📚 Capacitors in Parallel | 电容的并联
Capacitors are components that store electric charge and energy. When capacitors are connected side by side across the same two terminals, they form a parallel combination. This arrangement appears frequently in CIE A-Level Physics, especially in circuit analysis, energy storage and problem solving. Parallel capacitors share the same potential difference, and their individual capacitances add directly to give a larger total capacitance.
电容器是储存电荷和电能的元件。当多个电容器并排连接在同一两个端点之间时，就构成并联组合。这种连接方式在 CIE A-Level 物理中经常出现，尤其在电路分析、能量储存和解题中。并联电容器两端电势差相同，各电容值直接相加，得到更大的等效电容。
1. Definition and Circuit Setup | 定义与电路连接
In a parallel connection, all the positive plates are joined to one common terminal, and all the negative plates are joined to the other common terminal. This means every capacitor is connected directly across the same two points of the circuit.
在并联连接中，所有正极板接到同一个公共端点，所有负极板接到另一个公共端点。这意味着每个电容器都直接跨接在电路的相同两点之间。
A simple circuit diagram would show two or more capacitor symbols drawn side by side between the same two wires. No capacitor is in the path of another, so the current divides among the branches.
简单电路图会在同一对导线之间并排画出两个或多个电容符号。没有电容位于另一电容的路径中，因此电流在支路之间分配。
2. Key Rule: Same Voltage | 关键规律：电压相同
Because all parallel branches are connected across the same two points, the potential difference across each capacitor is exactly the same as the supply voltage, provided the capacitors are ideal and steady conditions have been reached.
由于所有并联支路都连接在相同两个点之间，当电容器为理想元件且达到稳态时，每个电容器两端的电势差与电源电压完全相同。
We write this as:
我们写作：
V = V₁ = V₂ = V₃ = …
This identical voltage is the most important condition in parallel capacitor problems. It means that the smallest working voltage rating in the group may limit the safe supply voltage.
相同的电压是并联电容问题中最重要的条件。这意味着组合中最小的工作电压额定值可能限制安全供电电压。
3. Total Charge is the Sum | 总电荷为各电容电荷之和
The charge stored by each capacitor is given by Q = C V. Since the voltage is the same for all branches, the charges are:
每个电容器储存的电荷由 Q = C V 给出。由于各支路电压相同，电荷分别为：
Q₁ = C₁V, Q₂ = C₂V, Q₃ = C₃V
The total charge supplied by the battery is the sum of the charges on the individual capacitors:
电源提供的总电荷等于各电容器上电荷之和：
Q_total = Q₁ + Q₂ + Q₃ + …
This charge addition occurs because the battery must supply charge to every branch. More branches mean more charge can be stored at the same voltage.
电荷相加的原因是电源必须向每个支路提供电荷。支路越多，在相同电压下能储存的电荷就越多。
4. Deriving Total Capacitance | 推导等效电容
Let C_total be the equivalent capacitance of the parallel combination. By definition, the total charge is Q_total = C_total V.
设 C_total 为并联组合的等效电容。根据定义，总电荷为 Q_total = C_total V。
Substituting the individual charges gives:
代入各电容的电荷得到：
C_total V = C₁V + C₂V + C₃V
Dividing through by the common voltage V yields:
两边同除以共同电压 V，得到：
C_total = C₁ + C₂ + C₃
This is a direct addition. The equivalent capacitance is always greater than any individual capacitance in the parallel group.
这是直接相加。等效电容总是大于并联组中任何一个单独的电容。
5. Formula for n Capacitors | n 个电容的公式
For n capacitors connected in parallel, the general formula is:
对于 n 个并联电容器，一般公式为：
C_total = C₁ + C₂ + C₃ + … + Cₙ
or using summation notation:
或用求和符号表示：
C_total = Σ Cᵢ
If all n capacitors have the same capacitance C, the total capacitance simplifies to:
如果所有 n 个电容器的电容都相同为 C，则总电容简化为：
C_total = n C
This linear addition can make a very large equivalent capacitance using several smaller, cheaper capacitors in parallel.
这种线性相加可以用几个较小、较便宜的电容器并联得到很大的等效电容。
6. Energy Stored in Parallel Combination | 并联组合储存的能量
The energy stored in a capacitor is given by:
电容器储存的能量由下式给出：
E = ½ C V²
For each capacitor, Eᵢ = ½ Cᵢ Vᵢ².
对于每个电容器，Eᵢ = ½ Cᵢ Vᵢ²。
Since V is the same for all parallel capacitors, the total stored energy is:
由于所有并联电容器两端电压相同，总储存能量为：
E_total = ½ C₁V² + ½ C₂V² + ½ C₃V² = ½ (C₁ + C₂ + C₃) V² = ½ C_total V²
Thus the parallel combination behaves like a single capacitor of capacitance C_total for energy storage as well. Increasing the parallel capacitance increases the energy stored at a given voltage, but the maximum safe energy is limited by the voltage rating.
因此并联组合在能量储存方面也表现得像一个电容为 C_total 的单一电容器。在给定电压下，增大并联电容会增加储存的能量，但最大安全能量受额定电压限制。
7. Parallel vs Series | 并联与串联对比
In series, charge is the same on each capacitor and the reciprocal capacitances add:
串联时，每个电容器上的电荷相同，电容倒数相加：
1/C_total = 1/C₁ + 1/C₂ + …
In parallel, voltage is the same on each capacitor and the capacitances add directly:
并联时，每个电容器上的电压相同，电容直接相加：
C_total = C₁ + C₂ + …
A useful comparison table:
一个有用的对比表：
| Quantity | Series | Parallel |
|---|---|---|
| Charge Q | Same on all | Adds: Q_total = Q₁ + Q₂ |
| Voltage V | Adds: V_total = V₁ + V₂ | Same on all |
| Total capacitance | Less than smallest C | Greater than largest C |
This comparison highlights the opposite rules: series reduces effective capacitance, while parallel increases it.
这一对比突出了相反的规律：串联减小等效电容，而并联增大等效电容。
8. Worked Example | 典型例题
Example: A 10 μF capacitor and a 20 μF capacitor are connected in parallel across a 12 V battery. Calculate the total capacitance, the charge on each capacitor, the total charge supplied by the battery, and the total energy stored.
例题：一个 10 μF 的电容和一个 20 μF 的电容并联在 12 V 电池上。计算总电容、每个电容上的电荷、电池提供的总电荷以及储存的总能量。
Step 1: Total capacitance.
第 1 步：总电容。
C_total = C₁ + C₂ = 10 μF + 20 μF = 30 μF
Step 2: Charge on each capacitor.
第 2 步：每个电容上的电荷。
Q₁ = C₁V = 10 μF × 12 V = 120 μC
Q₂ = C₂V = 20 μF × 12 V = 240
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