A-Level Physics Paper 3 Formula Derivations – Key Insights from the Jun19 Exam Report | A-Level物理试卷3公式推导——2019年6月考试报告关键洞见

📚 A-Level Physics Paper 3 Formula Derivations – Key Insights from the Jun19 Exam Report | A-Level物理试卷3公式推导——2019年6月考试报告关键洞见

The June 2019 A-Level Physics Paper 3 exam report revealed that a significant number of candidates lost marks not through flawed practical work, but by failing to produce clear, logical formula derivations when processing experimental data. Whether you are determining g from a pendulum, resistivity from a wire, or Young modulus from a stress-strain investigation, the ability to manipulate equations and show the link between raw measurements and the final physical quantity is essential. This article dissects the most common derivations required, highlights frequent errors flagged by examiners, and provides a structured approach to help you write out derivations that score full marks.

2019年6月A-Level物理试卷3的考官报告显示，大量考生失分并非因为实验操作失误，而是因为在处理实验数据时未能给出清晰、有逻辑的公式推导。无论你是通过单摆测定g、通过导线测定电阻率，还是通过应力-应变实验测定杨氏模量，具备变换方程并在原始测量量与最终物理量之间建立明确联系的能力至关重要。本文将剖析最常见的推导类型，指出考官标记的常见错误，并提供一套结构化方法，帮助你写出能获满分的推导过程。

1. Why Formula Derivation Matters in Paper 3 | 为什么公式推导在试卷3中至关重要

Paper 3 practical examinations assess not only your ability to collect data but also your understanding of the underlying physics. A correct derivation demonstrates that you can connect the theory to the specific apparatus and measurements at hand. Examiners award marks for stating the initial theoretical relationship, substituting measured variables, and rearranging to obtain an expression for the unknown quantity. Skipping steps or jumping straight to a final formula often leads to errors and lost credit.

试卷3的实验考试不仅考查你收集数据的能力，还考查你对背后物理原理的理解。正确的推导表明你能将理论与具体的实验装置和测量量联系起来。考官会为陈述初始理论关系、代入测量变量、重新整理得到未知量表达式的步骤给分。跳过步骤或直接写出最终公式常常导致错误和失分。

In the June 2019 report, chief examiners stressed that candidates should show all working clearly, because the derivation itself carries marks independent of the final numerical answer. Even if you later miscalculate a value, a correct derivation can still earn significant credit.

在2019年6月的报告中，主考官强调考生应清晰地展示所有推导过程，因为推导本身有独立于最终数值答案的分数。即使你之后计算数值出错，正确的推导仍能获得可观的分数。

2. Linearising Equations for Graphical Analysis | 将方程线性化以进行图像分析

The most powerful tool in Paper 3 is the straight-line graph. You are often required to plot measured quantities such that the gradient or intercept yields the desired physical constant. To do this, you must first rearrange the theoretical equation into the form y = mx + c, where y and x are the variables you will plot, and m and c contain the constants you seek.

试卷3中最强大的工具是直线图像。你经常需要绘制某些测量量的散点图，使得斜率或截距给出所需的物理常数。为此，你必须首先将理论方程重排为 y = mx + c 的形式，其中 y 和 x 是你要绘制的变量，m 和 c 包含你要寻找的常量。

For example, the period T of a simple pendulum is T = 2π √(l/g). Squaring both sides gives T² = (4π²/g) l. Comparing with y = mx, we identify y = T², x = l and gradient m = 4π²/g. Therefore, g can be found from g = 4π² / m. This step is mandatory before plotting; always write it explicitly in your answer.

例如，单摆的周期 T 为 T = 2π √(l/g)。两边平方可得 T² = (4π²/g) l。与 y = mx 对比，我们得到 y = T²，x = l，斜率 m = 4π²/g。因此，g 可通过 g = 4π² / m 求得。这一步在绘图前是必须的；请务必在答案中明确写出。

T² = (4π² / g) × l

3. Deriving g from a Simple Pendulum Experiment | 从单摆实验推导g

Imagine you have measured the length l of a pendulum and the corresponding period T for several lengths. The theoretical basis is T = 2π √(l/g). To derive a working formula for g, square both sides: T² = 4π² l / g. Multiply both sides by g: g T² = 4π² l. Then divide by T²: g = 4π² l / T². This expression allows you to calculate g for a single pair of l and T, but a graphical method is more reliable.

