📚 Formula Derivations from the Edexcel IAL Physics Unit 3 Examination Report (Jan 2020) | 基于爱德思IAL物理Unit 3考试报告（2020年1月）的公式推导
The January 2020 Edexcel International A-Level Physics Unit 3 (WPH13) examination report highlighted the importance of manipulating equations into linear forms, propagating uncertainties, and extracting physical quantities from graphs. The report noted that many students lost marks not because they could not recall formulas, but because they struggled to derive gradient expressions, interpret intercepts, or combine percentage uncertainties correctly. This article revisits the key formula derivations and data analysis techniques that were central to that paper, providing step-by-step explanations to help you master the experimental and analytical skills expected at A-Level.
2020年1月的爱德思国际A-Level物理Unit 3（WPH13）考试报告强调，将方程变形为线性形式、传递不确定度以及从图像中提取物理量至关重要。报告指出，许多学生失分并非因为记不住公式，而是因为难以推导斜率表达式、解读截距或正确合成百分比不确定度。本文重新梳理了该试卷涉及的核心公式推导与数据处理方法，通过分步讲解助你掌握A-Level要求的实验与分析技能。
1. Overview of the Unit 3 Exam and Key Derivations | 单元考试概览与核心推导
The Unit 3 paper tested practical skills through questions on familiar experiments such as determining g using a pendulum, measuring the resistivity of a wire, and investigating the behaviour of a thermistor. In each case, candidates needed to rearrange a nonlinear equation into the form y = mx + c, state what should be plotted, and relate gradient or intercept values to the target physical constant. The examiners’ report stressed that a thorough understanding of how to linearise an equation is essential, as is the ability to derive the expression for the uncertainty in the final result.
Unit 3 试卷通过常见实验考查实践技能，例如用单摆测定重力加速度g、测量导线电阻率、研究热敏电阻特性等。每种情形下，考生都需要将非线性方程变形为 y = mx + c 的形式，说明应绘制的图像，并将斜率或截距值与目标物理常数相关联。考官报告强调，透彻理解如何线性化方程至关重要，推导最终结果不确定度表达式的能力同样不可或缺。
2. Deriving g from a Simple Pendulum | 从单摆实验中推导重力加速度 g
The period T of a simple pendulum is given by T = 2π√(L/g). To extract g, we square both sides: T² = 4π²L/g. This is already linear in L if we treat T² as y and L as x: T² = (4π²/g) L. Thus a graph of T² against L yields a straight line through the origin with gradient = 4π²/g. Therefore g = 4π² / gradient. The report noted that students frequently forgot to square the period, or attempted to plot T vs L directly, which produces a curve and makes gradient analysis invalid.
单摆的周期 T = 2π√(L/g)。为求出g，两边平方得 T² = 4π²L/g。若将 T² 视作 y、L 视作 x，该式已是线性形式：T² = (4π²/g) L。因此绘制 T²-L 图像将得到一条过原点的直线，斜率 = 4π²/g。于是 g = 4π² / 斜率。报告指出，学生常常忘记将周期平方，或试图直接绘制 T-L 图，这得到的是曲线，无法用斜率分析。
3. Linearising the Pendulum Equation When the Intercept is Not Zero | 非零截距情形下的单摆方程线性化
If the pendulum length L includes an unknown offset (e.g. the radius of the bob), the effective length becomes L + ε, and the equation is T = 2π√((L + ε)/g). Squaring gives T² = (4π²/g) L + (4π²ε/g). This is of the form y = mx + c, where gradient m = 4π²/g and intercept c = 4π²ε/g. From the gradient we find g; from the intercept we can determine the systematic error ε. The January 2020 paper required students to predict the impact of such an offset on the derived g value.
若单摆长度 L 含有一个未知偏移量（例如摆球半径），有效长度变为 L + ε，周期公式即为 T = 2π√((L + ε)/g)。两边平方得到 T² = (4π²/g) L + (4π²ε/g)。这符合 y = mx + c 的形式，其中斜率 m = 4π²/g，截距 c = 4π²ε/g。由斜率可求 g；由截距可确定系统误差 ε。2020年1月的试卷要求预测此类偏移对所求 g 值的影响。
4. Deriving Resistivity from a Wire Experiment | 从导线实验中推导电阻率
The resistance of a uniform wire is R = ρL/A, where ρ is resistivity, L is length, and A is cross-sectional area. Rearranging gives R = (ρ/A) L. If R is plotted against L, the gradient is ρ/A. Therefore, ρ = gradient × A. The area A can be computed from the wire diameter d using A = πd²/4. The exam report stressed that students must measure the diameter at several points along the wire to obtain a reliable average and must calculate the percentage uncertainty in A from repeated diameter readings.
