A-Level Physics June 2018 Unit 3 Examiner's Report: Key Formula Derivations | A-Level物理2018年6月第三单元考官报告：关键公式推导

📚 A-Level Physics June 2018 Unit 3 Examiner’s Report: Key Formula Derivations | A-Level物理2018年6月第三单元考官报告：关键公式推导

The June 2018 Unit 3 examiner’s report for A-Level Physics highlighted that many candidates lost marks not because they lacked understanding of the underlying physics, but because they could not present clear, logical derivations of key experimental formulas. This article breaks down the most important derivations and common pitfalls, helping you turn raw data into reliable conclusions.

2018年6月A-Level物理第三单元的考官报告指出，许多考生失分并非因为不理解物理原理，而是因为他们无法清晰、有条理地推导关键的实验公式。本文详细解析最重要的推导过程和常见误区，帮助你从原始数据得出可靠结论。

1. Importance of Formula Derivation in Practical Assessments | 实验评估中公式推导的重要性

The examiner’s report stressed that students often jump straight to substituting numbers without showing how the working formula arises from fundamental definitions. In Unit 3 practical skills, you are expected to manipulate equations, linearise relationships, and combine variables correctly.

考官报告强调，学生经常会直接代入数字，却不展示计算公式是如何从基本定义推导出来的。在第三单元实验技能中，你需要能够对等式进行变换、将关系线性化，并正确地进行变量组合。

Marks are awarded for demonstrating the logical progression from a theoretical expression to the specific equation used to analyse your data. For instance, blindly quoting ‘g = 2s/t²’ without deriving it from s = ut + ½at² or explaining why u = 0 could cost you the communication marks.

展示从理论表达式到用于分析数据的特定公式的逻辑推进过程，是可以得分的。例如，盲目引用 ‘g = 2s/t²’ 而不从 s = ut + ½at² 推导，或不解释为什么 u = 0，可能会让你失掉表达分。

2. Deriving Resistivity from Ohm’s Law and Geometry | 由欧姆定律和几何推导电阻率

One of the most frequent derivations required in electricity practicals is the resistivity formula ρ = RA / L. Start with the definition of resistance R = V / I and the fact that resistance is proportional to length L and inversely proportional to cross-sectional area A.

电学实验中要求最多的推导之一就是电阻率公式 ρ = RA / L。从电阻的定义 R = V / I 出发，并利用电阻与长度 L 成正比、与横截面积 A 成反比这一事实。

The full relationship is R = ρL / A. To find ρ, multiply both sides by A and divide by L: ρ = RA / L. When a wire’s diameter d is given, area A is π(d/2)², a step that the examiner’s report noted was often omitted or miscalculated.

完整关系为 R = ρL / A。要求出 ρ，将两边同时乘以 A 再除以 L：ρ = RA / L。当给出导线直径 d 时，面积 A 为 π(d/2)²，考官报告指出这一步常被忽略或被错误计算。

ρ = R × (π d² / 4) / L or ρ = (π R d²) / (4 L)

Common mistake: using diameter directly as area, or forgetting to halve the diameter before squaring. Always show the substitution step in your write-up.

常见错误：直接用直径代表面积，或忘记先将直径除以2再平方。在你的实验报告中一定要写出代入步骤。

3. Determination of g Using Free Fall: Linearising s = ut + ½at² | 利用自由落体测定g：线性化公式

For an object dropped from rest (u = 0), the displacement-time relation simplifies to s = ½gt². To extract g from experimental data, you need to create a linear graph.

对于由静止开始下落的物体（u = 0），位移–时间关系简化为 s = ½gt²。要从实验数据中提取 g，你需要绘制线形图像。

Rearranging gives s = (g/2) t². Plotting s on the y-axis against t² on the x-axis yields a straight line through the origin with gradient = g/2. Then g = 2 × gradient. The examiner commented that many candidates plotted s vs t instead, obtaining a curve and attempting to force a straight line, which is invalid.

重排后得 s = (g/2) t²。将 s 作为 y 轴，t² 作为 x 轴绘图，可得到一条过原点的直线，其斜率为 g/2。那么 g = 2 × 斜率。考官评论说，许多考生绘制的是 s–t 图，得到一条曲线却试图硬画成直线，这是无效的。

If the object has an initial downward velocity, use the full equation s = ut + ½gt², which can be rearranged as s/t = u + (g/2)t. Plot s/t vs t, gradient = g/2.

如果物体具有向下的初速度，则使用完整公式 s = ut + ½gt²，可重排为 s/t = u + (g/2)t。绘制 s/t–t 图像，斜率为 g/2。

4. Young’s Modulus and the Gradient of a Force-Extension Graph | 杨氏模量与力–伸长图像梯度

Young’s modulus E is defined as stress/strain = (F/A) / (ΔL/L₀). When verifying Hooke’s law and determining E for a wire, the raw data is force F and extension ΔL.

杨氏模量 E 定义为应力/应变 = (F/A) / (ΔL/L₀)。在验证胡克定律并测定金属丝的杨氏模量时，原始数据是力 F 和伸长量 ΔL。

Rearranging the definition gives F = (EA / L₀) ΔL. Therefore, a graph of F (y-axis) against ΔL (x-axis) should be a straight line with gradient = EA / L₀. From this, E = gradient × (L₀ / A). Be careful to use the original length L₀, not the stretched length.

