📚 3 Capacitance | 电容
Capacitance is a cornerstone concept in both IB and Cambridge A Level Physics, describing a system's ability to store electric charge. From the simple parallel-plate capacitor to the transient behaviour of RC circuits, mastering capacitance opens the door to understanding energy storage, signal processing and timing applications in modern electronics.
电容是IB和剑桥A Level物理中的一个基石概念，描述系统储存电荷的能力。从简单的平行板电容器到RC电路的瞬态行为，掌握电容知识是理解现代电子学中能量存储、信号处理和定时应用的关键。
1. What is Capacitance? | 什么是电容？
nit of capacitance is the farad (F); one farad equals one coulomb per volt. In practical circuits, capacitors are usually rated in microfarads (μF), nanofarads (nF) or picofarads (pF).
电容（C）定义为储存在导体上的电荷量Q与导体两端电势差V之比。公式为 C = Q / V。电容的国际单位是法拉（F）；1法拉等于1库仑每伏特。在实际电路中，电容器的标称值常用微法（μF）、纳法（nF）或皮法（pF）表示。
C = Q / V
2. Capacitors – Storing Charge | 电容器——储存电荷
harged. The charge stored is directly proportional to the applied voltage, obeying Q = CV.
电容器是专门用来在电场中储存电荷和能量的元件。它通常由两块被绝缘体（电介质）隔开的导体板组成。当施加电压时，电子积聚在一块极板上，另一块极板带正电。储存的电荷量与外加电压成正比，满足 Q = CV。
3. The Parallel-Plate Capacitor | 平行板电容器
εᵣ is the relative permittivity (dielectric constant) of the material between the plates.
对于理想的平行板电容器，电容只取决于极板的几何尺寸和电介质材料。电容由公式 C = ε₀ εᵣ A / d 给出，其中A是极板的有效面积，d是极板间距，ε₀是真空介电常数（约8.85×10⁻¹² F·m⁻¹），εᵣ是两极板间材料的相对介电常数（介电常数）。
C = ε₀ εᵣ A / d
4. Role of the Dielectric | 电介质的作用
Inserting a dielectric between the plates increases the capacitance by a factor of εᵣ. The dielectric weakens the electric field for a given charge, enabling more charge to be stored at the same voltage. This occurs because the dielectric becomes polarised: its molecules align opposite to the external field, producing a field that partially cancels the applied field. The dielectric also prevents electrical breakdown up to a certain maximum field strength.
在两板之间插入电介质会使电容增大εᵣ倍。电介质减弱了给定电荷下的电场，因而可以在相同电压下储存更多电荷。这是因为电介质被极化：其分子沿与外电场相反的方向排列，产生一个部分抵消外电场的电场。电介质还能在一定最大场强下防止电击穿。
5. Energy Stored in a Capacitor | 电容器储存的能量
sides in the electric field between the plates and can be released very quickly, which makes capacitors essential in camera flashes and pulsed power supplies.
带电电容器中储存的能量U由 U = ½ Q V 给出。利用定义 Q = CV，可改写为 U = ½ C V² 或 U = Q² / (2C)。这些能量储存在极板间的电场中，能够极快地释放，因此电容器在照相机闪光灯和脉冲电源中至关重要。
U = ½ C V² U = ½ Q V U = Q² / (2C)
6. Capacitors in Series and Parallel | 电容器的串联与并联
When capacitors are connected in series, the total capacitance Ctotal is smaller than any individual capacitance, given by 1 / Ctotal capacitor is the same. For capacitors in parallel, the total capacitance is simply the sum: Ctotal = C₁ + C₂ + ... . In this case, the voltage V across each branch is identical.
电容器串联时，总电容Ctotal小于任一单个电容，满足 1 / Ctotal = 1/C₁ + 1/C₂ + ... 。每个电容器上的电荷量Q相同。并联时，总电容为各电容之和：Ctotal = C₁ + C₂ + ... 。此时各支路两端的电压V相同。
| Connection | Capacitance Formula | Charge/Voltage |
|---|---|---|
| Series | 1 / Ctotal = 1/C₁ + 1/C₂ + ... | Same Q, V splits |
| Parallel | Ctotal = C₁ + C₂ + ... | Same V, charges add |
7. Charging a Capacitor through a Resistor | 通过电阻给电容器充电
ulates, the capacitor voltage rises, reducing the current. The key equations are q = Q₀ (1 – e–t/RC ), V = V₀ (1 – e–t/RC ), and I = I₀ e–t/RC.
