📚 Capacitors | 电容
A capacitor is a passive electrical component that stores energy in an electric field. It consists of two conducting plates separated by an insulating material called a dielectric. Capacitors are used in countless electronic circuits, from smoothing power supplies to timing applications. Understanding how charge, voltage, capacitance, and energy relate is essential for GCSE CCEA Physics.
电容器是一种能够储存电场能量的无源电子元件。它由两个导体板以及中间隔开的绝缘介质（电介质）构成。电容器在无数电路中被使用，从电源平滑滤波到定时电路应用。理解电荷、电压、电容和能量之间的关系是 GCSE CCEA 物理的重要考点。
1. What is a Capacitor? | 什么是电容器？
A capacitor is a device designed to store electric charge temporarily. The two metal plates accumulate equal but opposite charges when connected to a voltage source. The dielectric prevents the charges from flowing directly between the plates, creating a potential difference that persists even after the source is removed.
电容器是一种用于暂时储存电荷的器件。当连接到电压源时，两个金属板会积累等量异号电荷。电介质阻止电荷直接在极板间流动，从而建立电势差，即便断开电源后该电势差仍能保持一段时间。
2. Capacitance Definition | 电容的定义
Capacitance (C) is the amount of charge a capacitor can store per unit of potential difference across its plates. It is measured in farads (F). The formula is: C = Q / V, where Q is the charge stored in coulombs (C) and V is the potential difference in volts (V). One farad represents one coulomb of charge stored per volt.
电容（C）表示电容器每单位电势差所能储存的电荷量，单位是法拉（F）。公式为 C = Q / V，其中 Q 是储存的电荷量（单位库仑，C），V 是极板间的电势差（单位伏特，V）。1 法拉表示每伏特电压可储存 1 库仑电荷。
C = Q / V
(C: capacitance in farads; Q: charge in coulombs; V: voltage in volts)
3. Energy Stored in a Capacitor | 电容器储存的能量
When a capacitor is charged, work is done to separate the charges on the plates, and this work is stored as electrical potential energy. The energy stored (E) can be calculated using the formula: E = ½ Q V. Since Q = C V, this can also be written as E = ½ C V² or E = Q² / (2C). The energy is measured in joules (J).
当电容器充电时，外力做功将电荷分离在极板上，所做的功以电势能形式储存。储存的能量（E）可用公式 E = ½ Q V 计算。由于 Q = C V，也可写作 E = ½ C V² 或 E = Q² / (2C)。能量的单位是焦耳（J）。
E = ½ Q V = ½ C V² = Q² / (2C)
(E: energy in joules; Q: charge; V: voltage; C: capacitance)
4. Charging a Capacitor | 电容器的充电过程
When a capacitor is connected to a battery through a resistor, charge begins to flow onto the plates. Initially, the current is large because the potential difference across the capacitor is zero. As charge builds up, the capacitor voltage increases, opposing the battery voltage, so the current decreases exponentially. The voltage across the capacitor rises gradually and asymptotically approaches the battery voltage.
当电容器通过一个电阻连接到电池时，电荷开始流向极板。初始时刻电流很大，因为电容器两端的电压为零。随着电荷积累，电容器的电压上升，与电池电压相对抗，因此电流呈指数衰减。电容器两端的电压逐渐上升，渐近地趋近电池电压。
VC(t) = V₀ (1 – e⁻ᵗ/ᴿᶜ)
(V₀: supply voltage; R: resistance; C: capacitance; t: time)
5. Discharging a Capacitor | 电容器的放电过程
When the battery is removed and the capacitor is connected across a resistor, the stored charge flows through the resistor, creating a current. The voltage across the capacitor decreases exponentially from its initial value V₀ to zero. The discharge current also falls exponentially, and the capacitor is considered fully discharged after a time of about 5 RC.
当移走电池并将电容器连接到一个电阻两端时，储存的电荷通过电阻流动，形成电流。电容器两端的电压从其初始值 V₀ 开始呈指数下降，直至零。放电电流同样指数衰减，大约经过 5 倍 RC 时间常数之后，电容器可视为完全放电。
VC(t) = V₀ e⁻ᵗ/ᴿᶜ
(V₀: initial voltage; R: resistance; C: capacitance; t: time)
6. Time Constant (RC) | 时间常数 (RC)
The product of resistance and capacitance, RC, is called the time constant (τ) and has units of seconds. It indicates how quickly a capacitor charges or discharges. After one time constant, the voltage during charging reaches about 63% of the supply voltage; during discharging, it falls to about 37% of its initial value. A larger resistance or capacitance increases the time constant, slowing the charge/discharge process.
电阻与电容的乘积 RC 被称为时间常数（τ），单位是秒。它表示电容器充放电的快慢。经过一个时间常数后，充电时电压约达到电源电压的 63%；放电时电压约降至初始值的 37%。较大的电阻或电容会增大时间常数，从而减慢充放电过程。
τ = R × C
(τ in seconds; R in ohms; C in farads)
7. Factors Affecting Charging and Discharging | 影响充放电的因素
The rate at which a capacitor charges or discharges depends on the resistance in the circuit and the capacitance itself. A higher capacitance stores more charge, so for the same resistance, it takes longer to fill or empty. A higher resistance limits the current, so the process becomes slower. In a practical circuit, altering either R or C changes the time constant directly, which is useful in timing circuits.
