Capacitor Networks | 电容器网络

📚 Capacitor Networks | 电容器网络

A capacitor network is a combination of two or more capacitors connected in series, in parallel, or in a mixed arrangement. To analyse such a network, you need to find the equivalent capacitance and then work backwards to determine the charge, voltage and energy for each capacitor.

电容网络是由两个或多个电容器以串联、并联或混联方式连接而成的组合。分析这类网络需要先求出等效电容，再逆向求解每个电容器的电荷、电压和能量。

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1. Capacitance and Charge Storage | 电容与电荷储存

A capacitor stores separated charge and therefore stores electric potential energy. Its capacitance C is defined by C = Q / V, where Q is the magnitude of the charge stored on either plate and V is the potential difference across the plates.

电容器储存分离电荷，因此储存电势能。其电容 C 定义为 C = Q / V，其中 Q 是任一极板上储存的电荷量，V 是两极板之间的电势差。

C = Q / V

The SI unit of capacitance is the farad (F). One farad is one coulomb per volt. In school experiments, common values are usually microfarads (µF), nanofarads (nF) or picofarads (pF).

电容的国际单位是法拉（F）。1 法拉等于每伏特 1 库仑。在实验和试题中，常见电容通常是微法（µF）、纳法（nF）或皮法（pF）。

For a fixed capacitor, charge Q is directly proportional to voltage V as long as the working voltage is not exceeded. The constant of proportionality is the capacitance C.

对固定电容器，只要不超过额定工作电压，电荷 Q 与电压 V 成正比。比例常数就是电容 C。

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2. Capacitors in Parallel | 并联电容器

In a parallel network, the positive plates are joined together and the negative plates are joined together. Each capacitor has exactly the same potential difference V across it because all positive plates are at the same potential and all negative plates are at the same potential.

在并联网络中，各电容器的正极板相连，负极板也相连。每个电容器两端的电势差 V 完全相同，因为所有正极板处于同一电势，所有负极板也处于同一电势。

The total charge supplied by the battery is the sum of the charges on the individual capacitors: Q_total = Q₁ + Q₂ + Q₃ + … . Using Q = CV gives C_eq V = C₁V + C₂V + C₃V + … .

电源提供的总电荷等于各电容器上电荷之和：Q_total = Q₁ + Q₂ + Q₃ + …。由 Q = CV 可得 C_eq V = C₁V + C₂V + C₃V + …。

C_eq = C₁ + C₂ + C₃

Capacitors in parallel therefore add directly. The equivalent capacitance is larger than any individual capacitance, and a larger capacitance stores more charge at the same voltage.

因此，并联电容器的等效电容直接相加。等效电容比任意单个电容都大，在相同电压下，电容越大，储存的电荷越多。

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3. Capacitors in Series | 串联电容器

In a series network, capacitors are connected end to end in a single loop. The current that charges one capacitor must charge the next capacitor, so the same magnitude of charge Q is stored on every capacitor in series.

在串联网络中，电容器首尾相连形成单一回路。给一个电容器充电的电流必然也给下一个电容器充电，因此串联中每个电容器储存的电荷量 Q 大小相同。

The total supply voltage is divided across the capacitors: V_total = V₁ + V₂ + V₃ + … . Substituting V = Q / C gives Q / C_eq = Q / C₁ + Q / C₂ + Q / C₃ + … .

总电源电压被分配在各电容器上：V_total = V₁ + V₂ + V₃ + …。代入 V = Q / C 可得 Q / C_eq = Q / C₁ + Q / C₂ + Q / C₃ + …。

1 / C_eq = 1 / C₁ + 1 / C₂ + 1 / C₃

The reciprocal rule shows that the equivalent capacitance of series capacitors is always smaller than the smallest individual capacitance.

倒数规则表明，串联电容器的等效电容总是小于其中最小的单个电容。

For two capacitors in series, the product-over-sum shortcut applies: C_eq = (C₁C₂) / (C₁ + C₂). For three or more capacitors, use the reciprocal rule systematically.

对于两个电容器串联，可用“积除以和”的简算公式：C_eq = (C₁C₂) / (C₁ + C₂)。三个或更多电容器串联时，应系统地使用倒数规则。

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4. Deriving Series and Parallel Rules | 串并联规则的推导

The series and parallel rules come from two fundamental conservation principles: conservation of charge and conservation of energy, expressed through Kirchhoff’s laws. For parallel branches, the voltage across each branch is the same because every branch connects the same two nodes.

串并联规则来自两个基本守恒原理：电荷守恒和能量守恒，并以基尔霍夫定律表示。对并联支路，各支路电压相同，因为每条支路都连接相同的两个节点。

Charge conservation at a junction means the total charge from the source splits among the parallel capacitors. Hence Q_total = Q₁ + Q₂ and therefore C_eq = C₁ + C₂.

节点处的电荷守恒意味着来自电源的总电荷在并联电容器之间分配。因此 Q_total = Q₁ + Q₂，从而 C_eq = C₁ + C₂。

For series capacitors, the same charging current passes through each component, so each capacitor gains the same magnitude of charge. The total work done per unit charge, that is the voltage, must be the sum of the individual voltages.

对串联电容器，相同的充电电流流过每个元件，因此每个电容器获得相同大小的电荷。每单位电荷做的总功，即总电压，必然等于各电压之和。

These derivations are frequently tested. Avoid memorising blindly: first decide whether the components share the same two nodes or sit in the same single loop.

这些推导经常出现在考试中。不要死记

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

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