📚 A-Level Physics Unit 4 Insert Jan 2020: Experimental Investigation | A-Level 物理：Unit 4 实验探究（2020年1月附加资料）
The January 2020 Unit 4 insert for A-Level Physics presented a set of experimental data and guidance for an investigation into the discharge of a capacitor through a resistor. This article unpacks the key experimental concepts, analysis techniques, and typical exam questions that arise from such an investigation, helping students to master the required practical skills and data-handling methods.
2020年1月的A-Level物理Unit 4附加资料提供了一组实验数据和指导，用于研究电容器通过电阻放电的过程。本文将详细解析该实验的关键概念、分析方法以及由此衍生的典型考试问题，帮助学生掌握必备的实验技能与数据处理方法。
1. Experimental Context and Aims | 实验背景与目标
The insert formed part of a practical-based question where students were asked to analyse voltage–time data from a capacitor discharging through a fixed resistor. The primary aim was to determine the time constant of the RC circuit and, using that value, to calculate the capacitance of the capacitor if the resistance was already known.
该附加资料是一道实验题的一部分，要求学生分析一个电容器通过固定电阻放电的电压–时间数据。主要目标是测定RC电路的时间常数，并利用该值在已知电阻的情况下计算出电容器的电容。
A secondary aim was to assess understanding of exponential decay, graphical analysis using logarithms, and identification of sources of uncertainty in the data collection process.
次要目标是考查学生对指数衰减的理解、对数图像分析的能力，以及对数据采集过程中不确定度来源的识别。
2. Theoretical Foundation: Exponential Decay | 理论基础：指数衰减
For a capacitor discharging through a resistor, the voltage across the capacitor decreases exponentially with time. The governing equation is:
电容器通过电阻放电时，其两端电压随时间呈指数衰减。控制方程如下：
V = V₀ e−t/RC
The product R × C is called the time constant, often denoted by the Greek letter τ (tau).
其中V是t时刻的电压，V₀是t = 0时的初始电压，R是电阻，C是电容。乘积R × C称为时间常数，通常用希腊字母τ表示。
After one time constant (t = τ = RC), the voltage falls to approximately 37% its initial value. This property is frequently used to estimate τ directly from a voltage–time graph.
经过一个时间常数（t = τ = RC）后，电压降至初始值的约37%。这一特性常被用来直接从电压–时间图中估算τ。
3. Equipment Used in the January 2020 Setup | 2020年1月实验装置所用的设备
Based on typical Unit 4 practical inserts from this period, the experiment was likely assembled using the following apparatus:
根据这一时期典型的Unit 4实验附加资料，该实验很可能使用了如下装置：
A large-value electrolytic capacitor (e.g. 4700 μF) to produce a discharge slow enough for manual or digital recording.
一个大容值电解电容器（如4700 μF），使放电过程足够缓慢，以便手动或数字记录。
A fixed resistor of known resistance (e.g. 10 kΩ), connected in series with the capacitor.
一个已知阻值的固定电阻（如10 kΩ），与电容器串联。
A high-impedance digital voltmeter or a voltage sensor connected to a data logger, allowing voltage readings to be taken at regular intervals without significantly affecting the circuit.
高输入阻抗的数字电压表或连接数据记录仪的电压传感器，可在固定时间间隔读取电压值，而不会对电路产生明显影响。
A single-pole double-throw switch to allow the capacitor to be fully charged by a battery and then rapidly switched to discharge through the resistor.
一个单刀双掷开关，使电容器先由电池充满电，再迅速切换到通过电阻放电。
A d.c. power supply set to a suitable voltage, typically 6.0 V, to charge the capacitor fully before each run.
一个设置为合适电压的直流电源（通常为6.0 V），在每次实验前为电容器充满电。
4. The Insert Data Table: V against t | 附加资料数据表：电压随时间变化
acitor through a resistor of known resistance, R = 15.0 kΩ. The data are reproduced below.