当电容器通过电阻从直流电源（电动势V₀）充电时，电荷和电压呈指数规律增长。初始时刻电流最大（I₀ = V₀ / R），因为电容器相当于短路。随着电荷积累，电容器电压升高，电流减小。关键方程为 q = Q₀ (1 – e–t/RC )、V = V₀ (1 – e–t/RC ) 以及 I = I₀ e–t/RC。
8. Discharging a Capacitor | 电容器放电
s exponentially. The charge, voltage and current all follow an exponential decay: q = Q₀ e–t/RC, V = V₀ e–t/RC. The current reverses direction during discharge, so its magnitude follows |I| = I₀ e–t/RC. Eventually, all stored energy is dissipated as heat in the resistor.
当带电电容器通过电阻放电时，电荷、电压和电流均呈指数衰减：q = Q₀ e–t/RC，V = V₀ e–t/RC。放电过程中电流方向反转，其大小满足 |I| = I₀ e–t/RC。最终所有储存的能量都以热的形式消耗在电阻上。
9. The Time Constant τ | 时间常数 τ
he circuit. It has the unit of seconds. When t = τ during charging, the voltage reaches (1 – 1/e) ≈ 63% of its final value. During discharging, the voltage falls to 1/e ≈ 37% terises how quickly the circuit responds; a larger τ means slower charging or discharging.
乘积RC用希腊字母τ表示，称为电路的时间常数，单位是秒。充电过程中，当 t = τ 时，电压达到终值的 (1 – 1/e) ≈ 63%。放电过程中，在 t = τ 时，电压降至初始值的 1/e ≈ 37%。时间常数表征电路响应的快慢；τ值越大，充电或放电越慢。
τ = RC
10. Exponential Equations in RC Circuits | RC电路中的指数方程
g equations are solutions to the first-order differential equation obtained from Kirchhoff's voltage law. For a charging capacitor, the net voltage is V₀ – q/C – iR = 0, leading to dq/dt = (V₀ – takes about 5τ for a capacitor to be considered fully charged or fully discharged (over 99% of the final charge).
充电和放电方程是通过基尔霍夫电压定律列出一阶微分方程求解得到的。对于充电电容，净电压关系为 V₀ – q/C – iR = 0，由此推得 dq/dt = (V₀ – q/C) / R。求解即得前述的指数形式。无论在哪种过程中，每隔一个时间常数τ，相关量就变化e倍。一般认为，经过约5τ的时间，电容器即可视为已完全充电或完全放电（电荷变化超过99%）。
11. Graphical Analysis of Charge, Current and Voltage | 电荷、电流和电压的图形分析
Plotting q, V and I against time reveals characteristic exponential curves. For charging: q–t and V–t rise from zero and asymptotically approach Q₀ or V₀; I–t starts at a maximum and decays to zero. For discharging: q–t and V–t fall from initial values towards zero, while the magnitude of I decreases exponentially. The gradient of a q–t graph at any instant represents the current. The time constant can be determined from the graph by finding the time at which the quantity has changed to 37% of its initial amplitude (discharge) or 63% of its final value (charge).
将q、V和I对时间作图，可以得到特征性的指数曲线。充电时：q–t和V–t曲线从零开始，逐渐趋近于Q₀或V₀；I–t曲线起始值最大并衰减至零。放电时：q–t和V–t从初始值向零衰减，电流的大小按指数规律减小。q–t图在任意时刻的斜率代表该时刻的电流。由图上可确定时间常数：放电曲线中量值降至初始幅度的37%，或充电曲线中量值升至终值的63%所对应的时间。
12. Practical Applications of Capacitors | 电容器的实际应用
Capacitors are found in virtually all electronic circuits. Large-value electrolytic capacitors smooth rectified a.c. in power supplies by storing charge and releasing it during voltage dips. In timing circuits, the predictable charging curve of an RC combination generates precise delays (e.g. in flashing lights or oscillators). Capacitors are also used for coupling and decoupling signals, touch-sensitive screens, energy storage for pulsed lasers, and in defibrillators to deliver a controlled electric shock.
电容器几乎出现在所有电子电路中。大容量的电解电容通过储存电荷并在电压下降时释放，从而在电源中平滑整流后的交流电。在定时电路中，RC组合可预测的充电曲线能够产生精确的延时（如闪烁灯或振荡器）。电容器还用于信号耦合与去耦、触摸感应屏幕、脉冲激光器储能以及除颤器中提供受控电击等方面。
Published by TutorHao | Physics Revision Series | aleveler.com
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