假设你测量了摆长 l 以及对应的不同摆长下的周期 T。理论基础是 T = 2π √(l/g)。要推导 g 的工作公式，将两边平方：T² = 4π² l / g。两边乘以 g：g T² = 4π² l。然后除以 T²：g = 4π² l / T²。该表达式允许你用单组 l 和 T 计算 g，但图像法更为可靠。

For the graphical derivation, rewrite as T² = (4π²/g) l. The gradient of a T² vs l graph is m = Δ(T²)/Δl. Equate this to 4π²/g: m = 4π²/g. Rearranging gives g = 4π² / m. In your write-up, state these steps and then use the measured gradient to compute g. Remember to convert l to metres and T to seconds.

对于图像推导，改写为 T² = (4π²/g) l。T² 对 l 图像的斜率是 m = Δ(T²)/Δl。令其等于 4π²/g：m = 4π²/g。重新整理得 g = 4π² / m。在你的实验报告中，陈述这些步骤，然后使用测得的斜率计算 g。记住将 l 转换为米，T 转换为秒。

g = 4π² / gradient

4. Deriving Resistivity of a Wire | 推导导线的电阻率

Resistivity ρ is a material property, and the resistance R of a wire is related to its length l and cross-sectional area A by R = ρ l / A. In many experiments, you measure the diameter d of the wire with a micrometer, so the area is A = π(d/2)² = πd²/4. Substituting gives R = ρ l / (πd²/4) = 4ρ l / (πd²).

电阻率 ρ 是材料属性，导线的电阻 R 与其长度 l 和横截面积 A 的关系为 R = ρ l / A。在许多实验中，你用千分尺测量导线的直径 d，因此截面积为 A = π(d/2)² = πd²/4。代入得 R = ρ l / (πd²/4) = 4ρ l / (πd²)。

If you plot R against l, the gradient is m = ΔR/Δl. Rearrange the resistance formula to R = (4ρ / πd²) × l, showing that m = 4ρ / πd². Therefore, ρ = (m π d²) / 4. Where R is obtained from V/I measurements, you can also use R = V/I directly. Always measure d at several points along the wire and use the mean value in your derivation.

如果你绘制 R 对 l 图，斜率 m = ΔR/Δl。重排电阻公式为 R = (4ρ / πd²) × l，可见 m = 4ρ / πd²。因此，ρ = (m π d²) / 4。当 R 由 V/I 测量获得时，也可以直接使用 R = V/I。务必在导线多个位置测量 d，并在推导中使用平均值。

ρ = (π d² × gradient) / 4

5. Deriving Young Modulus from a Stretching Wire | 从拉伸导线推导杨氏模量

Young modulus E is defined as tensile stress divided by tensile strain: E = (F/A) / (Δl/l) = F l / (A Δl). For a wire of diameter d, A = πd²/4, so E = (F l) / (πd²/4 × Δl) = 4 F l / (π d² Δl). Here F is the applied force, l the original length, and Δl the extension.

杨氏模量 E 的定义是拉伸应力除以拉伸应变：E = (F/A) / (Δl/l) = F l / (A Δl)。对于直径为 d 的导线，A = πd²/4，因此 E = (F l) / (πd²/4 × Δl) = 4 F l / (π d² Δl)。此处 F 为施加的力，l 为原长，Δl 为伸长量。

When you plot applied force F (y-axis) against extension Δl (x-axis), the gradient is k = F/Δl. Substituting into the expression gives E = 4 l k / (π d²). It is critical to use the original, unstretched length l for the wire, and to collect diameter readings from several orientations to minimise uncertainty.

