📚 AS Physics Unit 5 June 2019 Experimental Investigation: EMF and Internal Resistance | AS物理Unit5 2019年6月实验探究：电动势与内阻
The June 2019 AS Physics Unit 5 examination featured an experimental investigation that required students to analyse and evaluate a classic circuit experiment. The task involved determining the electromotive force (EMF) and internal resistance of a battery using a voltmeter, an ammeter, a variable resistor, and a switch. This article provides a detailed walkthrough of the experimental design, data collection, graphical analysis, uncertainty treatment, and evaluative commentary, reflecting the exact demands of that question paper.
2019年6月AS物理第五单元的考试中，有一道实验探究题要求学生分析并评估一个经典电路实验。该任务需要使用电压表、电流表、可变电阻器和开关来测定电池的电动势（EMF）和内阻。本文将详细解析实验设计、数据采集、图像分析、不确定度处理以及评价性讨论，完整再现该试卷的考查要求。
1. Overview of the Question | 题目概览
ying the resistance. Candidates were asked to plot a graph, determine the EMF and internal resistance, calculate uncertainties, and discuss sources of error and possible improvements. Understanding the underlying equation ε = V + Ir was essential to interpret the gradient and intercept of the V-I graph.
题目给了电路图和一组操作说明。一名学生通过改变电阻记录了六组路端电压V和电流I的数据。考生需要绘制图像、求出电动势和内阻、计算不确定度，并讨论误差来源与改进方案。掌握基本公式 ε = V + Ir 是解读V-I图线斜率和截距的关键。
2. Apparatus and Setup | 实验器材与装置
The circuit consisted of a battery of unknown EMF ε and internal resistance r connected in series with an ammeter, a variable resistor (rheostat), and a switch. A high-resistance voltmeter was connected directly across the battery terminals to measure the terminal potential difference V. The variable resistor allowed the current I to be altered in controlled steps.
电路由一个未知电动势ε和内阻r的电池、一个电流表、一个可变电阻（滑线变阻器）和一个开关串联组成。一块高阻电压表并联在电池两端，以测量路端电压V。调节可变电阻可以按设定的步长改变电流I。
- Battery (approx. 6 V) | 电池（约6 V）
- Analogue ammeter (range 0–2 A, resolution 0.01 A) | 模拟电流表（量程0–2 A，分度值0.01 A）
- Analogue voltmeter (range 0–10 V, resolution 0.01 V) | 模拟电压表（量程0–10 V，分度值0.01 V）
- Rheostat (0–50 Ω) | 滑线变阻器（0–50 Ω）
- SPST switch and connecting wires | 单刀单掷开关及导线
3. Experimental Procedure | 实验步骤
The student closed the switch and adjusted the rheostat to obtain a series of current values. At each setting, both the ammeter and voltmeter readings were recorded after allowing the readings to stabilise. The switch was briefly opened between measurements to minimise heating of the battery. Six datasets were gathered, covering a wide range of terminal voltages.
学生闭合开关，调节变阻器得到一系列电流值。在每个设定值下，待读数稳定后记录电流表和电压表的示数。每两次测量之间短暂断开开关，以减少电池发热。共采集了六组数据，覆盖了较宽的路端电压范围。
4. Raw Data Table | 原始数据表
| Current I / A | Terminal p.d. V / V |
|---|---|
| 0.20 | 5.65 |
| 0.40 | 5.30 |
| 0.60 | 4.95 |
| 0.80 | 4.60 |
| 1.00 | 4.25 |
| 1.20 | 3.90 |
nd ±0.01 V for voltage. These values were used later to construct error bars on the graph.
表格列出了六组测量值。每台仪器的分度值给出电流绝对不确定度 ±0.01 A，电压绝对不确定度 ±0.01 V。这些数值随后用于在图上绘制误差棒。
5. Data Processing and Graph Plotting | 数据处理与绘图
er. The points followed a clear linear trend with a negative gradient, confirming that V = – lines (with maximum and minimum gradients) were added to facilitate uncertainty analysis.
利用原始数据，在坐标纸上绘制了路端电压V（纵轴）随电流I（横轴）变化的图像。各数据点呈现清晰的线性负相关趋势，验证了 V = –r I + ε。每个点都加上了表示±0.01 V和±0.01 A的误差棒。画出一条最佳拟合线，并添加了两条极端可接受的直线（具有最大和最小斜率），以便后续不确定度分析。
6. Determining EMF and Internal Resistance | 确定电动势和内阻
rapolation gave an intercept value of ε = 6.00 V. The gradient of the line, m, is equal to – the graph, the gradient of the best-fit line was calculated using two on-the-line points:
最佳拟合线的y轴截距对应I=0时的电动势。外推得到截距 ε = 6.00 V。图线斜率 m = –r。使用线上两点计算最佳拟合线斜率：
m = (3.90 V – 5.65 V) ÷ (1.20 A – 0.20 A) = –1.75 V A⁻¹
Hence, the internal resistance r = –m = 1.75 Ω. The equation of the line is V = 6.00 – 1.75I, which satisfies ε = V + Ir.
