📚 A-Level OCR Physics: June 2023 Paper 2 Formula Derivations | OCR A-Level物理2023年6月卷2公式推导
OCR A-Level Physics June 2023 Paper 2 put a strong emphasis on understanding and deriving fundamental physics equations. This article revisits the key derivations that appeared in the exam, breaking them down step by step so you can master the logic behind each formula and feel fully prepared for future assessments.
OCR A-Level物理2023年6月卷2对理解并推导基本物理方程的考查非常突出。本文重新梳理了卷子中出现的关键公式推导，一步步拆解背后的逻辑，帮助你彻底掌握每个公式的来龙去脉，为今后的考试做好充分准备。
1. Deriving the Wave Equation v = fλ | 推导波速公式 v = fλ
Start with the definition of wave speed: speed is the distance travelled per unit time. For a wave, the distance travelled in one complete cycle is the wavelength λ.
从波速的定义出发：速度是单位时间内传播的距离。对于一个波,在一个完整周期内传播的距离就是波长λ。
The time taken to travel one wavelength is the period T of the wave.
传播一个波长所需的时间就是波的周期T。
Therefore, wave speed v = distance / time = λ / T.
因此，波速 v = 距离 / 时间 = λ / T。
Since frequency f is the number of cycles per second, it is the reciprocal of the period: f = 1 / T.
由于频率f是每秒的周期数，它与周期互为倒数：f = 1 / T。
Substituting 1 / f for T gives v = λ × f, which is the wave equation.
将1 / f 替换T得到 v = λ × f，这就是波动方程。
v = f λ
2. Deriving Snell’s Law from Huygens’ Principle | 从惠更斯原理推导斯涅尔定律
Consider a plane wavefront travelling from medium 1 to medium 2 at an angle. Wave speed changes from v₁ to v₂ at the boundary.
考虑一列平面波阵面以一定角度从介质1进入介质2。波速在界面处由v₁变为v₂。
In a time Δt, point A on the wavefront travels to point B in medium 1, so AB = v₁ Δt. Simultaneously, point C in medium 2 travels to point E, where CE = v₂ Δt.
在时间Δt内，波阵面上的点A在介质1内运动到点B，因此AB = v₁ Δt。与此同时，介质2中的点C运动到点E，CE = v₂ Δt。
The new wavefront is BE. Using geometry, the angles of incidence θ₁ and refraction θ₂ are related by sinθ₁ = AB / CB and sinθ₂ = CE / CB.
新波阵面是BE。利用几何关系，入射角θ₁和折射角θ₂满足 sinθ₁ = AB / CB， sinθ₂ = CE / CB。
Dividing these gives sinθ₁ / sinθ₂ = AB / CE = v₁ / v₂ = constant. This is Snell’s law.
两式相除得到 sinθ₁ / sinθ₂ = AB / CE = v₁ / v₂ = 常数。这就是斯涅尔定律。
sinθ₁ / sinθ₂ = v₁ / v₂ = n₂ / n₁
3. Deriving the Critical Angle for Total Internal Reflection | 推导全内反射临界角
Total internal reflection occurs when light travels from a denser to a rarer medium and the angle of refraction reaches 90°.
全内反射发生在光从光密介质射向光疏介质且折射角达到90°时。
Using Snell’s law: n₁ sinθc = n₂ sin90°, where θc is the critical angle.
应用斯涅尔定律：n₁ sinθc = n₂ sin90°，其中θc就是临界角。
Since sin90° = 1, we have sinθc = n₂ / n₁. For light leaving a material of refractive index n into air (n₂ ≈ 1), sinθc = 1 / n.
因为 sin90° = 1，所以 sinθc = n₂ / n₁。当光从折射率为n的材料进入空气（n₂ ≈ 1）时，sinθc = 1 / n。
Thus the critical angle is simply the inverse sine of the ratio of the refractive indices.
因此，临界角就是折射率之比的反三角函数值。
θc = sin⁻¹( n₂ / n₁ )
4. Deriving the Double-Slit Fringe Separation Δx = λD / s | 推导双缝干涉条纹间距 Δx = λD / s
Consider two coherent sources separated by a distance s. On a screen at distance D, bright fringes appear where the path difference is nλ.