当你以施加的力 F（纵轴）对伸长量 Δl（横轴）作图时，斜率为 k = F/Δl。代入表达式可得 E = 4 l k / (π d²)。关键是要使用导线未拉伸时的原长 l，并从多个方向采集直径读数以减小不确定度。

E = (4 × original length × gradient) / (π d²)

6. Verifying Hooke’s Law and Finding the Spring Constant | 验证胡克定律并求弹簧劲度系数

Hooke’s law states that the force F exerted by a spring is directly proportional to its extension x, provided the elastic limit is not exceeded: F = k x. In a typical experiment, you add masses and measure the resulting extension. The weight Mg provides the force, so Mg = k x. Rearranging gives k = Mg / x, but a graph of M against x or F against x is better.

胡克定律指出，在不超过弹性限度的情况下，弹簧施加的力 F 与其伸长量 x 成正比：F = k x。在典型实验中，你添加砝码并测量相应的伸长量。重量 Mg 提供力，因此 Mg = k x。重排可得 k = Mg / x，但更佳的方法是绘制 M 对 x 图或 F 对 x 图。

Plotting F (y-axis) versus x (x-axis) yields a straight line through the origin with gradient k. The derivation is straightforward: start from F = k x, identify that y = F, m = k, so k = gradient. If you plot mass M vs x, then gradient m’ = k/g, so k = g × gradient. Examiners expect you to state this relationship before reading the gradient from your graph.

绘制 F（纵轴）对 x（横轴）图，得到一条通过原点的直线，斜率即为 k。推导十分直接：从 F = k x 出发，确定 y = F，m = k，因此 k = 斜率。如果你绘制质量 M 对 x 图，那么斜率 m’ = k/g，因此 k = g × 斜率。考官期望你在从图像读取斜率之前陈述这个关系。

k = gradient of F vs x graph

7. Using Logarithms to Handle Power Law Relationships | 用对数处理幂律关系

Some experiments involve relationships of the form y = a xⁿ, such as the illumination from a point source following E = k / r². To determine the constants, logarithms are used. Taking logs to base 10 (or natural log) gives log y = log a + n log x. Comparing with y = mx + c gives gradient = n and intercept = log a.

有些实验涉及形如 y = a xⁿ 的关系，比如点光源的照度遵循 E = k / r²。为了确定常数，需使用对数。取以10为底的对数（或自然对数）得到 log y = log a + n log x。与 y = mx + c 对比，可得斜率 = n，截距 = log a。

For instance, to verify the inverse square law for light, measure illuminance E at different distances r. The theory states E = (constant) / r². Taking logs: log E = log(constant) – 2 log r. A plot of log E against log r should yield a gradient of -2. The derivation of this linearised form must be shown; exam reports regularly note that candidates lose marks by failing to take logs properly or by misplacing the negative sign.

例如，要验证光的平方反比定律，需在不同距离 r 处测量照度 E。理论表明 E = (常数) / r²。取对数：log E = log(常数) – 2 log r。以 log E 对 log r 作图应得到斜率为 -2 的直线。这个线性化形式的推导必须展示；考试报告经常指出，考生因未能正确取对数或错放负号而失分。

log y = n log x + log a

8. Building Formulas from Tabulated Data | 从表格数据构建公式

Sometimes the question provides a table of raw data and asks you to deduce the relationship between variables. Begin by stating a proposed general relationship, such as y ∝ x², or y = k / x, based on the observed trend. Then calculate suitable test columns (e.g., y vs x²) to check for proportionality.

有时题目会提供一个原始数据表，要求你推导变量间的关系。先根据观察到的趋势提出一个假设的一般关系，例如 y ∝ x² 或 y = k / x。然后计算合适的检验列（例如 y 对 x²）以检查是否成正比。

To formalise the derivation, write the proportional relation and introduce a constant: y = k x². Rearrange to k = y / x². Show that k is approximately constant for all data pairs, then state the final formula with the mean value of k and appropriate units. This step-by-step construction demonstrates your ability to move from experiment to mathematical model.

为了规范推导，写出比例关系并引入常数：y = k x²。重排为 k = y / x²。展示对于所有数据对，k 近似为常数，然后以 k 的平均值和适当单位写出最终公式。这种逐步构建的方式展示了你从实验走向数学模型的能力。

Examiners reward candidates who clearly show the test for constant k and who comment on the uncertainty range. A simple “k is constant within experimental uncertainty” followed by the derived equation is sufficient.

