动量联系起来。
λ = h / p
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