📚 A-Level CIE Physics: Capacitance Key Points | A-Level CIE 物理：电容 考点精讲
Capacitance is a core topic in CIE A-Level Physics, linking electric fields, energy storage and exponential decay in RC circuits. This article walks you through every essential concept, equation and typical exam pitfall you need to master.
电容是 CIE A-Level 物理的核心主题之一，它将电场、能量存储和 RC 电路中的指数衰减联系起来。本文将带你梳理每一个关键概念、重要公式和常见考题陷阱，帮助你彻底掌握。
1. Capacitance Definition | 电容定义
Capacitance (C) is the ability of a pair of conductors to store charge per unit potential difference. It is defined by the ratio C = Q / V, where Q is the magnitude of charge on either plate and V is the p.d. between them. The SI unit is the farad (F), which is equivalent to C V−1.
电容 (C) 是一对导体在单位电势差下储存电荷的能力，定义为 C = Q / V，其中 Q 为任一极板上的电荷量，V 为两极板间的电势差。国际单位是法拉 (F)，等价于 C V−1。
2. Parallel Plate Capacitor | 平行板电容器
For an ideal parallel plate capacitor with plate area A and separation d in a vacuum, the capacitance is given by C = ε0 A / d, where ε0 = 8.85 × 10−12 F m−1 is the permittivity of free space. This formula shows that C increases with larger plates and smaller separation.
对于极板面积 A、间距 d 的真空平行板电容器，电容公式为 C = ε0 A / d，其中 ε0 = 8.85 × 10−12 F m−1 是真空介电常数。该公式表明，极板面积越大、间距越小，电容越大。
3. Dielectrics | 电介质
Inserting a dielectric material between the plates increases the capacitance by a factor εr (relative permittivity). The capacitance becomes C = εr ε0 A / d. Dielectrics work by polarising in the electric field, which reduces the effective field and the p.d. for the same charge, thus raising C. Typical values of εr range from 2 to 80.
在极板间插入电介质会使电容乘以相对介电常数 εr，公式变为 C = εr ε0 A / d。电介质在电场中极化，削弱了有效电场和对应电势差，从而在相同电荷下提升了电容。εr 的典型值在 2 到 80 之间。
4. Energy Stored in a Capacitor | 电容器储存的能量
A charged capacitor stores electrical potential energy. The energy E can be expressed in three equivalent forms: E = ½ Q V, E = ½ C V2 and E = ½ Q2 / C. You must be able to derive these from the work done in charging (area under the Q-V graph).
带电的电容器储存电势能。能量 E 有三种等价表达式：E = ½ Q V、E = ½ C V2 和 E = ½ Q2 / C。你必须能从充电过程做功（Q-V 图像下的面积）出发推导这些公式。
A common exam question asks you to calculate the energy lost when two capacitors are connected. The lost energy is dissipated as heat or radiation, highlighting that some energy is always wasted unless charge is moved infinitely slowly.
常见考题会要求计算两个电容器连接后的能量损失。这部分损失的能量以热或辐射形式耗散，说明除非电荷无限缓慢移动，否则总会有能量被浪费。
5. Charging a Capacitor | 电容器的充电过程
When a capacitor is connected in series with a resistor R and a d.c. supply of e.m.f. E, the charge q on the plates grows exponentially. The charging equation is q = Q0 (1 − e−t / (RC)), where Q0 = C E is the final charge. The p.d. across the capacitor follows V = E (1 − e−t / (RC)). Initially the current is maximum (I0 = E / R) and decays exponentially.
当电容器与电阻 R 和电动势为 E 的直流电源串联时，极板上的电荷 q 按指数规律增长。充电方程为 q = Q0 (1 − e−t / (RC))，其中 Q0 = C E 是最终电荷。电容器两端的电压遵循 V = E (1 − e−t / (RC))。初始电流最大 (I0 = E / R)，然后呈指数衰减。
You should be able to sketch the charge–time, voltage–time and current–time graphs for charging. All show a rapid change at first, gradually approaching a steady value.