2020年1月的附加资料给出了一个数据表，记录了电容器通过已知阻值R = 15.0 kΩ的电阻单次放电过程中每10秒的电位差值。数据重现如下。
| Time t / s | Voltage V / V |
|---|---|
| 0 | 6.00 |
| 10 | 4.42 |
| 20 | 3.25 |
| 30 | 2.39 |
| 40 | 1.76 |
| 50 | 1.30 |
| 60 | 0.96 |
that the expected τ = 15 000 × C will be in the range of tens of seconds, making manual timing practical.
这组数据是RC放电实验的典型代表，并能用多种方法求出时间常数。所选的电阻值能保证预期τ = 15 000 × C在几十秒的范围内，使人工计时变得可行。
5. Data Collection Procedure Step by Step | 数据采集步骤详解
Although the insert primarily tests data analysis, the practical method behind the data is essential knowledge. The following steps would produce the table given in the insert.
尽管附加资料主要考查数据分析，但数据背后的实验方法是必备知识。以下步骤可以生成资料中给出的数据表。
Step 1: Connect the capacitor, resistor, voltmeter, switch and power supply in the correct discharge circuit.
步骤1：将电容器、电阻、电压表、开关和电源正确连接成放电电路。
on and leave it until the voltmeter reading stabilises at the supply voltage, V₀ = 6.00 V.
步骤2：将开关拨到充电位置，直到电压表读数稳定在电源电压V₀ = 6.00 V。
Step 3: Start the stopwatch (or initiate data logging) at the exact moment the switch is thrown to the discharge position.
步骤3：在开关拨到放电位置的瞬间启动秒表（或开始数据记录）。
uce random error, the experiment should be repeated at least three times and the average voltage at each time calculated.
步骤4：在t = 10 s、20 s、30 s……60 s时记录电压表读数。为减少随机误差，实验应至少重复三次，并计算每个时间点的平均电压。
Step 5: Ensure that the voltmeter probes are connected across the capacitor only, and that the meter has an impedance much larger than R to avoid loading the circuit.
步骤5：确保电压表探头只跨接在电容器两端，且电表的阻抗远大于R，以避免给电路带来负载效应。
6. Calculating the Time Constant Directly | 直接计算时间常数
From the dataset, we can estimate the time constant by finding the time at which the voltage drops to 37% of its initial value. 37% of 6.00 V is approximately 2.22 V.
从数据集中，我们可以通过寻找电压降至初始值37%的对应时间来估算时间常数。6.00 V的37%约为2.22 V。
uals 2.22 V is approximately 31.3 s. Therefore, the experimental time constant τ ≈ 31.3 s.
查看表格，V在30 s时为2.39 V，在40 s时为1.76 V。通过在这两点之间进行线性插值，V等于2.22 V的时间约为31.3 s。因此，实验时间常数τ ≈ 31.3 s。
15.0 × 10³ Ω, we obtain C = τ / R = 31.3 / (15.0 × 10³) ≈ 2.09 × 10⁻³ F, or about 2090 μF.
利用τ = RC，已知R = 15.0 × 10³ Ω，可得C = τ / R = 31.3 / (15.0 × 10³) ≈ 2.09 × 10⁻³ F，即约2090 μF。
This direct method is quick but relies heavily on the accuracy of two data points and is sensitive to random errors in voltage measurement.
这种直接方法速度快，但严重依赖于两个数据点的准确性，并对电压测量中的随机误差非常敏感。
7. Graphical Analysis Using Logarithms | 使用对数进行图形分析
ing the exponential decay equation. Taking the natural logarithm of both sides of V = V₀ e−t/RC gives:
更可靠的方法是对方程进行线性化处理。对V = V₀ e−t/RC两边取自然对数，得到：
ln V = ln V₀ − t/(RC)
against t should yield a straight line with gradient m = −1/RC and y-intercept c = ln V₀.