重排定义式得 F = (EA / L₀) ΔL。因此，绘制 F（y 轴）– ΔL（x 轴）图像应为一条直线，其斜率为 EA / L₀。由此可得 E = 斜率 × (L₀ / A)。注意要使用原长 L₀，而不是拉伸后的长度。

E = (gradient × L₀) / (π d² / 4)

The report observed errors where students used the wire’s diameter at maximum load rather than the initial diameter; stress and strain are defined using original dimensions.

报告观察到有学生使用最大负载时的导线直径而非初始直径来计算，这是错误的；应力和应变均使用原始尺寸来定义。

5. Effective Spring Constants for Series and Parallel Combinations | 串联和并联弹簧的有效劲度系数

When springs are combined, deriving the effective spring constant starts from the same extension or same force principle. For springs in series, the force is the same in each, but extensions add.

当弹簧组合在一起时，有效劲度系数的推导始于“力相同”或“伸长量相同”的原理。对于串联弹簧，每根弹簧受力相同，但伸长量相加。

Let total extension x_total = x₁ + x₂. Using F = kx, we have x₁ = F/k₁, x₂ = F/k₂. Thus total extension = F/k_eq = F/k₁ + F/k₂. Cancel F to get 1/k_eq = 1/k₁ + 1/k₂.

设总伸长量 x_total = x₁ + x₂。利用 F = kx，有 x₁ = F/k₁，x₂ = F/k₂。因此总伸长量 = F/k_eq = F/k₁ + F/k₂。约去 F 得 1/k_eq = 1/k₁ + 1/k₂。

1 / k_eq = 1 / k₁ + 1 / k₂ (series)

For parallel springs, the extensions are equal (x) but the total force F = F₁ + F₂ = k₁x + k₂x = (k₁+k₂)x. Hence k_eq = k₁ + k₂. Examiners noted that mixing up these two derivations remains a very common error.

对于并联弹簧，伸长量相同（x），但总力 F = F₁ + F₂ = k₁x + k₂x = (k₁+k₂)x。因此 k_eq = k₁ + k₂。考官指出，将这两种推导混淆仍然是非常普遍的错误。

6. The Simple Pendulum and T² vs L Graph | 单摆与T²–L图像

The period of a simple pendulum for small amplitudes is given by T = 2π √(L/g). This equation is not linear, so to determine g from a graph we square both sides.

小振幅单摆的周期由 T = 2π √(L/g) 给出。该方程是非线性的，所以为了从图像中测定 g，我们将两边同时平方。

Squaring yields T² = (4π²/g) L. This is of the form y = mx, where y = T², x = L, and gradient m = 4π²/g. Then g = 4π² / gradient. The examiner’s report stressed the need to label axes with the correct compound units, e.g., T² / s² and L / m.

平方后得 T² = (4π²/g) L。这是 y = mx 的形式，其中 y = T²，x = L，斜率 m = 4π²/g。因此 g = 4π² / 斜率。考官报告强调，需要在坐标轴上标注正确的复合单位，例如 T²/s² 和 L/m。

A frequent pitfall: measuring pendulum length to the centre of the bob but not adding the radius of the bob. The effective length L is from the pivot to the centre of mass, which must be clearly described in the derivation as L = string length + bob radius.

一个常见的陷阱：测量摆长时只到摆球的上端，却没有加上摆球的半径。有效摆长 L 是从悬挂点到质心的距离，在推导中必须明确描述 L = 绳长 + 摆球半径。

7. Young’s Double Slit: Deriving λ = ax D and Its Uncertainties | 杨氏双缝干涉：推导λ = ax/D及其不确定度

The double slit formula λ = ax / D arises from the path difference geometry. For the n-th bright fringe (usually n=1 for the first order maximum), path difference = nλ = d sinθ.

双缝公式 λ = ax/D 源于光程差的几何关系。对于第 n 级亮纹（通常 n=1 为第一级极大），光程差 = nλ = d sinθ。

For small angles, sinθ ≈ tanθ = x/D, where x is the fringe separation measured from the central maximum and D is the slit-to-screen distance. Substituting gives nλ = d (x/D). For the separation between adjacent fringes Δx, using n=1 we get λ = d Δx / D. Often the symbol ‘a’ is used for slit spacing, hence λ = a Δx / D.

对于小角度，sinθ ≈ tanθ = x/D，其中 x 为从中央极大测得的条纹位置，D 为双缝到屏幕的距离。代入得 nλ = d (x/D)。对于相邻条纹间距 Δx，将 n=1 代入即可得 λ = d Δx / D。通常用符号 ‘a’ 表示缝间距，因此 λ = a Δx / D。

λ = a Δx / D

The report highlighted that candidates often measured several fringe spacings to find an average Δx, but then used n=1 directly. The correct step is to measure the width of N fringes, get Δx = width / N, and use λ = a Δx / D.

