📚 Investigating Capacitor Discharge: Determining the Time Constant | 探究电容器放电：测定时间常数
In the January 2023 Edexcel International A-Level Physics Unit 4 paper (WPH04/01), the experimental investigation focused on the discharge of a capacitor through a resistor, using a voltage sensor and data logger to record the decaying potential difference. This classic experiment enables students to determine the time constant of the RC circuit and to evaluate the capacitance, while practising skills in data logging, graphical analysis and uncertainty estimation. Here we present a complete walkthrough of the investigation, linking directly to the themes and techniques required by the examination.
在2023年1月爱德思国际A-Level物理单元4试卷（WPH04/01）中，实验探究部分聚焦于电容器通过电阻的放电过程，要求学生利用电压传感器和数据记录仪记录衰减的电压。这一经典实验让学生能够测定RC电路的时间常数并推算电容值，同时锻炼数据记录、图像分析以及不确定度估算的能力。下文将完整呈现该探究的操作流程与分析方法，紧扣试卷所考察的技能与原理。
1. Experimental Aim | 实验目的
To investigate how the potential difference across a capacitor decreases with time when it discharges through a fixed resistor, and to use the data to determine the time constant of the circuit and the capacitance of the capacitor.
探究电容器通过固定电阻放电时两端电压随时间衰减的规律，并利用实验数据确定电路的时间常数以及电容器的电容值。
2. Theoretical Background | 理论背景
ischarge of a capacitor through a resistor follows an exponential decay described by V = V0 e–t/RC, where V is the voltage at time t, V0 e. The product RC is called the time constant τ. When t = τ, the voltage falls to about 37% of V0. Taking natural logarithms gives a linear relationship: ln V = ln V0 – t/(RC). A graph of ln V against t yields a straight line with gradient –1/RC and intercept ln V0.
电容器通过电阻放电遵循指数衰减规律：V = V0 e–t/RC，其中V为t时刻的电压，V0为初始电压，R为电阻，C为电容。乘积RC称为时间常数τ。当t = τ时，电压降至初始值的约37%。对等式两边取自然对数可得线性关系：ln V = ln V0 – t/(RC)。绘制ln V对t的图像将得到一条斜率为–1/RC、截距为ln V0的直线。
3. Apparatus Required | 所需器材
V), a voltage sensor (±0.01 V), a data logger with a computer, connecting leads, a double‑throw switch or a push‑to‑make switch for rapid disconnection, and a stopwatch if manual timing is used.
一个电容器（例如2200 μF电解电容）、一个固定电阻（例如100 kΩ）、电池或直流电源（例如6 V）、电压传感器（精度±0.01 V）、带电脑的数据记录仪、导线、一个双掷开关或快速断开按钮开关，若需手动计时则准备秒表。
4. Experimental Set‑Up | 实验装置
Connect the capacitor and resistor in series with the switch and the power supply so that the capacitor can first be charged by connecting it to the supply and then be allowed to discharge through the resistor by flipping the switch. The voltage sensor is connected across the capacitor and linked to the data logger. The data logger is set to record the voltage every 1 second for approximately 200 seconds, triggered just as the switch moves to the discharge position.
将电容器、电阻、开关和电源串联，确保开关可先将电容器连接到电源进行充电，随后通过切换开关使电容器经电阻放电。电压传感器并联在电容器两端，并与数据记录仪相连。设置数据记录仪在开关切换至放电位置的瞬间开始，每1秒记录一次电压，持续约200秒。
5. Procedure | 实验步骤
First, with the switch in the ‘charge’ position, the capacitor is allowed to charge fully to the supply voltage for at least 30 seconds. Then the switch is flipped to the ‘discharge’ position, simultaneously starting the data logger. The recorded voltage‑time data are saved for analysis. The experiment is repeated three times to allow evaluation of random uncertainties. For safety, the capacitor is discharged completely before handling.
首先将开关置于“充电”位置，让电容器充分充电至电源电压至少30秒。然后将开关切换到“放电”位置，同时启动数据记录仪。保存记录到的电压–时间数据以供分析。实验重复三次，以便评估随机误差。为安全起见，操作前需将电容器完全放电。
6. Data Recording | 数据记录
A table is constructed with columns for time t/s, voltage V/V, and ln(V/V). The voltage readings are taken from the data logger output at regular intervals, e.g. every 10 s up to 180 s. An example of processed data for a 2200 μF capacitor and 100 kΩ resistor is shown below.
设计数据表，包含时间 t/s、电压 V/V 以及 ln(V/V) 三列。从数据记录仪输出中按固定时间间隔（如每10秒直至180秒）读取电压值。下表展示了一组用2200 μF电容器与100 kΩ电阻获得的处理数据示例。
| Time t / s | Voltage V / V | ln(V / V) |
|---|---|---|
| 0 | 5.95 | 1.783 |
| 20 | 4.73 | 1.554 |
| 40 | 3.72 | 1.314 |
| 60 | 2.93 | 1.075 |
| 80 | 2.31 | 0.837 |
| 100 | 1.81 | 0.595 |
| 120 | 1.43 | 0.357 |
| 140 | 1.13 | 0.122 |
| 160 | 0.89 | –0.117 |
| 180 | 0.70 | –0.357 |
Data are recorded to an appropriate number of significant figures, and the natural logarithms are calculated to three decimal places to preserve precision.