这符合y = c + mx的形式，因此绘制ln V对t的图像应得到一条直线，其斜率m = −1/RC，y轴截距c = ln V₀。
For the insert data, the ln V values are calculated and tabulated below:
对于资料中的数据，ln V值计算并列于下表：
| Time t / s | Voltage V / V | ln(V / V) |
|---|---|---|
| 0 | 6.00 | 1.79 |
| 10 | 4.42 | 1.49 |
| 20 | 3.25 | 1.18 |
| 30 | 2.39 | 0.871 |
| 40 | 1.76 | 0.565 |
| 50 | 1.30 | 0.262 |
| 60 | 0.96 | −0.041 |
close to the direct estimate. The capacitance is then C = 32.3 / 15 000 ≈ 2.15 × 10⁻³ F (2150 μF).
绘制这些点并画出最佳拟合线，可得斜率约为−0.031 s⁻¹。由于斜率 = −1/RC，得到RC = 1/0.031 ≈ 32.3 s，与直接估算值非常接近。由此计算电容C = 32.3 / 15 000 ≈ 2.15 × 10⁻³ F（2150 μF）。
The log method uses all data points, thereby reducing the impact of random errors and making it the preferred technique in A-Level analysis.
对数方法使用了所有数据点，从而减少了随机误差的影响，是A-Level分析中首选的技巧。
8. Determining Capacitance if Resistance is Known | 已知电阻时计算电容值
y 2150 μF, as calculated. If the capacitor was labelled as 2200 μF with a tolerance of ±10%, the experimental value falls well within the manufacturer's specification.
已知R = 15.0 kΩ，并由斜率求得τ = 32.3 s，则电容C = τ / R。如上计算，结果约为2150 μF。如果电容器标称值为2200 μF、容差为±10%，则实验值完全在制造商的规格范围内。
Exam questions often ask students to compare their experimental capacitance with the stated value and to discuss possible reasons for any discrepancy, such as leakage current within the capacitor or the accuracy of the resistance used.
考试题常常要求学生将实验电容值与标示值进行比较，并讨论可能造成差异的原因，例如电容器内部的漏电流或所用电阻的准确度。
9. Investigating the Effect of Resistance on Discharge | 探究电阻对放电的影响
Although the insert focused on one fixed resistor, a common extension is to investigate how the time constant changes when different resistors are used. The discharge equation shows that τ is directly proportional to R.
虽然附加资料聚焦于一个固定电阻，常见的拓展是探究使用不同电阻时时间常数如何变化。放电方程表明τ与R成正比。
If the experiment is repeated with R = 7.5 kΩ, the gradient of the ln V– lower resistance allows the capacitor to discharge more rapidly, as the current is larger.
若改用R = 7.5 kΩ重复实验，则ln V–t图的斜率会变得更陡，τ将减半。这证实了较低的电阻会使电容器放电更快，因为电流更大。
In a practical investigation, it is essential to ensure that the capacitor is fully recharged before each run and that the same initial voltage is used to maintain consistency. Temperature should also be kept constant because capacitance can drift with temperature changes.
在实际探究中，必须确保每次实验前电容器已完全充电，并使用相同的初始电压以保持一致性。温度也应保持恒定，因为电容可能会随温度变化而发生漂移。
10. Common Sources of Error and Uncertainty | 常见误差与不确定度来源
Several factors can affect the reliability of the data in this experiment. The following are frequently cited in mark schemes:
有几个因素会影响本实验数据的可靠性。以下是在评分标准中经常提到的：
Parallel leakage resistance within the capacitor effectively reduces the overall resistance, causing the voltage to decay slightly faster than predicted by the nominal RC value.
电容器内部的并联泄漏电阻有效降低了总电阻，导致电压衰减速度略快于标称RC值所预测的。
Finite impedance of the voltmeter draws a small current, altering the effective discharge path. A digital voltmeter with an input impedance of at least 10 MΩ should be used.
电压表有限的阻抗会引入微小电流，改变了实际的放电路径。应使用输入阻抗至少为10 MΩ的数字电压表。
tainty, typically ±0.2 s. Using a data logger with a voltage sensor eliminates this error.
