📚 A-Level Physics: Jun 18 Insert 4 – EMF and Internal Resistance Experiment | A-Level物理：2018年6月插页4 电池电动势与内阻实验探究
In the A-Level Physics specification, one of the most important core practical investigations involves measuring the electromotive force (e.m.f.) and internal resistance of a cell or battery. The June 2018 examination insert 4 provided students with a typical experimental setup, a partially completed data table, and graph grid for this activity. By analysing the voltage-current relationship, candidates are expected to determine the e.m.f. ε and internal resistance r of the power source. This article breaks down the theory, method, data analysis, and common pitfalls associated with this experiment, ensuring you can approach similar problems with confidence.
在A-Level物理大纲中，最核心的实验探究之一就是测量电池的电动势和内阻。2018年6月考试的插页4为学生提供了一个典型的实验装置图、一张部分完成的数据表格以及图像坐标纸。通过分析电压与电流的关系，考生需要求出电源的电动势 ε 和内阻 r。本文详细解析该实验的理论、方法、数据分析及常见误区，帮助你自信地应对同类题目。
1. Introduction to the Experiment | 实验介绍
The aim of this investigation is to determine the e.m.f. ε and internal resistance r of a dry cell by plotting a graph of terminal potential difference (V) against current (I). When a current flows, the terminal voltage V is less than ε due to the potential drop across the internal resistance, described by the equation V = ε − Ir. By treating this as a linear function y = c + mx, the y-intercept gives ε and the negative gradient gives r.
本次实验的目的是通过描绘路端电压 V 随电流 I 变化的图像，测量一节干电池的电动势 ε 和内阻 r。当回路中有电流时，由于内阻上的压降，路端电压 V 会小于 ε，其关系式为 V = ε − Ir。将该式视为线性函数 y = c + mx，其纵轴截距即为 ε，斜率绝对值即为 r。
2. Equipment List | 设备清单
The following apparatus is typically provided, as shown on the insert:
通常提供的仪器如下所示：
- One or more dry cells (1.5 V nominal)
- 可调变阻器（例如 0–50 Ω, 2 A）
- A digital ammeter (0–2 A range, resolution 0.01 A)
- 数字电流表（量程 0–2 A，分辨率 0.01 A）
- A digital voltmeter (0–5 V range, resolution 0.01 V)
- 数字电压表（量程 0–5 V，分辨率 0.01 V）
- Connecting leads and a single-pole switch
- 连接导线以及一个单刀开关
Some insert versions also include a fixed resistor to limit current, a heat sink, or a thermistor for temperature-dependent investigations. In the standard internal resistance experiment, the variable resistor allows the current to be changed smoothly while recording corresponding values of V and I.
部分插页版本还会提供限流固定电阻、散热片或热敏电阻用于温度相关的探究。在标准的内阻实验中，变阻器可平稳改变电流，同时记录对应的 V 和 I 值。
3. Circuit Diagram | 电路图
The circuit consists of a cell connected in series with an ammeter and a variable resistor; a voltmeter is connected directly across the terminals of the cell. A switch is included to prevent the cell from discharging when not taking measurements. The insert usually illustrates this arrangement with standard circuit symbols. Note that the voltmeter must be a high-resistance device to ensure negligible current flows through it, so that the ammeter reading is effectively the current through the cell.
电路由电池、电流表、变阻器串联组成，电压表直接并联在电池两端。开关用来防止不记录数据时电池持续放电。插页中通常会用标准电路符号画出该连接方式。注意电压表必须具有高阻抗，以保证其中流过的电流可忽略不计，这样电流表的读数才是流过电池的实际电流。
4. Procedure | 实验步骤
Follow these steps to obtain a set of V–I data:
按以下步骤获取一组 V–I 数据：
- Construct the circuit with the switch open and the variable resistor set to its maximum resistance.
- 打开开关，将变阻器调至最大阻值，搭建电路。
- Close the switch and quickly record the voltmeter reading V and ammeter reading I. Open the switch immediately to minimise cell heating.
