📚 PH02 January 2023 Experimental Investigation: EMF and Internal Resistance | PH02 2023年1月实验探究题：电动势与内阻测定
This article breaks down Question 1 from the Edexcel International AS Physics Paper PH02 (January 2023). The question is a classic experimental investigation task that requires students to determine the electromotive force (EMF) and internal resistance of a cell using a variable resistor, ammeter, voltmeter and a set of repeated readings. We dissect every step, from data tabulation to drawing the best‑fit line, calculating the gradient, handling uncertainties and evaluating the procedure.
本文详细解析爱德思国际AS物理PH02试卷（2023年1月）的第1题。这道题是一道经典的实验探究任务，要求学生利用可变电阻、电流表、电压表和一组重复读数来测定电池的电动势和内阻。我们将逐步拆解每一步，包括数据制表、画最佳拟合直线、计算斜率、处理不确定度以及评价实验过程。
1. Experiment Scenario and Aims | 实验场景与目标
terminals. By adjusting the variable resistor, the current I in the circuit and the terminal potential difference V are recorded. The relationship between V and I is given by V = E − I r, so a graph of V against I should yield a straight line where the y-intercept equals the EMF E and the gradient equals −r.
该实验围绕一节电池展开，电池与电流表、可变电阻以及并联在电池两端的电压表串联连接。通过调节可变电阻，记录电路中的电流 I 和电池端电压 V。V 和 I 的关系由 V = E − I r 给出，因此绘制 V对 I 的图像应得到一条直线，其 y 轴截距等于电动势 E，斜率等于 −r。
2. Circuit Setup and Key Variables | 电路连接与关键变量
The circuit must be arranged so that the voltmeter reads the terminal voltage directly across the cell, not including the ammeter or connecting wires. The independent variable is the current I, which is varied by changing the resistance of the variable resistor. The dependent variable is the terminal voltage V. Control variables include the temperature of the cell and the condition of the electrical contacts.
电路必须确保电压表直接测量电池两端的端电压，不包括电流表或连接导线的电压降。自变量是通过改变可变电阻阻值来调节的电流 I。因变量是端电压 V。控制变量包括电池的温度和电接触的状况。
A common student mistake is to place the voltmeter across the variable resistor instead of the cell; this would give a graph that does not show the cell’s internal resistance correctly. Carefully checking the circuit diagram before starting is essential.
学生常犯的错误是将电压表并联在可变电阻两端而非电池两端；这样得到的图像无法正确反映电池内阻。实验开始前仔细核对电路图至关重要。
3. Collecting and Recording Raw Data | 收集与记录原始数据
In the PH02 question, the student adjusts the variable resistor to set six different values of current I: 0.20 A, 0.40 A, 0.60 A, 0.80 A, 1.00 A and 1.20 A. For each current setting, three repeat readings of terminal voltage V are taken. The raw data are shown in the table below.
在PH02题目中，学生调节可变电阻设定了六个不同的电流 I 值：0.20 A、0.40 A、0.60 A、0.80 A、1.00 A 和1.20 A。每设定一个电流值，都记录三次端电压 V 的重复读数。原始数据如下表所示。
| Current I / A | V₁ / V | V₂ / V | V₃ / V |
|---|---|---|---|
| 0.20 | 1.41 | 1.42 | 1.43 |
| 0.40 | 1.33 | 1.34 | 1.35 |
| 0.60 | 1.25 | 1.26 | 1.27 |
| 0.80 | 1.17 | 1.18 | 1.19 |
| 1.00 | 1.09 | 1.10 | 1.11 |
| 1.20 | 1.01 | 1.02 | 1.03 |
Notice that the readings for each current setting are very close to each other, indicating good precision. The table also includes extra columns that the student is required to complete, such as the mean V.
注意，每个电流值下的读数彼此非常接近，表明精度良好。表格还包括学生需要完成的额外列，例如平均电压 Vₘ。
4. Calculating Mean Values | 计算平均值
42 + 1.43) ÷ 3 = 1.42 V. The same process gives the other means: 1.34 V, 1.26 V, 1.18 V, 1.10 V, and 1.02 V. Recording these values to the same number of decimal places as the raw data (2 decimal places) maintains consistency.
对于每个电流值，平均端电压 Vₘ 是三次读数之和除以 3。例如，在 I = 0.20 A 时，Vₘ = (1.41 + 1.42 + 1.43) ÷ 3 = 1.42 V。同样的过程得出其他平均值：1.34 V、1.26 V、1.18 V、1.10 V 和 1.02 V。将这些值记录到与原始数据相同的小数位数（两位小数）可保持一致性。
e uncertainty is approximately ±0.01 V. This is a useful skill for the evaluation section.