报告强调，考生常测量多个条纹间距求平均 Δx，但在公式中错误地直接使用 n=1。正确的步骤是测量 N 条条纹的总宽度，求得 Δx = 总宽度 / N，再使用 λ = a Δx / D。

For uncertainty analysis, the fractional uncertainty in λ is the sum of fractional uncertainties in a, Δx, and D: Δλ/λ = Δa/a + Δ(Δx)/Δx + ΔD/D, since the quantities are multiplied and divided.

对于不确定度分析，由于各量相乘或相除，λ 的相对不确定度为 Δλ/λ = Δa/a + Δ(Δx)/Δx + ΔD/D。

8. Capacitor Discharge and Logarithmic Linearisation | 电容器放电与对数线性化

The exponential decay of voltage in an RC circuit is V = V₀ e^(−t / RC). To verify this relationship and determine the time constant RC, you must convert it to straight-line form.

RC电路中电压的指数衰减公式为 V = V₀ e^(−t/RC)。为了验证这一关系并测定时间常数 RC，必须将其转化为直线形式。

Take the natural logarithm of both sides: ln V = ln V₀ − (1/RC) t. This matches y = mx + c, with y = ln V, x = t, gradient m = −1/RC, and intercept c = ln V₀.

对两边取自然对数：ln V = ln V₀ − (1/RC) t。这与 y = mx + c 的形式吻合，其中 y = ln V，x = t，斜率 m = −1/RC，截距 c = ln V₀。

gradient = − 1 / (RC) → RC = − 1 / gradient

The examiner observed that some students plotted log₁₀ instead of ln, resulting in a different gradient (log₁₀ V = log₁₀ V₀ − t/(2.303 RC)). Both are acceptable if clearly stated, but mixing the two without correction leads to errors.

考官注意到，部分学生使用 log₁₀ 而非 ln 绘图，导致斜率不同（log₁₀ V = log₁₀ V₀ − t/(2.303 RC)）。如果事先明确说明，两种方法均可接受；但若混淆而不进行修正则会导致错误。

9. Error Propagation in Derived Quantities | 导出量的误差传播

Once a formula is derived, you need to estimate the uncertainty in the final result. The examiner’s report pointed out that many candidates still struggle with combining absolute and percentage uncertainties correctly.

一旦公式推导完成，就需要估算最终结果的不确定度。考官报告指出，许多考生仍然不能正确地进行绝对不确定度和百分比不确定度的组合。

For a product or quotient, like ρ = RA / L, add the percentage uncertainties: %U(ρ) = %U(R) + %U(A) + %U(L). For a quantity raised to a power, multiply the percentage uncertainty by that power.

对于积或商的形式，如 ρ = RA / L，将百分比不确定度相加：%U(ρ) = %U(R) + %U(A) + %U(L)。对于有幂次的量，将百分比不确定度乘以相应的幂次。

If a constant is involved, e.g., g = 4π² / gradient, the uncertainty in g comes solely from the gradient uncertainty: Δg/g = Δ(gradient)/gradient. The constant 4π² has no uncertainty.

如果涉及常数，例如 g = 4π² / 斜率，g 的不确定度完全来自斜率的不确定度：Δg/g = Δ(斜率)/斜率。常数 4π² 没有不确定度。

The report recommended always showing the derivation of the uncertainty formula step by step, just as you show the main formula derivation. For instance, for λ = a Δx / D, explicitly state that the combination is multiplicative, so percentage uncertainties add.

报告建议，正如展示主公式推导一样，始终要逐步展示不确定度公式的推导。例如，对于 λ = a Δx / D，应明确说明其组合为乘法关系，因此百分比不确定度相加。

10. Examiner Recommendations for Presenting Derivation Steps | 考官关于推导步骤表达的建议

The exam board expects a structured approach: state the fundamental physical law, define all symbols, rearrange algebraically, and then explain the significance of the newly derived equation in the context of your graph or data analysis.

考试委员会期望考生采用结构化的步骤：陈述基本物理定律，定义所有符号，进行代数重排，然后在你所绘制的图像或数据分析的语境中，解释新推导公式的含义。

Use words, not just equations. For example: ‘Since the graph of F against ΔL is a straight line through the origin, Hooke’s law is obeyed, and the gradient k is the spring constant.’ This links the mathematics to the physical interpretation.

不仅要写公式，还要用文字加以阐述。例如：’由于 F–ΔL 图像是一条过原点的直线，表明遵循胡克定律，且斜率 k 就是弹簧的劲度系数。’ 这样就将数学与物理诠释联系了起来。

Avoid skipping steps. If you need to go from T = 2π √(L/g) to g = 4π²L / T², show squaring both sides: T² = 4π² L/g, then multiply by g and divide by T². This clarity earns marks even if the final numerical answer has a small error.

避免跳跃步骤。如果你需要从 T = 2π √(L/g) 推导到 g = 4π²L / T²，要展示两边平方：T² = 4π² L/g，然后乘以 g 再除以 T²。即使最终的数值答案有微小误差，这样清晰的推导也能得分。

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A-Level Physics June 2018 Unit 3 Examiner’s Report: Key Formula Derivations | A-Level物理2018年6月第三单元考官报告：关键公式推导