均匀导线的电阻为 R = ρL/A，其中 ρ 是电阻率，L 是长度，A 是横截面积。整理得 R = (ρ/A) L。若绘制 R-L 图像，斜率即为 ρ/A。因此，ρ = 斜率 × A。面积 A 可通过导线直径 d 用 A = πd²/4 算出。考试报告强调，学生必须沿导线多点测量直径，获得可靠的平均值，并依据多次直径读数计算 A 的百分比不确定度。
5. Deriving Internal Resistance and e.m.f. from a Terminal p.d. Graph | 从路端电压图像推导内阻与电动势
For a cell with e.m.f. E and internal resistance r, the terminal potential difference V across a load is V = E − Ir. This is already a linear equation: if V is plotted on the y-axis and current I on the x-axis, the gradient is −r and the y-intercept is E. The examiners’ report emphasised that many students incorrectly stated the gradient as r (omitting the negative sign) or misidentified the intercept as the short-circuit current. Correctly labelling axes and using the linear form V = −r I + E is crucial.
对于电动势为 E、内阻为 r 的电池，路端电压 V 满足 V = E − Ir。这已经是一个线性方程：若将 V 绘在 y 轴，电流 I 绘在 x 轴，斜率为 −r，y 轴截距为 E。考官报告指出，许多学生错误地将斜率表述为 r（遗漏负号），或将截距误认为短路电流。正确标注坐标轴并使用线性形式 V = −r I + E 至关重要。
6. Logarithmic Plotting to Derive an Unknown Power Law | 对数作图推导未知幂律关系
A common Unit 3 task is to determine the relationship between two variables when it is of the form y = k xⁿ. Taking natural logarithms (or log base 10) yields ln y = n ln x + ln k. Plotting ln y against ln x gives a straight line with gradient n and intercept ln k. The report highlighted that students often forget to include error bars on the log-transformed axes or fail to express the final equation with the constant k found from e^(intercept). Example: if y is the period of oscillation and x is the mass added to a spring, a log-log plot can reveal whether the relationship follows T ∝ √m.
Unit 3 常见任务是确定两个变量之间形如 y = k xⁿ 的关系。取自然对数（或以10为底的对数）可得 ln y = n ln x + ln k。绘制 ln y 对 ln x 的图像，得到斜率为 n、截距为 ln k 的直线。报告强调，学生经常忘记在对数变换后的轴上添加误差棒，或未能用 e^(截距) 求得常数 k 并写出最终方程。例如，若 y 为振动周期，x 为加在弹簧上的质量，双对数图可揭示是否满足 T ∝ √m。
7. Deriving the Viscosity of a Liquid from Terminal Velocity | 从终端速度推导液体黏度
When a small sphere falls at terminal velocity v through a viscous liquid, Stokes’ law gives 6πηrv = (4/3)πr³(ρₛ − ρₗ)g, where η is viscosity, r is sphere radius, ρₛ is sphere density, and ρₗ is liquid density. Simplifying: v = [2r²(ρₛ − ρₗ)g] / (9η). To find η, we rearrange: η = [2r²(ρₛ − ρₗ)g] / (9v). If an experiment varies r and measures v, a linear plot of v against r² can be used, with gradient = 2(ρₛ − ρₗ)g / (9η). The report noted that students must be careful with unit conversions, especially when r is measured in mm.
当小球在黏性液体中以终端速度 v 下落时，斯托克斯定律给出 6πηrv = (4/3)πr³(ρₛ − ρₗ)g，其中 η 为黏度，r 为球半径，ρₛ 为球密度，ρₗ 为液体密度。化简得 v = [2r²(ρₛ − ρₗ)g] / (9η)。为求 η，变形得 η = [2r²(ρₛ − ρₗ)g] / (9v)。若实验改变 r 并测量 v，可绘制 v 对 r² 的线性图，斜率 = 2(ρₛ − ρₗ)g / (9η)。报告指出，学生必须注意单位换算，尤其当 r 以毫米为单位时。
8. Deriving the Spring Constant from a Mass-Spring System | 从弹簧振子系统推导劲度系数
For a mass m oscillating vertically on a spring, the period T = 2π√(m/k) if the spring mass is negligible. Squaring gives T² = (4π²/k) m. A graph of T² against m yields a straight line with gradient = 4π²/k, so k = 4π² / gradient. However, if the spring has significant mass mₛ, the effective mass becomes m + (mₛ/3), leading to T² = (4π²/k) m + (4π²mₛ)/(3k). In this case, the intercept can be used to estimate the spring’s effective mass, a concept tested indirectly in the January 2020 paper through a data analysis question.