重复读数可以让你评估 V 的随机不确定度。一个快速估算是取极差的一半。对于 I = 0.20 A，极差为 0.02 V，因此不确定度约为 ±0.01 V。这是在评价部分很有用的技能。
5. Plotting the Graph of V against I | 绘制 V 对 I 的图像
Use graph paper or software to plot the six mean terminal voltage values on the y-axis against the current on the x-axis. Choose scales that allow the plotted points to occupy at least half the grid in each direction. Label axes clearly as 'Terminal voltage V / V' and 'Current I / A'. Plot each point as a small cross or circled dot, and ensure the points are accurately placed.
使用坐标纸或软件，将六个平均端电压值画在 y 轴上，电流画在 x 轴上。选择能使描点在每个方向上至少占据格子一半的标度。坐标轴清晰标注为 'Terminal voltage V / V' 和 'Current I / A'。每个描点用小十字叉或带圆圈的圆点标记，并保证描点位置准确。
pical plot will have the highest V at the smallest I and a steady decrease as I increases.
六个描点应呈现明显的负线性趋势，这验证了 V = E − I r 的关系成立。典型的图像在电流最小时 V 最大，并随着 I 增加而稳定下降。
6. Drawing the Line of Best Fit | 绘制最佳拟合直线
Using a transparent ruler, draw a single straight line that passes as close as possible to all the plotted points. The line should have roughly equal numbers of points on either side. Avoid forcing the line through the origin unless the intercept is actually (0,0). The best-fit line is continuous and extends beyond the first and last data points to allow determination of the y-intercept.
使用透明直尺，画一条尽可能接近所有描点的单一直线。直线两侧应有大致相等数量的点。除非截距确实在 (0,0)，否则不要强行使直线通过原点。最佳拟合直线是连续的，并延伸到第一个和最后一个数据点之外，以便确定 y 轴截距。
Once the line is drawn, pick two points on the line (not data points) that are far apart to calculate the gradient. For example, (0.10 A, 1.46 V) and (1.30 A, 0.98 V) could be used. Record these coordinates clearly.
当直线画好后，在直线上挑选两个相距较远的点（不是原数据点）来计算斜率。例如，可以使用 (0.10 A, 1.46 V) 和 (1.30 A, 0.98 V)。清晰地记录这些坐标。
7. Determining the EMF from the y‑intercept | 由 y 轴截距确定电动势
.50 V. According to the equation V = E − I r, when I = 0, V = E. Hence the EMF of the cell is 1.50 V.
延伸最佳拟合直线，使其与 y 轴相交（即 I = 0 A 处）。读取此时 V 的截距值。在我们的示例中，直线在 V = 1.50 V 处与 y 轴相交。根据方程 V = E − I r，当 I = 0 时，V = E。因此电池的电动势为 1.50 V。
V = E − I r
Always quote the EMF with its unit (V) and to the same precision as the graph allows, typically 2 or 3 significant figures. Here 1.50 V is appropriate.
始终为电动势标出单位（V），并按照图像能够达到的精度给出数值，通常取 2 或 3 位有效数字。这里的 1.50 V 是合适的。
8. Calculating Internal Resistance from the Gradient | 由斜率计算内阻
The gradient m of the V–I graph is calculated as ΔV / ΔI. Using the two chosen on‑ (−0.48) / 1.20 = −0.40 V A⁻¹. Since the gradient = −r, the internal resistance r = 0.40 Ω.
V–I 图像的斜率 m 计算为 ΔV / ΔI。使用所选的两个线上点：(0.10 A, 1.46 V) 和 (1.30 A, 0.98 V)，斜率 = (0.98 − 1.46) / (1.30 − 0.10) = (−0.48) / 1.20 = −0.40 V A⁻¹。由于斜率 = −r，内阻 r = 0.40 Ω。
Be careful to include the negative sign in your gradient calculation but remember that internal resistance is a positive quantity. The unit is ohms (Ω). The value of r can also be checked by selecting another pair of points; a similar result confirms the line is well drawn.
小心地在斜率计算中保留负号，但要记住内阻是一个正数。单位是欧姆 (Ω)。r 的值也可以通过另选一对点来核对；相近的结果可以确认直线画得很好。
9. Uncertainty Analysis Using Max and Min Slopes | 利用最大和最小斜率进行不确定度分析
To estimate the uncertainty in EMF and internal resistance, draw two additional lines: the steepest and shallowest possible straight lines that are still acceptable fits to the data points. These are often called the 'worst‑fit' lines. Their slopes and y‑intercepts give the ranges for r and E.