- 闭合开关，快速记录电压表读数 V 和电流表读数 I，然后立即断开开关以减少电池发热。
- Decrease the resistance of the variable resistor slightly, close the switch again, and record the new pair of values. Repeat for at least six different current settings.
- 略微减小变阻器阻值，再次闭合开关，记录新的一组数值。至少改变六次不同的电流值重复测量。
- Ensure that the current does not exceed the safe rating of the cell or the ammeter; typically keep I below 1.5 A for a standard dry cell.
- 确保电流不超过电池或电流表的安全额定值；对于标准干电池，通常保持电流低于 1.5 A。
- Open the switch between readings to allow the cell to cool, reducing temperature-related changes in internal resistance.
- 每次读数之间断开开关，让电池冷却，减小内阻随温度的变化。
5. Data Recording Table | 数据记录表
Insert 4 typically presents a table with columns for I (in amperes) and V (in volts), sometimes with an additional column for 1/I or for calculated power. Below is a representative data set that could appear in such an examination context.
插页4通常会展示一张包含电流 I（单位安培）和电压 V（单位伏特）两列的表格，有时会增加 1/I 或计算功率的列。下面是一组可能出现在考试情境中的代表性数据。
| I / A | V / V |
|---|---|
| 0.20 | 1.40 |
| 0.40 | 1.28 |
| 0.60 | 1.16 |
| 0.80 | 1.04 |
| 1.00 | 0.92 |
| 1.20 | 0.80 |
You should record all values to the same number of decimal places as the instrument resolution allows. A graph of V (vertical axis) against I (horizontal axis) is then plotted on the grid provided.
所有数据应根据仪器分辨率记录到相同的小数位数。随后在提供的坐标纸上绘制 V（纵轴）对 I（横轴）的图线。
6. Graphical Analysis | 图像分析
The relationship V = ε − Ir is a straight-line equation. Plot V on the y-axis and I on the x-axis. The y-intercept is the e.m.f. ε (the voltage when I = 0 A). The gradient of the line is −r, so the magnitude of the gradient equals the internal resistance r.
关系式 V = ε − Ir 是一个直线方程。将 V 标在纵轴，I 标在横轴。纵轴截距就是电动势 ε（即 I = 0 A 时的电压）。图线的斜率为 −r，因此斜率的绝对值等于内阻 r。
A best-fit line should be drawn through the plotted points. Avoid forcing the line through the origin; it should be the line that minimises the vertical deviations. If a data point lies far from the line, it is likely anomalous and should be repeated if time allows, or excluded when drawing the best-fit line.
应在数据点之间画出最佳拟合直线。避免强制让直线通过坐标原点；该直线应使各点垂直偏差最小。如果某个数据点远远偏离直线，很可能是异常点，若时间允许应重测，或在画最佳拟合线时将其排除。
7. Calculation of EMF and Internal Resistance | 电动势和内阻的计算
Using the straight line equation, we apply two methods: graphical and algebraic. Graphically, read the y-intercept directly from the graph where I = 0. From the sample data, extending the best-fit line gives an intercept of approximately 1.52 V. For the gradient, choose two widely separated points on the line, for example (0.20 A, 1.40 V) and (1.00 A, 0.92 V):
利用直线方程，我们可以采用图像法和代数法两种方式。图像法直接从图上读出 I = 0 时的纵轴截距。根据示例数据，延长最佳拟合线得到截距约为 1.52 V。对于斜率，选取直线上间隔较大的两个点，例如 (0.20 A, 1.40 V) 和 (1.00 A, 0.92 V)：
gradient = ΔV ÷ ΔI = (0.92 − 1.40) ÷ (1.00 − 0.20) = −0.48 ÷ 0.80 = −0.60 V A⁻¹
Therefore, r = |−0.60| = 0.60 Ω, and ε = 1.52 V. Always express r with the unit ohm (Ω) and ε with volts (V).
因此，r = |−0.60| = 0.60 Ω，而 ε = 1.52 V。始终用欧姆 (Ω) 表示内阻，用伏特 (V) 表示电动势。
8. Uncertainty and Error Analysis | 不确定度与误差分析
Every measurement carries uncertainty. The ammeter and voltmeter have an instrumental resolution limited to ± the least significant digit. For example, a 0.01 A resolution gives an absolute uncertainty of ±0.01 A. When plotting the graph, draw lines of worst fit to estimate the uncertainty in gradient and intercept.
