📚 A-Level Physics Unit 4 Jan 2019 Experimental Investigation | A-Level 物理 Unit 4 2019年1月实验探究
In the January 2019 Edexcel IAL Physics Unit 4 (WPH04/01) paper, one of the core questions centred on an experimental investigation into the discharge of a capacitor through a resistor. This type of practical-based question is a staple of the Unit 4 syllabus, testing your ability to plan, collect data, analyse a graph, and determine a physical constant – in this case, the time constant RC of the circuit. This article will walk you through the thinking, technique, and exam-ready answers required to master such experimental tasks. We will reconstruct the likely experimental setup, table of results, graphical analysis, and error evaluation, giving you a complete revision guide for similar past-paper challenges.
在2019年1月的爱德思IAL物理Unit 4（WPH04/01）试卷中，有一道核心题目围绕电容通过电阻放电的实验探究展开。这类基于实验操作的题目是Unit 4考纲的常客，重点考查你设计实验、收集数据、分析图像并确定物理量（此处为电路的时间常数RC）的能力。本文将全程解析应对这类实验任务所需的思路、技巧和应试答案。我们将重现可能的实验装置、记录表格、图像分析以及误差评估，为你提供一份应对同类真题的完整复习指南。
1. Understanding the Question Structure | 理解题目结构
Questions on experimental physics in Unit 4 typically follow a predictable pattern. First, they ask you to identify the independent and dependent variables, comment on the control of other factors, and list the apparatus. Next, you will be guided to record readings in a table, often with pre-printed headings. A graph is then plotted, from which a gradient or intercept is used to find a target quantity. Finally, you will discuss uncertainties, limitations, and realistic improvements. For the Jan 2019 capacitor discharge question, the independent variable was time t, the dependent variable was the voltage V across the capacitor, and the goal was to determine the time constant RC.
Unit 4实验物理题通常遵循一个可预测的模式。首先，它要求你指出自变量和因变量，评论其他因素的控制，并列出所用仪器。接下来，题目会引导你把读数记录在表格中，表格通常已给出表头。然后你需要绘制图像，利用图像梯度或截距求出目标物理量。最后，你要讨论不确定度、局限性和可行的改进方案。对于2019年1月的电容放电题，自变量是时间t，因变量是电容器两端的电压V，实验目标是确定时间常数RC。
2. Experimental Context: Capacitor Discharge Through a Resistor | 实验背景：电容通过电阻放电
When a charged capacitor is connected across a resistor, the voltage across its plates decays exponentially with time. The fundamental equation governing this process is V = V₀ exp(–t / RC), where V₀ is the initial voltage at t = 0, R is the resistance, and C is the capacitance. The product RC, which has units of seconds, is called the time constant. After a time equal to one time constant, the voltage falls to about 37% of its original value. By measuring V at various times and plotting a suitable linear graph, we can obtain RC without needing to know R and C individually.
当充电后的电容器与一个电阻连接时，其两端的电压随时间按指数规律衰减。描述这一过程的基本方程为V = V₀ exp(–t / RC)，其中V₀是t=0时刻的初始电压，R是电阻值，C是电容值。乘积RC具有时间量纲，称为时间常数。经过一个时间常数后，电压将降至初始值的约37%。通过在不同时刻测量电压V并绘制合适的线性化图像，我们可以在不单独知道R和C的情况下求出RC。
3. Expected Apparatus and Circuit Setup | 预期仪器与电路搭建
The typical list of apparatus for this investigation includes: a d.c. power supply (e.g. 6 V), a large electrolytic capacitor (commonly 1000 μF), a high-resistance resistor (e.g. 33 kΩ), a voltmeter (digital or analogue), a single-pole double-throw (SPDT) switch to charge and discharge, a stopwatch, and connecting leads. It is good practice to mount the components on a circuit board and to use a switch that allows quick disconnection from the supply while simultaneously completing the discharge loop, thus minimising the delay between starting the timer and the actual start of discharge.
这项实验常用的仪器清单包括：一台直流电源（例如6 V），一只大容量电解电容（通常为1000 μF），一只高阻值电阻（例如33 kΩ），一块电压表（数字或模拟），一个单刀双掷（SPDT）开关用于充电和放电，一块秒表以及若干连接导线。将元件安装在电路板上并使用能迅速断开电源同时闭合放电回路的开关是良好的做法，这样可以最大程度缩短启动秒表与放电真正开始之间的延迟。
4. Step-by-Step Data Collection Procedure | 逐步数据收集流程
Begin by connecting the capacitor and resistor in series with the power supply and voltmeter, ensuring the voltmeter is connected in parallel with the capacitor. Use the SPDT switch to connect the capacitor to the supply until the voltmeter reading stabilises at the supply voltage, V₀. Then, throw the switch to connect the capacitor solely to the resistor, and simultaneously start the stopwatch. Record the voltmeter reading at regular time intervals – every 10 s is a sensible choice for an RC of about 30–40 s. Continue until the voltage drops to about one-tenth of V₀, to have enough points for a reliable graph.