考虑两个相距s的相干光源。在距离为D的屏幕上，当光程差为nλ时出现亮条纹。
For small angles, the path difference between rays from the two slits to a point on the screen is approximately s × θ, where θ is the angle measured from the centre.
在小角度下，从双缝到屏幕上某点的两束光之间的光程差近似为 s × θ，其中θ是从中心线量起的角度。
Also, the linear displacement y on the screen from the central maximum is related by tanθ ≈ θ = y / D.
同时，屏幕上距中央亮纹的线位移y满足 tanθ ≈ θ = y / D。
For the first-order bright fringe (n=1), we set path difference sθ = λ. Substituting θ = y / D gives s × y / D = λ, so y = λD / s.
对一级亮纹（n=1），令光程差 sθ = λ。代入 θ = y / D 得到 s × y / D = λ，因此 y = λD / s。
Thus the fringe separation Δx between adjacent bright fringes is the same value: Δx = λD / s.
因此相邻亮纹的间距Δx便是同一个数值：Δx = λD / s。
Δx = λD / s
5. Deriving the Diffraction Grating Equation d sinθ = nλ | 推导衍射光栅方程 d sinθ = nλ
A diffraction grating consists of many equally spaced slits, with slit separation d. When a plane wave passes through, each slit acts as a coherent source.
衍射光栅由许多等间距的狭缝组成，缝间距为d。当平面波通过时，每条缝相当于一个相干光源。
For a bright fringe at angle θ, the path difference between waves from adjacent slits must be an integer multiple of the wavelength: d sinθ = nλ.
对于角度θ处的亮纹，相邻狭缝发出的波必须满足光程差为波长的整数倍：d sinθ = nλ。
This simple geometry comes from the right-angled triangle where the path difference is the side opposite angle θ in a triangle of hypotenuse d.
这一简单的几何关系源自一个直角三角形，其中光程差是斜边为d、角度为θ的三角形中对边。
Hence the grating equation is derived as d sinθ = nλ, where n is the order of the maximum.
由此推导出光栅方程 d sinθ = nλ，其中n是极大值的级数。
d sinθ = nλ
6. Deriving the Capacitor Discharge Equation Q = Q₀ e^(-t/RC) | 推导电容器放电方程 Q = Q₀ e^(-t/RC)
In an RC circuit, the rate of decrease of charge Q on the capacitor is proportional to the current I, which itself is Q / (RC) from V = IR and Q = CV.
在RC电路中，电容器上电荷Q的减少速率与电流I成正比，而根据V = IR 和 Q = CV，I = Q / (RC)。
Thus dQ / dt = – Q / (RC). This first-order differential equation can be solved by separating variables.
因此 dQ / dt = – Q / (RC)。这一阶微分方程可以通过分离变量来求解。
Separate: dQ / Q = – dt / (RC). Integrate both sides from initial charge Q₀ at t=0 to Q at time t.
分离变量：dQ / Q = – dt / (RC)。从t=0时的初始电荷Q₀到t时刻的Q对两边积分。
∫ dQ / Q = -∫ dt / (RC) gives ln(Q) – ln(Q₀) = – t / (RC). Exponentiating yields Q = Q₀ e^(-t/RC).
∫ dQ / Q = -∫ dt / (RC) 得到 ln(Q) – ln(Q₀) = – t / (RC)。两边取指数得 Q = Q₀ e^(-t/RC)。
The product RC is called the time constant τ, representing the time for the charge to fall to 1/e of its initial value.
乘积RC称为时间常数τ，表示电荷衰减至初始值1/e所需的时间。
Q = Q₀ e^(-t/RC)
7. Deriving the Energy Stored in a Capacitor E = ½CV² | 推导电容器储存的能量 E = ½CV²
Charging a capacitor means transferring charge Q against an increasing potential difference. The work done dW to add a small charge dq is V dq, where V = q / C.
给电容器充电意味着在逐渐升高的电势差下迁移电荷Q。迁移小量电荷dq所做的功 dW = V dq，其中V = q / C。
Integrate from q=0 to q=Q: W = ∫ V dq = ∫ (q / C) dq = (1 / C) ∫₀꙰ q dq = (1 / C) × ½ Q² = ½ Q² / C.