考官会奖励那些清晰展示常数 k 检验过程并评论其不确定范围的考生。一句简单的“在实验不确定度范围内，k 为常数”，再跟上推导出的方程就足够了。

9. Common Derivation Mistakes Highlighted by the Jun19 Report | 2019年6月报告指出的常见推导错误

Confusing diameter with radius when calculating cross-sectional area. Many used A = πd² instead of π(d/2)², leading to a factor of 4 error. Always write A = πd²/4 explicitly.

计算横截面积时混淆直径与半径。许多人用了 A = πd² 而不是 π(d/2)²，导致因子4的错误。务必明确写出 A = πd²/4。
Forgetting to convert units, especially cm to m. A swing of 85 cm must be entered into the formula as 0.85 m; otherwise the derived g becomes unrealistic and marks are deducted.

忘记换算单位，尤其是厘米转米。85 厘米的摆长必须以 0.85 米代入公式；否则推导出的 g 会不切实际并被扣分。
Misidentifying graph axes in the linearisation step. Candidates sometimes plotted l vs T² but still treated the gradient as 4π²/g, failing to invert the relationship. Ensure your y and x match your rearranged equation.

在线性化步骤中错误识别图像轴。考生有时绘制了 l 对 T² 图，却仍将斜率当作 4π²/g，而未能反转关系。确保你的 y 和 x 与重排后的方程匹配。
Neglecting to propagate uncertainties correctly in the derivation. When g = 4π² l / T², the fractional uncertainty must include the doubled contribution from T: Δg/g = √[(Δl/l)² + (2ΔT/T)²]. Omitting the factor of 2 on ΔT was a frequent error.

在推导中未能正确传递不确定度。当 g = 4π² l / T² 时，相对不确定度必须包含 T 贡献的两倍效应：Δg/g = √[(Δl/l)² + (2ΔT/T)²]。漏掉 ΔT 前面的因子2是个常见错误。

10. Structured Approach to Presenting Your Derivation | 展示推导的结构化方法

To secure all derivation marks, follow these five steps every time: (1) Write the fundamental theoretical equation relevant to the experiment. (2) Identify which variables you have measured directly and substitute them with their symbols. (3) Rearrange the equation so that the quantity you want to find is the subject, or into a linear form y = mx + c if you are plotting a graph. (4) For graphical methods, explicitly state how the gradient (or intercept) relates to the target constant. (5) Finally, before plugging in numbers, write down the derived working formula in its simplest form, with all constants (such as π) clearly visible.

为确保拿到所有推导分数，每次都遵循以下五步：(1) 写出与实验相关的基本理论方程。(2) 确定哪些变量是你直接测量的，并用符号代入。(3) 重新整理方程，使你想求的量成为公式主体，或者若需绘图则转化为 y = mx + c 的线性形式。(4) 对于图像法，明确指出斜率（或截距）如何与目标常数关联。(5) 最后，在代入数字之前，写出最简形式的推导工作公式，并让所有常数（如 π）清晰可见。

This systematic method not only impresses examiners but also reduces slips. The June 2019 report specifically praised answers where candidates wrote the linearised equation beside the graph and labelled the axes accordingly. Neat, logical derivations are a mark of a strong practical physicist.

这种系统方法不仅会给考官留下深刻印象，还能减少差错。2019年6月的报告特别赞扬了那些在图像旁写出线性化方程并据此标记轴标签的答卷。整洁、有逻辑的推导是优秀实验物理学家的标志。

11. Uncertainty Propagation in Derived Quantities | 导出量的不确定度传递

Once you have derived a formula for the physical constant, you often need to estimate its uncertainty. For a quantity Q that depends on measured variables a, b, and c, the absolute uncertainty ΔQ

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A-Level Physics Paper 3 Formula Derivations – Key Insights from the Jun19 Exam Report | A-Level物理试卷3公式推导——2019年6月考试报告关键洞见