为了估算电动势和内阻的不确定度，额外画出两条直线：一条是仍能大致吻合数据点的最陡可接受直线，另一条是最平缓可接受直线。这两条线常被称为“最差拟合线”。它们的斜率和 y 轴截距给出了 r 和 E 的可能范围。
A⁻¹. The uncertainty in EMF is about ±0.02 V, and the uncertainty in r is about ±0.04 Ω. Express final results as E = 1.50 ± 0.02 V and r = 0.40 ± 0.04 Ω.
假设最陡直线的截距为 1.52 V、斜率为 −0.44 V A⁻¹，最平缓直线的截距为 1.48 V、斜率为 −0.36 V A⁻¹。那么，电动势的不确定度约为 ±0.02 V，内阻的不确定度约为 ±0.04 Ω。最终结果可表示为 E = 1.50 ± 0.02 V 和 r = 0.40 ± 0.04 Ω。
Always check that your uncertainty estimate is consistent with the scatter of your points. If the points lie almost perfectly on a straight line, a very small uncertainty can be justified; if the points show noticeable spread, the uncertainty will be larger.
始终检查你的不确定度估算是否与数据点的离散程度相符。如果描点几乎完全落在一条直线上，较小的不确定度是合理的；如果描点呈现出明显的散布，不确定度就会更大。
10. Evaluation of the Experiment | 实验评价
A good evaluation identifies the main sources of error and assesses their impact. In this experiment, the dominant random error comes from reading the voltmeter, especially if the needle or digital display fluctuates. Taking repeat readings helps reduce this. A possible systematic error is a voltmeter with a non‑zero offset or a high internal resistance voltmeter drawing a tiny current, though modern digital voltmeters have very high impedance.
优秀的评价要找出主要的误差来源并评估其影响。在此实验中，主要的随机误差来自读取电压表时的读数，尤其是当指针或数字显示发生波动时。进行重复读数有助于减小这一误差。可能的系统误差包括电压表存在非零零点偏移，或者虽内阻很高但仍会汲取微小电流的电压表影响测量，不过现代数字电压表阻抗非常高。
The internal resistance of the ammeter may also introduce a slight systematic error if it is not negligible compared with r, but in most AS experiments r is much larger. Another point is that the cell's internal resistance may change as the cell discharges during the experiment; the current settings should be changed quickly and the circuit opened between readings to minimise this effect.
电流表的内阻如果与 r 相比不可忽略，也会带来轻微的系统误差，但在大多数 AS 实验中 r 要大得多。另一点是，电池的内阻可能在实验过程中因放电而改变；应快速改变电流设定，并在两次读数之间断开电路，以最大限度减少这种影响。
11. Suggested Improvements and Follow‑up Investigations | 改进建议与后续探究
One improvement is to use a potentiometer to measure the true EMF without drawing current from the cell, providing an independent check on the graphical value. Another is to replace the single cell with a series of cells to investigate how internal resistance varies with cell configuration and age.
一项改进是使用电位差计来测量真正的电动势而不从电池取电流，以此独立验证图像所得的值。另一个改进是，用一组串联的电池代替单节电池，以探究内阻如何随电池的组合方式和老化程度而变化。
Adding a data logger to record V and I simultaneously can greatly reduce human reaction time errors and allow collection of many more data points, improving the reliability of the best‑fit line. However, students should still understand the manual method for examination purposes.
添加数据采集器来同时记录 V 和 I 可以大大减少人为反应时间误差，并允许采集更多数据点，提高最佳拟合直线的可靠性。不过，出于考试目的，学生仍应理解手动操作方法。
12. Key Examiner Tips for AS Experimental Questions | 给AS实验题的关键考官提示
Always read the question carefully to see what you are asked to calculate: EMF, internal resistance, or both. If an equation is provided, write it down and identify which graph variables correspond to y, x, gradient and intercept. Use the exact values requested from your line of best fit, and show your working clearly when calculating the gradient.
始终仔细阅读题目，搞清楚是要你计算电动势、内阻还是两者都要。如果给出了公式，把它写下来，并找出哪个图像变量对应 y、x、斜率和截距。使用从最佳拟合直线上读出的精确数值，并在计算斜率时清晰地展示计算过程。
Never connect the data points dot‑to‑dot; always draw a single, smooth best‑fit straight line. Scale your axes evenly and label them with quantity and unit, just as you see in mark schemes. A poorly drawn graph is one of the most common reasons for losing marks in this section.
绝不能将数据点逐点连接；始终画出一条单一、平滑的最佳拟合直线。均匀地标记坐标轴，并像在评分方案中看到的那样标注物理量和单位。画图不佳是本节失分最常见的原因之一。
Finally, always relate your evaluation comments back to the actual readings and the scatter you observe. Generic statements about 'human error' without specifics rarely gain credit.
最后，始终将你的评价评论与你观察到的实际读数和点的散布联系起来。没有具体细节的关于“人为误差”的空泛陈述很少能得分。
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