启动和停止秒表时的人为反应时间会引入随机计时不确定度，通常为±0.2 s。采用配有电压传感器的数据记录仪可消除此误差。
Poor connections or dirty contacts can introduce variable contact resistance, causing fluctuations in the discharge rate.
不良的接线或脏污的触点会引入可变的接触电阻，导致放电速率波动。
The resistor's actual value may differ from its colour-coded value due to tolerance (e.g. ±5%), directly affecting the calculated capacitance.
电阻的实际值可能因容差（如±5%）而不同于其色环标注值，这直接影响计算出的电容值。
11. Improving the Experiment | 改进实验
To obtain more reliable values for the time constant and capacitance, students are expected to suggest realistic improvements. Replacing manual timing with a data logger and voltage sensor sampling at 1-second intervals dramatically increases the number of data points and allows a more precise determination of the gradient.
为了获得更可靠的时间常数和电容值，学生应能够提出切实可行的改进方案。用数据记录仪和电压传感器以每秒一次的频率采样，替代人工计时，可大幅增加数据点数量，从而更精确地确定斜率。
The experiment should be repeated several times with the same capacitor and resistor to allow calculation of a mean time constant and an estimate of the spread of results, which can be quoted as the experimental uncertainty.
应对同一个电容器和电阻重复进行多次实验，以便计算出平均时间常数并估算结果的离散程度，这可作为实验不确定度给出。
Measuring the actual resistance with an ohmmeter before the experiment, rather than assuming the colour-code value, eliminates a systematic error. Similarly, allowing the capacitor to discharge fully and checking that the initial voltage is exactly repeatable helps to ensure consistency.
在实验前用欧姆表测量电阻的实际值，而不是假设色环标称值，可以消除一个系统误差。同样，让电容器充分放电并检查初始电压是否能够准确重现，有助于确保一致性。
12. Exam-Style Questions and Model Answers | 考试风格问题与范答
Typical questions based on the January 2020 insert include:
基于2020年1月附加资料的典型问题包括：
Q: Describe how you would use the data in the table to obtain a value for the time constant without drawing a graph.
问：描述如何不通过作图而利用表中的数据得到时间常数的值。
A: Find the time at which the voltage falls to approximately 2.22 V (37% of 6.00 V). From the table, V is between 2.39 V and 1.76 V at 30 s and 40 s. Interpolation gives approximately 31.3 s, so the time constant is about 31 s.
答：找出电压降至约2.22 V（6.00 V的37%）所对应的时间。从表中可知，V在30 s时为2.39 V，在40 s时为1.76 V。通过插值得到约31.3 s，因此时间常数约为31 s。
Q: Explain why taking the natural logarithm of the voltage values and plotting a ln V–t graph is more reliable than the direct method.
问：解释为什么对电压值取自然对数并绘制ln V–t图像比直接法更可靠。
A: The ln V–t graph uses all seven data points to determine the gradient, so the effect of a single anomalous reading is reduced. The straight line also allows easy identification of systematic departures from the theoretical exponential decay, such as those caused by capacitor leakage.
答：ln V–t图像使用了全部七个数据点来确定斜率，因此单个异常读数的影响被减小。直线形状还便于识别与理论指数衰减的系统性偏离，例如由电容器漏电引起的偏离。
Q: The manufacturer's value for the capacitance is 2200 μF ± 10%. Evaluate the agreement between the experimental result and this value.
问：制造商给出的电容值为2200 μF ± 10%。评估实验结果与该值之间的一致性。
A: The experimental capacitance from the gradient is about 2150 μF. The manufacturer's tolerance range is 1980–2420 μF. The measured value lies well within this range; the percentage difference is only about 2.3%. The agreement is excellent and validates the method.
答：由斜率得到的实验电容约为2150 μF。制造商的容差范围为1980–2420 μF。测量值在此范围内；百分比差异仅为约2.3%。一致性极好，验证了该方法的有效性。
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