📚 Deriving Key Formulae from A-Level Physics Insert 4 (June 2018) | A-Level物理核心公式推导（2018年6月插入页4）
The 2018 June A-Level Physics data sheet (Insert 4) provides a concise collection of essential equations that underpin mechanics, waves, electricity and materials. Simply memorising these formulae is not enough – understanding their physical origin and derivation deepens conceptual mastery and makes it easier to apply them in unfamiliar contexts. In this article, we step through the logical reasoning and mathematical derivations behind ten of the most fundamental formulae from that insert, connecting each one to core principles such as Newton’s laws, conservation of energy, wave interference and electromagnetism.
2018年6月A-Level物理数据表（插入页4）集中列出了力学、波、电学和材料等领域的核心方程。仅仅记住这些公式是不够的——理解它们的物理来源和推导过程能加深概念掌握，使我们在陌生情境中也能灵活应用。本文将逐一梳理该插入页中十个最基础公式的逻辑推理与数学推导，将每个公式与牛顿定律、能量守恒、波的干涉和电磁学等核心原理紧密联系起来。
1. Equations of Uniformly Accelerated Motion | 匀加速运动方程
The four kinematic equations found on the insert are not independent laws but direct consequences of the definitions of acceleration and average velocity under constant acceleration. Starting with the definition of acceleration, a = (v − u) / t, we immediately obtain v = u + at. Since velocity changes linearly with time, the average velocity is (u + v) / 2, and displacement is simply average velocity multiplied by time: s = ((u + v) / 2) t.
插入页上的四个运动学方程并非独立定律，而是恒定加速度下加速度和平均速度定义的直接结果。由加速度的定义 a = (v − u) / t 立即得到 v = u + at。由于速度随时间线性变化，平均速度为 (u + v) / 2，因此位移等于平均速度乘以时间：s = ((u + v) / 2) t。
Substituting v = u + at into s = ((u + v) / 2) t yields s = ut + ½at². To eliminate t, solve v = u + at for t, giving t = (v − u) / a, and insert into the displacement equation to obtain v² = u² + 2as. These four relationships form the bedrock of linear motion analysis.
将 v = u + at 代入 s = ((u + v) / 2) t 得到 s = ut + ½at²。消去时间 t：由 v = u + at 解出 t = (v − u) / a，再代入位移方程，即得 v² = u² + 2as。这四个关系构成了直线运动分析的基石。
2. Newton’s Second Law and Impulse | 牛顿第二定律与冲量
Newton’s second law in its most powerful form is F = dp/dt, where p = mv is linear momentum. For a constant force, integrating over the time of interaction Δt gives F Δt = Δp = mv − mu. This impulse–momentum relationship, often listed as F Δt = Δ(mv), explains how a force applied over time changes an object’s momentum, and underpins vehicle safety and collision analysis.
牛顿第二定律最强大的形式是 F = dp/dt，其中 p = mv 为线动量。对于恒定力，对作用时间 Δt 积分得到 F Δt = Δp = mv − mu。冲量–动量关系式常列为 F Δt = Δ(mv)，它解释了力在一定时间内如何改变物体的动量，是汽车安全和碰撞分析的基础。
3. Work and Kinetic Energy | 功与动能
When a constant net force F acts on an object over a displacement s, the work done is W = F s. Using v² = u² + 2as and Newton’s second law F = ma, we rewrite displacement as s = (v² − u²) / (2a). Then F s = ma × (v² − u²) / (2a) = ½m(v² − u²). Defining kinetic energy as KE = ½mv², this shows that the net work done equals the change in kinetic energy: W_net = ΔKE.
当恒定合外力 F 作用在物体上产生位移 s 时，所做的功为 W = F s。利用 v² = u² + 2as 和牛顿第二定律 F = ma，将位移写为 s = (v² − u²) / (2a)。于是 F s = ma × (v² − u²) / (2a) = ½m(v² − u²)。定义动能 KE = ½mv²，即可见合外力做功等于动能的变化：W_net = ΔKE。
4. Gravitational Potential Energy Near the Earth’s Surface | 地表附近的重力势能
The work done against a uniform gravitational field g when lifting a mass m through a vertical height Δh is W = F Δh = mg Δh. This work is stored as gravitational potential energy, so ΔGPE = mg Δh. The data sheet therefore gives GPE = mgh relative to a chosen zero reference level, a direct consequence of the definition of work in a uniform field.
在均匀重力场 g 中将质量为 m 的物体竖直提升高度 Δh，克服重力所做的功为 W = F Δh = mg Δh。这份功以重力势能的形式储存，因此 ΔGPE = mg Δh。因而数据表上给出以选定零势能面为参考的 GPE = mgh，这是均匀场中功的定义的直接结果。
5. Elastic Potential Energy of a Spring | 弹簧的弹性势能
Hooke’s law states F = k x, where k is the spring constant and x the extension. Since the force increases linearly from zero to F_max = k x, the average force during stretching is ½k x. The work done (and hence the energy stored) is average force × displacement: Eₑₗ = (½k x) × x = ½k x². This result can also be obtained from the area under the force–extension graph, a triangle of base x and height k x.