电容器充放电的速率取决于电路中的电阻和电容本身。电容越大，储存的电荷越多，在相同电阻下，充放电所需的时间更长。电阻越大，限制电流的作用越强，过程也变得越慢。在实际电路中，改变 R 或 C 会直接改变时间常数，这一定时特性十分有用。
Temperature can also slightly affect the capacitance value in some types of capacitors, but for GCSE, the primary focus is on R and C.
温度也会对某些类型电容器的电容值产生轻微影响，但在 GCSE 阶段主要关注 R 和 C。
8. Capacitors in Series and Parallel | 电容器的串联与并联
When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitances: Ctotal = C₁ + C₂ + C₃ + … . The voltage across each capacitor is the same. In series, the total capacitance is less than the smallest individual capacitance and is calculated using the reciprocal formula: 1/Ctotal = 1/C₁ + 1/C₂ + 1/C₃ + … . This is opposite to the rules for resistors.
电容器并联时，总电容等于各个电容之和：Ctotal = C₁ + C₂ + C₃ + …，每个电容器两端的电压相同。串联时，总电容小于最小的单个电容，使用倒数公式计算：1/Ctotal = 1/C₁ + 1/C₂ + 1/C₃ + …。这与电阻的串并联规则恰好相反。
Parallel: Ctotal = C₁ + C₂ + …
Series: 1/Ctotal = 1/C₁ + 1/C₂ + …
9. Practical Investigations | 实验探究
GCSE experiments often involve charging a capacitor through a fixed resistor and recording the voltage across it at regular time intervals using a voltmeter or data logger. From the collected data, a voltage-time graph can be plotted, showing the exponential rise. The time constant can be estimated from the graph by finding the time taken for the voltage to reach 63% of the final value during charging, or 37% during discharging. Comparing different R and C values demonstrates the effect on the curve shape.
GCSE 实验通常包括通过固定电阻给电容器充电，并使用电压表或数据记录仪每隔固定时间记录一次电压。利用收集的数据可以绘制电压 – 时间图表，显示指数增长曲线。时间常数可根据图表估算，即找出充电过程中电压达到终值 63% 或放电过程中降至 37% 所需的时间。对比不同 R 和 C 值可以观察其对曲线形状的影响。
10. Uses of Capacitors | 电容器的应用
Capacitors have a wide range of applications: they smooth out fluctuations in DC power supplies by storing charge when the voltage rises and releasing it when it drops. In camera flash units, a capacitor quickly discharges through a bulb to produce a bright flash. They are also used in timing circuits, coupling and decoupling in audio systems, and as sensors in touch screens. The ability to store and release energy rapidly makes them indispensable in electronics.
电容器应用广泛：在直流电源中平滑波动，当电压升高时储存电荷，下降时释放电荷。在相机闪光灯中，电容器通过灯泡快速放电产生明亮闪光。它们还用于定时电路、音频系统中的耦合与去耦，以及触摸屏中的传感器。快速储存和释放能量的能力使电容器成为电子设备中不可或缺的元件。
11. Key Formulas Summary | 关键公式总结
It is essential to know the relationships between charge, voltage, capacitance, and energy. Below is a summary table of the main equations and their meanings.
熟悉电荷、电压、电容与能量之间的关系至关重要。以下是主要公式及其含义的总结表格：
| Relationship | Formula |
|---|---|
| Charge–Voltage–Capacitance | C = Q / V |
| Energy Stored | E = ½ Q V = ½ C V² = Q² / (2C) |
| Time Constant | τ = R C |
| Charging Voltage | V = V₀ (1 – e⁻ᵗ/ᴿᶜ) |
| Discharging Voltage | V = V₀ e⁻ᵗ/ᴿᶜ |
Memorising these and being able to rearrange them is crucial for exam success.
熟记并能够灵活变换这些公式是考试取得好成绩的关键。
12. Graphical Analysis of Charging and Discharging | 充放电曲线图解分析
The charging curve of voltage against time is an increasing exponential that starts steep and levels off at V₀. The discharge curve is a decreasing exponential starting at V₀ and tending to zero. The current during charging is a decreasing exponential from a maximum to zero, and during discharging the current flows in the opposite direction but also decays exponentially. Understanding how to read these graphs and extract the time constant by drawing tangents or using the 63%/37% rule is an important skill.
电压随时间变化的充电曲线是一条指数增长曲线，开始很陡，然后逐渐趋于平缓直至 V₀。放电曲线是一条从 V₀ 开始趋向零的指数衰减曲线。充电过程中电流是从最大值指数衰减至零，放电时电流方向相反但同样指数衰减。理解如何阅读这些图形，以及通过画切线或 63%/37% 规则提取时间常数，是一项重要技能。
The area under a current-time graph during charging represents the total charge stored, reinforcing the link Q = I × t for constant current, but for a varying current integration can be approximated by counting squares or using data loggers.
充电时电流 – 时间图下所包围的面积代表储存的总电荷，这加强了恒流情况下的 Q = I × t 关系，而对于变化电流则可通过数格或数据记录仪近似获得。
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