因此，内阻 r = –m = 1.75 Ω。直线的方程为 V = 6.00 – 1.75I，满足 ε = V + Ir。
7. Uncertainty Calculations | 不确定度计算
Uncertainty in the internal resistance was estimated using the extreme gradient method. The maximum gradient line was drawn through the error bars to give the steepest plausible negative slope, and the minimum gradient line gave the shallowest. The maximum gradient obtained was –1.80 V A⁻¹ and the minimum was –1.70 V A⁻¹. The corresponding r values are 1.80 Ω and 1.70 Ω. The absolute uncertainty Δr is half the range:
内阻的不确定度通过极端斜率法估算。穿过误差棒画出最大负斜率线（最陡）和最小负斜率线（最缓）。所得最大斜率为 –1.80 V A⁻¹，最小斜率为 –1.70 V A⁻¹。对应的内阻值分别为 1.80 Ω 和 1.70 Ω。绝对不确定度 Δr 取极差的一半：
Δr = (1.80 – 1.70) ÷ 2 = ±0.05 Ω
ertainty in the EMF, based on the intercept spread, was ±0.02 V, giving ε = 6.00 ± 0.02 V.
因此，内阻可表示为 r = 1.75 ± 0.05 Ω。根据截距的分散情况，电动势的不确定度为 ±0.02 V，即 ε = 6.00 ± 0.02 V。
8. Evaluation of Errors | 误差评估
The largest source of uncertainty stemmed from the analogue meters' ccounted for in the error bars. Systematic errors could arise if the voltmeter had a non-ideal (low) internal resistance, causing it to draw some current and lower the measured terminal voltage slightly. Additionally, heating of the battery during high currents could temporarily change its internal resistance, introducing a drift error. The connection wires' own resistance, although small, also contributed to a systematic overestimation of r.
最大的不确定度来源是模拟电表的有限分辨率。读取指针位置的随机误差可达±半个分格，这已在误差棒中考虑。若电压表的内阻不够理想（偏低），会分走少量电流，使测得的端电压略微偏低，从而引入系统误差。此外，大电流时电池发热会暂时改变其内阻，带来漂移误差。连接导线自身电阻虽小，也会使内阻的系统性偏大。
9. Improvements and Further Investigation | 改进与延伸探究
To reduce random errors, a digital voltmeter and digital ammeter with higher resolution could be used. Repeating each reading several times and averaging would also improve precision. To minimise heating effects, the circuit should be switched on only while readings are taken, and a heat sink could be placed near the battery. Using a potentiometer method would eliminate the need for current measurement altogether and give a more direct determination of the EMF without drawing current. For a more comprehensive investigation, the experiment could be extended to study how internal resistance changes with temperature or state of charge.
为减小随机误差，可改用分辨率更高的数字电压表和数字电流表。每次读数重复多次并取平均值，也能提高精密度。为减少发热效应，仅该数时闭合电路，并可在电池旁放置散热片。采用电位差计法可彻底避免电流测量，在不取用电流的情况下更直接地测定电动势。若想作更全面的探究，可将实验扩展至研究内阻随温度或荷电状态的变化。
10. Common Examination Pitfalls | 常见考试易错点
Many students mistakenly assume that the gradient of the V-I graph equals –1/r, forgetting the relationship V = ε – s the EMF or misreading the scale on the graph are also frequent mistakes. This June 2019 paper rewarded clearly labelled axes, correct units, and explicit calculation steps.
不少学生错误地认为V-I图的斜率等于 –1/r，忘记了关系式 V = ε – Ir。有些学生画最佳拟合线时忽略了误差棒，导致不确定度值不切实际。忘记将截距作为电动势，或读错图像刻度，也是常见错误。2019年6月的这张试卷明确奖励坐标轴清晰标注、单位正确、计算步骤明确。
11. Conclusion | 结论
The Unit 5 investigation from June 2019 is a classic example of how a straightforward electrical experiment can assess a wide range of practical skills: circuit assembly, precise measurement, graphical analysis, uncertainty propagation, and critical evaluation. Mastering this experiment provides a solid foundation for tackling similar questions on terminal potential difference, power transfer, and battery modelling in AS and A-level physics.
2019年6月Unit 5实验探究是一个经典范例，用简单的电路实验考查了多项实验技能：电路连接、精确测量、图像分析、不确定度传递和批判性评价。掌握这个实验能为应对AS和A-level物理中路端电压、功率传输和电池模型等类似题目打下扎实基础。
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