胡克定律指出 F = k x，其中 k 为劲度系数，x 为伸长量。由于力从零线性增大至 F_max = k x，拉伸过程中的平均力为 ½k x。所做的功（即储存的能量）为平均力乘以位移：Eₑₗ = (½k x) × x = ½k x²。这一结果也可由力–伸长图下面积（底 x、高 k x 的三角形）得出。
6. Young Modulus | 杨氏模量
Stress is defined as the applied force per unit cross-sectional area, σ = F / A, while strain is the fractional extension, ε = ΔL / L. Young modulus E is the ratio of stress to strain in the linear elastic region: E = σ / ε = (F / A) / (ΔL / L). This formula, often rewritten as E = (F L) / (A ΔL), characterises the stiffness of a material independently of its dimensions.
应力定义为施加的力与横截面积之比，σ = F / A；应变则是相对伸长量，ε = ΔL / L。杨氏模量 E 是线弹性区内应力与应变之比：E = σ / ε = (F / A) / (ΔL / L)。该公式常改写为 E = (F L) / (A ΔL)，它表征材料本身与尺寸无关的刚度。
7. Refractive Index and Snell’s Law | 折射率与斯涅尔定律
When a wave passes from medium 1 to medium 2, its frequency remains constant while its speed changes. For light, refractive index n is defined as the ratio of the speed of light in vacuum to the speed in the medium: n = c / v. At the boundary, the wavefronts satisfying Huygens’ principle lead to n₁ sinθ₁ = n₂ sinθ₂, where θ is the angle to the normal. This is Snell’s law, and using the definition of n it can be expressed equivalently as (sinθ₁) / v₁ = (sinθ₂) / v₂.
波从介质1进入介质2时，频率保持不变，而波速改变。对于光，折射率 n 定义为真空中光速与介质中光速之比：n = c / v。在界面处，满足惠更斯原理的波阵面推导出 n₁ sinθ₁ = n₂ sinθ₂，其中 θ 为与法线的夹角。这就是斯涅尔定律；利用 n 的定义可等价写成 (sinθ₁) / v₁ = (sinθ₂) / v₂。
8. The Diffraction Grating Equation | 衍射光栅方程
A diffraction grating consists of many equally spaced slits separated by a distance d. For constructive interference of light passing through adjacent slits, the path difference must be an integer multiple of the wavelength λ: d sinθ = nλ, where n = 0, ±1, ±2, … and θ is the angle of the nth-order maximum from the centre line. This formula is listed on the insert as a key condition for observing bright fringes.
衍射光栅由大量间距为 d 的等距狭缝组成。对于相邻狭缝通过的光发生相长干涉，其光程差必须为波长 λ 的整数倍：d sinθ = nλ，其中 n = 0, ±1, ±2, …，θ 为第 n 级明纹与中心线之间的夹角。该公式是数据表上观察亮条纹的关键条件。
9. Resistivity and Resistance | 电阻率与电阻
For a uniform conductor of length L and cross-sectional area A, resistance R is directly proportional to L and inversely proportional to A, giving R ∝ L / A. Introducing resistivity ρ as the constant of proportionality yields R = ρ L / A. This relationship is derived from the microscopic drift velocity model, where resistance arises from collisions between free electrons and the lattice, and ρ is a material-specific property that depends on temperature.
对于长为 L、横截面积为 A 的均匀导体，电阻 R 与 L 成正比、与 A 成反比，即 R ∝ L / A。引入比例常数——电阻率 ρ，得到 R = ρ L / A。该关系可由微观漂移速度模型推导：电阻源于自由电子与晶格的碰撞，ρ 是取决于温度的材料特性。
10. Parallel-Plate Capacitance | 平行板电容
The capacitance C is defined as the ratio of the charge stored on one plate to the potential difference between the plates: C = Q / V. For two parallel plates of area A separated by distance d, the uniform electric field strength is E = V / d. Using Q = ε₀ A E from Gauss’s law (or the relation σ = ε₀ E for a vacuum), we find Q = ε₀ A (V / d), hence C = ε₀ A / d. If a dielectric of relative permittivity εᵣ is inserted, the formula becomes C = εᵣ ε₀ A / d.
电容 C 定义为极板上储存的电荷量与板间电势差之比：C = Q / V。对于面积为 A、间距为 d 的两平行板，均匀电场强度为 E = V / d。根据高斯定律（或真空中 σ = ε₀ E），有 Q = ε₀ A E = ε₀ A (V / d)，因此 C = ε₀ A / d。若插入相对电容率为 εᵣ 的介质，公式变为 C = εᵣ ε₀ A / d。
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