首先将电容器、电阻与电源及电压表串联，确保电压表并联在电容器两端。利用单刀双掷开关将电容器接入电源，直到电压表读数稳定在电源电压V₀。然后迅速拨动开关，使电容器仅与电阻连接，同时启动秒表。按照固定的时间间隔记录电压表读数——对30–40秒左右的时间常数，每10秒记录一次是合理的。持续记录直到电压降至V₀的大约十分之一，以便获得足够的数据点绘制可靠图像。
5. Recording Data in a Structured Table | 在结构化表格中记录数据
The markscheme typically rewards a table with clear headings, consistent significant figures, and a column for the processed quantity that will be plotted. Below is a sample data set obtained with V₀ = 6.00 V, R = 33 kΩ, and C = 1000 μF (theoretical RC = 33 s). The third column gives the natural logarithm of the voltage, which linearises the exponential decay: ln V = ln V₀ – t / RC.
评分标准通常会奖励表头清晰、有效数字一致且包含即将绘制的处理后物理量的一栏表格。以下是一组示例数据，采用V₀ = 6.00 V，R = 33 kΩ，C = 1000 μF（理论RC = 33 s）。第三栏给出电压的自然对数，它将指数衰减线性化：ln V = ln V₀ – t / RC。
| Time t / s | Voltage V / V | ln(V / V) |
|---|---|---|
| 0 | 6.00 | 1.79 |
| 10 | 4.43 | 1.49 |
| 20 | 3.27 | 1.18 |
| 30 | 2.42 | 0.88 |
| 40 | 1.79 | 0.58 |
| 50 | 1.32 | 0.28 |
| 60 | 0.97 | -0.03 |
Notice that all voltage readings are quoted to two decimal places, consistent with a voltmeter of precision 0.01 V. The ln values are given to two decimal places, matching the plotting precision expected on standard graph paper. If your own data shows larger scatter, you might still round to two decimal places for plotting, but always note any anomalous points.
注意所有电压读数都保留到小数点后两位，这与精度为0.01 V的电压表一致。ln值也保留到小数点后两位，与标准坐标纸上预期的绘图精度匹配。如果你的数据离散性较大，绘图时可能仍取两位小数，但务必标注任何异常点。
6. Graphical Analysis: Plotting ln V Against Time | 图形分析：绘制 ln V 与时间的关系图
The next step is to plot a graph of ln(V / V) on the vertical axis against time t on the horizontal axis. Use a scale that occupies at least half the grid on both axes. Draw a line of best fit – a straight line for an ideal capacitor discharge. Since ln V = ln V₀ – (1/RC) t, the gradient of this line is equal to –1/RC. Therefore, a larger time constant gives a shallower negative slope. Be meticulous: label the axes with quantities and units, and mark plotted points with small crosses.
下一步是绘制纵轴为ln(V / V)、横轴为时间t的图形。使用的坐标刻度应至少占据网格线的一半。画一条最佳拟合线——理想情况下电容放电应为一条直线。根据ln V = ln V₀ – (1/RC) t，该直线的梯度等于–1/RC。因此，时间常数越大，负斜率的绝对值越小。请务必认真：坐标轴应标明物理量和单位，并用小十字标出数据点。
7. Determining the Time Constant from the Graph | 从图像确定时间常数
To obtain RC, select two well-separated points on the line of best fit (not necessarily data points). Calculate the gradient using:
gradient = Δ(ln V) / Δt
Then, since gradient = –1/RC, the time constant is RC = –1 / gradient. For the sample data above, using the points at t = 20 s (ln V = 1.18) and t = 50 s (ln V = 0.28) gives:
gradient = (0.28 – 1.18) / (50 – 20) = –0.90 / 30 = –0.0300 s⁻¹
RC = –1 / (–0.0300 s⁻¹) = 33.3 s
This experimental value agrees well with the theoretical value of 33.0 s, demonstrating a successful investigation. Always quote the final answer with the correct unit (seconds) and, if possible, compare it with the expected value calculated from the component markings.