每次测量都带有不确定度。电流表和电压表的仪器分辨率受限于最低有效位数的 ± 值。例如，0.01 A 分辨率带来 ±0.01 A 的绝对不确定度。作图时，画出最差拟合线以估算斜率和截距的不确定度。
Common sources of error include: heating of the cell increasing internal resistance; poor contact resistance at terminals; and voltmeter loading error if the meter resistance is not sufficiently high. To reduce random errors, repeat readings at each current setting and average. To minimise systematic errors, allow the cell to cool between measurements and ensure clean, tight connections.
常见误差来源包括：电池发热导致内阻增大；接线端接触电阻；以及电压表内阻不够高造成的负载误差。为减少随机误差，可在每个电流设置下重复读数并取平均值。为减小系统误差，测量间隙让电池冷却，并确保连接清洁、牢固。
9. Safety Precautions | 安全注意事项
Although this is a low-voltage experiment, safety remains essential. High currents can cause the cell to overheat and potentially leak, so avoid leaving the circuit closed for extended periods. Always start with the variable resistor at its maximum to prevent excessive initial current. Do not short-circuit the cell, as this can damage the cell and generate dangerous heat.
尽管这是低压实验，安全仍然至关重要。大电流可能导致电池过热甚至漏液，因此要避免长时间闭合回路。始终将变阻器置于最大阻值开始实验，以防初始电流过大。切勿将电池短路，否则可能损坏电池并产生危险的热量。
10. Extension Questions | 拓展问题
Examiners often link this practical to underlying theory. Be prepared to explain why the e.m.f. is sometimes defined as the work done per unit charge and how it differs from terminal p.d.. You may also be asked to explain the effect of temperature on internal resistance, or to analyse what happens if a second cell is added in series or parallel. A common question: ‘Explain why the voltmeter reading when the switch is open approximates ε.’ Because no current flows, Ir = 0, so V = ε.
考官常常将此实验与深层理论相联系。要能够解释为何电动势有时被定义为单位电荷所做的功，它与路端电压有何区别。还可能会问到温度对内阻的影响，或者分析若再串联或并联一节电池会发生什么。一个常见问题：‘解释为何开关断开时电压表读数近似等于 ε。’因为此时没有电流，Ir = 0，因此 V = ε。
11. Common Mistakes | 常见错误
Students frequently confuse the axes: plotting I against V gives a gradient of −1/r, not −r. Another mistake is forgetting that the y-intercept of a V–I graph is ε, not the gradient. When calculating gradient, using data points directly from the table instead of from the best-fit line will propagate random errors and yield an inaccurate r. Also, omitting units in final answers or quoting r to too many significant figures (it should reflect the precision of the instruments, typically 2 or 3 s.f.) loses marks.
学生经常搞错坐标轴：如果绘制 I 对 V 的图线，其斜率为 −1/r 而非 −r。另一个错误是忘记了 V–I 图的纵轴截距是 ε 而不是斜率。在计算斜率时，直接使用表格中的数据点而非最佳拟合线上的点，则会将随机误差传递过来，得到不准确的 r。此外，最终答案漏写单位或将 r 写成过多有效数字（应根据仪器精度取 2 或 3 位有效数字）也会丢分。
12. Conclusion | 结论
The internal resistance investigation is a staple of A-Level Physics practical assessment. By mastering the V = ε − Ir relationship, graph interpretation, and uncertainty handling, you can secure high marks in both the written paper and the practical endorsement. Whether the June 2018 insert 4 appeared in your examination or you are preparing for future assessments, the skills outlined here will serve you well.
内阻实验是A-Level物理实验评估的基本内容。掌握 V = ε − Ir 的关系、图像解读和不确定度处理，你就能在笔试和实践认证中稳拿高分。无论2018年6月的插页4是否出现在你的考卷上，或者你正在为未来的考核做准备，本文所述的技巧都将使你受益匪浅。
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