对于在弹簧上垂直振动的质量 m，若弹簧质量可忽略，周期 T = 2π√(m/k)。平方得 T² = (4π²/k) m。T²-m 图为一条直线，斜率 = 4π²/k，因此 k = 4π² / 斜率。然而，若弹簧具有不可忽略的质量 mₛ，有效质量变为 m + (mₛ/3)，从而 T² = (4π²/k) m + (4π²mₛ)/(3k)。此时截距可用于估算弹簧的有效质量，这一概念在2020年1月试卷的数据分析题中有所渗透。
9. Uncertainty Propagation in Derived Quantities | 导出量的不确定度传递
The report emphasised that candidates must be able to combine uncertainties. For a product or quotient, we add percentage uncertainties. For a power relationship like g = 4π²L/T², the percentage uncertainty in g is %U(g) = %U(L) + 2 × %U(T). If T is obtained from the time for 20 oscillations, %U(T) = (%U in timing) + (%U in counting), though the counting uncertainty is often a half oscillation. For a resistivity derived from ρ = gradient × (πd²/4), the percentage uncertainty in ρ is %U(ρ) = %U(gradient) + 2 × %U(d). The examiners reported frequent errors in doubling the diameter percentage uncertainty.
报告强调考生必须能够合成不确定度。对于乘除关系，百分比不确定度相加。对于幂关系，如 g = 4π²L/T²，g 的百分比不确定度为 %U(g) = %U(L) + 2 × %U(T)。若 T 由20次振动的时间求得，%U(T) = （计时%U）+（计数%U），但计数不确定度通常取半个周期。对于由 ρ = 斜率 × (πd²/4) 求得的电阻率，ρ 的百分比不确定度 %U(ρ) = %U(斜率) + 2 × %U(d)。考官报告指出，对直径百分比不确定度加倍处理时错误频发。
10. Using the Balmer Equation to Derive the Rydberg Constant | 用巴耳末公式推导里德伯常数
Although spectroscopic measurements are more common in Unit 4, the January 2020 paper included a simple data analysis task based on the Balmer series. The equation 1/λ = R (1/2² − 1/n²) can be rewritten as 1/λ = R/4 − R/n². For n = 3,4,5… a plot of 1/λ against 1/n² gives a straight line with gradient = −R and intercept = R/4. Two independent estimates of R can be obtained from gradient and intercept. The examiners noted that students confused the wavelength unit (nm to m conversion) and often omitted the negative sign, leading to a positive R value that, while numerically correct, contradicted the graph’s sign convention.
尽管光谱测量在Unit 4中更常见，2020年1月试卷包含了一个基于巴耳末系的简单数据分析题。1/λ = R (1/2² − 1/n²) 可改写为 1/λ = R/4 − R/n²。对于 n = 3,4,5…，绘制 1/λ 对 1/n² 的图像得到一条直线，斜率 = −R，截距 = R/4。从斜率和截距可各自独立估算 R。考官指出，学生混淆波长单位（nm到m的换算）且经常遗漏负号，导致 R 为正数，虽数值正确，却与图像符号约定矛盾。
11. Deriving the Time Constant from a Capacitor Discharge Graph | 从电容放电图像推导时间常数
The discharge of a capacitor through a resistor follows V = V₀ e^(−t/RC). Linearisation requires taking natural logs: ln V = ln V₀ − (1/RC) t. Plotting ln V against t yields a straight line with gradient = −1/RC, giving the time constant τ = RC = −1/gradient. The January 2020 paper asked students to compare the time constant from the gradient with that from the time for V to fall to 37% of V₀. The report recommended using the logarithmic method as it uses all data points and is less affected by a single reading error.
电容器通过电阻放电遵循 V = V₀ e^(−t/RC)。线性化需取自然对数：ln V = ln V₀ − (1/RC) t。绘制 ln V 对 t 的图像得到一条直线，斜率 = −1/RC，时间常数 τ = RC = −1/斜率。2020年1月试卷要求学生将来自斜率的时间常数与根据 V 降至 V₀ 的37% 所需时间作比较。报告建议使用对数方法，因为它利用了所有数据点，受单个读数误差影响较小。
12. Summary of Key Skills from the Examination Report | 总结考试报告中的核心技能
The report concluded that success in Unit 3 hinges on three competencies: (1) rearranging an equation into y = mx + c form and clearly identifying what to plot; (2) correctly translating gradient and intercept into physical quantities, paying attention to signs and powers; and (3) combining percentage uncertainties according to the rules for propagation. Practice these derivations using real experimental data, and always sketch the predicted graph line before answering a question.
报告总结，Unit 3的成功取决于三项能力：(1) 将方程变形为 y = mx + c 形式，并清晰说明该绘制什么图像；(2) 将斜率和截距正确转化为物理量，注意符号和幂次；(3) 根据传递规则合成百分比不确定度。用真实实验数据练习这些推导，并在答题前始终勾勒出预期的图线。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导