数据记录时注意保留合适的有效数字，自然对数值计算至小数点后三位以保持精度。
7. Graphical Analysis | 图像分析
Plot a graph of ln(V/V) on the y‑axis against time t on the x‑axis. The data points should show a clear linear trend. Draw a best‑fit straight line and determine its gradient. From the theory, gradient = –1/RC. The intercept on the vertical axis gives an estimate of ln V0, which can be compared with the measured initial voltage.
绘制纵轴为 ln(V/V)、横轴为时间 t 的图像。数据点应呈现清晰的线性趋势。画出最佳拟合直线并计算其斜率。根据理论，斜率 = –1/RC。纵轴截距可估算 ln V0，并与实测初始电压进行比较。
gradient = –1/RC
For the sample data, the gradient is approximately –0.0126 s⁻¹, leading to an RC value of about 79.4 s.
对于上述示例数据，斜率约为 –0.0126 s⁻¹，对应的 RC 值约为79.4 s。
8. Determining the Time Constant | 确定时间常数
me constant τ = RC can be found directly from the negative reciprocal of the gradient: τ = – ly, τ can be read from the discharge curve as the time taken for the voltage to drop to 37% of V0: if V0 = 5.95 V, 37% methods should give consistent values, and the gradient method is generally more accurate.
时间常数 τ = RC 可直接由斜率的负倒数求得：τ = –1/斜率。也可以从放电曲线上直接读取电压降至初始值37%所需的时间：若 V0 = 5.95 V，则37%为2.20 V，对应时间约为79 s。两种方法结果应一致，而斜率法通常更为准确。
9. Calculating the Capacitance | 计算电容值
⁻⁴ F = 794 μF. This can be compared with the nominal value of the capacitor (e.g. 2200 μF) — fect of leakage resistance within the capacitor which reduces the effective time constant.
已知电阻 R = 100 × 10³ Ω，由实验时间常数 τ 可得 C = τ / R。代入 τ = 79.4 s，得 C = 79.4 / 1.00 × 10⁵ = 7.94 × 10⁻⁴ F = 794 μF。这可以与电容器的标称值（如2200 μF）进行比较——二者若相差悬殊，可能表明元件规格有误，或更可能是电容器内部的泄漏电阻影响了有效时间常数。
10. Uncertainty Analysis | 不确定度分析
V), the timing interval of the data logger (±0.1 s), and the scatter of points on the ln V– is double that of the gradient. Additional contributions from the resistance tolerance (±5%) should be combined when quoting the final capacitance value.
不确定度来源包括电压传感器的分辨率（±0.01 V）、数据记录仪的时间间隔（±0.1 s）以及 ln V–t 图上数据点的离散性。通过作最大与最小拟合直线可求得斜率的不确定度。时间常数 τ 的百分不确定度约为斜率不确定度的两倍。在给出最终电容值时，还需结合电阻的允差（±5%）。
11. Common Errors and Improvements | 常见问题与改进
Leakage current through the dielectric of the electrolytic capacitor can cause the voltage to decay faster than predicted by the ideal RC model, reducing the measured time constant. Using a high‑quality capacitor or connecting a large value of R (e.g. 1 MΩ) minimises this effect. Stray capacitance in the circuit and the input impedance of the voltage sensor can also affect results; a buffer amplifier may improve accuracy. Repeating the measurement with reverse polarity can check for asymmetry.
电解电容介质中的泄漏电流会使电压衰减比理想RC模型更快，导致测量时间常数偏低。使用高质量电容器或接入大阻值电阻（如1 MΩ）可减轻此影响。电路中的杂散电容以及电压传感器的输入阻抗也会影响结果；使用缓冲放大器能提高精度。通过反向极性重复测量可检验是否存在不对称效应。
12. Conclusion | 结论
The capacitor discharge investigation confirms the exponential decay relationship V = V0 e–t/RC and demonstrates how linearisation by logarithms allows the time constant to be extracted from the gradient. Although the derived capacitance may deviate from the nominal value due to leakage effects, the method provides excellent insight into data‑logging techniques, uncertainty handling, and the practical limitations of real components — skills directly assessed in the Unit 4 examination.
电容器放电实验验证了电压按 V = V0 e–t/RC 指数衰减的规律，并展示了如何通过对数线性化从斜率中提取时间常数。尽管电容推算值可能因泄漏效应与标称值存在偏差，但本方法能让学生深入理解数据记录技术、不确定度处理以及实际器件的局限性——这些正是单元4考试直接考查的核心技能。
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