要得到RC，在最佳拟合线上挑选两个间隔较远的点（不一定为原始数据点）。用公式计算梯度：梯度 = Δ(ln V) / Δt。然后，因为梯度 = –1/RC，时间常数RC = –1 / 梯度。对于上述样本数据，选取t = 20 s (ln V = 1.18) 和 t = 50 s (ln V = 0.28)两点，计算得梯度 = –0.0300 s⁻¹，因此RC = 33.3 s。该实验值与理论值33.0 s吻合良好，表明实验探究成功。给出最终答案时务必标明正确单位（秒），并尽可能与根据元件标称值计算出的预期值进行比较。
8. Another Verification: Using the 37% Method | 另一种验证：37% 法
An alternative approach, often accepted as a check, is to read the time at which the voltage falls to 37% of V₀. From the initial voltage 6.00 V, 37% is about 2.22 V. In our data table, the voltage at t = 30 s is 2.42 V and at t = 40 s is 1.79 V; interpolation places the time for V = 2.22 V at roughly 33 s, which closely matches our gradient-based result. While examiners prefer the graphical method, mentioning both can demonstrate deeper understanding.
一种替代方法，常被用作验算，是读取电压降至V₀的37%所对应的时间。初始电压6.00 V的37%约为2.22 V。从数据表看，t = 30 s时电压为2.42 V，t = 40 s时为1.79 V；通过内插可以估算V = 2.22 V的时间大约为33 s，这与基于梯度的结果非常接近。尽管考官更偏爱作图法，但提及两者能够展现更深的理解。
9. Sources of Error and Their Impact | 误差来源及其影响
- Reaction time in starting the stopwatch: Even with a switch, a small delay between starting the timer and the actual discharge can cause systematic error, shifting the entire ln V graph slightly but not affecting the gradient significantly if the delay is constant.
- Voltmeter loading effect: A digital voltmeter has a very high but finite internal resistance, which can provide an additional discharge path and lower the effective time constant slightly.
- Capacitor leakage: Electrolytic capacitors can leak charge over time, causing the voltage to drop faster than predicted, particularly at long times.
- Rounding in ln values: Using two decimal places in ln V introduces small uncertainties that propagate into the gradient calculation.
- 启动秒表的反应时间：即使使用开关，启动计时与放电实际开始之间的微小延迟仍会引入系统误差，使整条ln V图像轻微平移，但如果延迟恒定，对梯度无明显影响。
- 电压表的负载效应：数字电压表的内阻极大但并非无限，会形成一个额外的放电通路，略微降低有效时间常数。
- 电容器漏电：电解电容随时间会泄漏电荷，导致电压下降比预期更快，尤其在长时间段更明显。
- ln值舍入：ln V取两位小数会引入微小不确定度，并传递到梯度计算中。
10. Recommended Improvements for Greater Accuracy | 提高精度的改进建议
To reduce timing uncertainties, use a data‑logger with a voltage sensor that records V every fraction of a second automatically. This eliminates reaction-time error and produces many more data points for a smoother graph. Choose a larger time constant (e.g. by increasing R or C) so that the discharge proceeds more slowly, making manual timing errors less significant. Also, measure R and C individually using a multimeter and compare the product with the experimental RC to validate the result. Finally, repeat the discharge run three times and average the gradients to minimise random errors.
为减小计时不确定度，可使用配有电压传感器的数据采集器，自动以几分之一秒的间隔记录V。这能消除反应时间误差，并为获得更平滑的图像提供大量数据点。选择一个更大的时间常数（例如增大R或C），使放电进行得更缓慢，从而削弱手动计时误差的影响。此外，用万用表分别测量R和C，将乘积与实验得出的RC进行比较，以验证结果。最后，重复放电过程三次并对所得梯度取平均值，以减小随机误差。
11. Answering the Jan 2019 Exam-Style Question | 回答 2019 年 1 月真题风格的问题
In your examination answer, be concise but complete. For the ‘plan’ part, write a step-by-step recipe using bullet points. For the table, ensure units are in the header and data are consistent. For the graph, include a clear line of best fit and draw a large triangle to show your gradient calculation. State the final RC value with units, and compare it with the theoretical RC. In the evaluation section, identify at least two sources of error and suggest practical improvements linked to each. Use the mark scheme’s language: ‘systematic error’, ‘random error’, ‘parallax’, ‘zero error’, etc., where appropriate.
在考试作答中，要简洁但完整。对于“设计”部分，用项目符号分步写出方案。对于表格，确保表头包含单位且数据前后一致。对于图像，画出清晰的的最佳拟合线，并用一个大三角形展示你的梯度计算。给出带有单位的最终RC值，并与理论RC进行比较。在评估环节，至少指出两项误差来源，并针对每一项提出切实的改进措施。适当使用评分标准中的术语，如“系统误差”“随机误差”“视差”“零点误差”等。
12. Key Takeaways for Unit 4 Practical Questions | Unit 4 实验题的核心要点
- Always linearise exponential relationships using natural logarithms — here, ln V vs t yields a straight line.
- Label axes with quantity and unit, e.g. ‘ln(V / V)’ and ‘t
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