你应该能够画出充电时的电荷–时间、电压–时间和电流–时间图像。它们都呈现出开始变化快，随后逐渐趋于稳定值的特征。
6. Discharging a Capacitor | 电容器的放电过程
When a charged capacitor is discharged through a resistor, the charge, voltage and current all decay exponentially. The discharge equation is q = Q0 e−t / (RC), where Q0 is the initial charge. Similarly, V = V0 e−t / (RC) and I = I0 e−t / (RC).
已充电的电容器通过电阻放电时，电荷、电压和电流均按指数规律减少。放电方程为 q = Q0 e−t / (RC)，其中 Q0 为初始电荷。类似地，V = V0 e−t / (RC)，I = I0 e−t / (RC)。
A quick way to check if a graph represents exponential decay is to note that the time for the quantity to halve is constant. This half-life is t½ = RC ln 2 ≈ 0.693 RC.
快速判断图像是否为指数衰减的方法是：数值减半所需的时间恒定。这个半衰期为 t½ = RC ln 2 ≈ 0.693 RC。
7. Time Constant and RC Circuits | 时间常数与 RC 电路
The product RC, called the time constant τ, has units of seconds (Ω × F = s). τ is the time taken for the charge (or voltage) to rise to 63% of its final value during charging, or to fall to 37% of its initial value during discharging. A large τ means a slow charge/discharge.
乘积 RC 称为时间常数 τ，单位为秒 (Ω × F = s)。τ 是充电过程中电荷（或电压）上升到最终值 63%，或放电过程中下降到初始值 37% 所需的时间。τ 越大，充放电越慢。
Experimentally, τ can be found from the gradient of a ln(V) versus time graph or by reading the time at 0.37 V0 from a discharge curve. CIE expects you to handle data-logging methods and interpret such graphs.
实验中，可利用 ln(V)–时间 图像的斜率求得 τ，或从放电曲线上读取 0.37 V0 对应的时间。CIE 要求你掌握数据记录方法并能够解释此类图像。
8. Capacitors in Series and Parallel | 电容器的串联与并联
For capacitors in parallel, the total capacitance is the sum: Ctotal = C1 + C2 + C3 + … . The p.d. across each capacitor is the same, and the charges add up.
电容器并联时，总电容等于各电容之和：Ctotal = C1 + C2 + C3 + … 。每个电容器两端的电势差相等，电荷量相加。
For capacitors in series, the reciprocal total capacitance is the sum of reciprocals: 1 / Ctotal = 1 / C1 + 1 / C2 + … . The charge on each capacitor is the same, and the p.d.s add up to the supply voltage. Always check which arrangement stores more energy for a given voltage – parallel usually stores more.
电容器串联时，总电容的倒数等于各电容倒数之和：1 / Ctotal = 1 / C1 + 1 / C2 + … 。每个电容器上的电荷量相同，电势差之和等于电源电压。记得检查在给定电压下哪种连接储存的能量更多——并联通常储能更大。
9. Practical Capacitors and Applications | 实际电容器及应用
Real capacitors are not ideal; they have leakage current and a maximum working voltage. The capacitance value is usually marked on the body, often in microfarads (μF), nanofarads (nF) or picofarads (pF). Electrolytic capacitors are polarised and must be connected the correct way round.
实际电容器并非理想元件，存在漏电流和最大工作电压。电容值通常标注在元件外壳上，单位多为微法 (μF)、纳法 (nF) 或皮法 (pF)。电解电容器有极性，连接时必须正负极正确。
Capacitors are used for smoothing rectified a.c., in timing circuits, for touch screens, in flash photography and as energy storage in defibrillators. In each case, their ability to store and release charge controllably makes them indispensable.
电容器常用于整流电源的滤波、定时电路、触摸屏、闪光灯和心脏除颤器的能量存储。它们能够可控地储存和释放电荷，使得这些应用成为可能。
Published by TutorHao | Physics Revision Series | aleveler.com
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