从q=0到q=Q积分：W = ∫ V dq = ∫ (q / C) dq = (1 / C) ∫₀꙰ q dq = (1 / C) × ½ Q² = ½ Q² / C。
Substituting Q = CV gives W = ½ (CV)² / C = ½ C V². This work is stored as electrical potential energy in the capacitor.
代入Q = CV 得 W = ½ (CV)² / C = ½ C V²。这份功以电势能的形式储存在电容器中。
Thus the energy stored is E = ½ C V². The same result can be expressed as ½ Q V or ½ Q² / C.
因此储存的能量为 E = ½ C V²。该结果也可表达为 ½ Q V 或 ½ Q² / C。
E = ½ C V²
8. Deriving the Radioactive Decay Law and Half-Life | 推导放射性衰变规律及半衰期
Radioactive decay is a random process where the number of nuclei N decreases at a rate proportional to N: dN/dt = – λ N.
放射性衰变是一种随机过程，原子核数目N的减少速率与N成正比：dN/dt = – λ N。
Separation of variables gives dN / N = – λ dt. Integrating from N₀ at t=0 to N at t yields ln(N / N₀) = – λ t.
分离变量得 dN / N = – λ dt。从t=0时的N₀积分到t时的N得到 ln(N / N₀) = – λ t。
Exponentiating gives the exponential decay law N = N₀ e^(-λ t). The decay constant λ is related to the half-life T₁/₂.
两边取指数得到指数衰减定律 N = N₀ e^(-λ t)。衰变常数λ与半衰期T₁/₂有关。
By setting N = N₀ / 2, we get ½ = e^(-λ T₁/₂), so taking natural logs: -ln 2 = -λ T₁/₂, hence T₁/₂ = ln 2 / λ.
令N = N₀ / 2，得到½ = e^(-λ T₁/₂)，取自然对数：-ln 2 = -λ T₁/₂，因此 T₁/₂ = ln 2 / λ。
N = N₀ e^(-λ t) , T₁/₂ = ln 2 / λ
9. Deriving the de Broglie Wavelength λ = h / p | 推导德布罗意波长 λ = h / p
De Broglie proposed that moving particles exhibit wave-like behaviour, with wavelength inversely proportional to momentum.
德布罗意提出运动粒子表现出波动性，其波长与动量成反比。
Starting from Einstein’s photon energy E = h f and the mass–energy equivalence E = m c², a photon’s momentum is p = E / c = h f / c.
从爱因斯坦的光子能量E = h f 和质能等价E = m c²出发，光子的动量 p = E / c = h f / c。
Since f / c = 1 / λ for a photon, p = h / λ. Generalising to any particle with momentum p = m v gives λ = h / p.
对光子而言 f / c = 1 / λ，因此 p = h / λ。将这一关系推广到任意动量为 p = m v 的粒子，即得 λ = h / p。
This derivation bridges the particle and wave models and is fundamental to quantum mechanics.
这一推导在粒子和波动的模型之间架起了桥梁，是量子力学的基础。
λ = h / p
10. Deriving the Photoelectric Effect Equation eV_s = hf – ϕ | 推导光电效应方程 eV_s = hf – ϕ
In the photoelectric effect, a photon of energy h f is absorbed by an electron. The electron uses an energy equal to the work function ϕ to escape the metal.
在光电效应中，一个能量为h f的光子被电子吸收。电子需要消耗相当于逸出功ϕ的能量才能逃离金属。
If the photon energy exceeds ϕ, the remaining energy becomes the electron’s maximum kinetic energy: KE_max = h f – ϕ.
如果光子能量大于ϕ，剩余的能量转化为电子的最大动能：KE_max = h f – ϕ。
A stopping potential V_s can be applied to reduce the photocurrent to zero. The work done by this potential is e V_s, which equals KE_max.
施加遏止电势V_s可使光电流降至零。该电势做的功 e V_s 等于 KE_max。
Hence e V_s = h f – ϕ, the photoelectric effect equation. This derivation beautifully confirms energy conservation.
由此得到 e V_s = h f – ϕ，即光电效应方程。这一推导完美地印证了能量守恒原理。
e V_s = h f – ϕ
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