Combination of Capacitors

Why Combine Capacitors?

Capacitors are often connected together in circuits to achieve a specific total (or effective) capacitance $C$ that is not available from a single capacitor. The effective capacitance depends entirely on how the individual capacitors $C_1, C_2, \dots, C_n$ are connected. There are two fundamental ways to connect them: series and parallel.

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1. Capacitors in Parallel

When capacitors are connected in parallel, the left plates of all capacitors are connected to one common point (say, the positive terminal of a battery), and the right plates are connected to another common point (the negative terminal).

Key idea: The voltage across each capacitor is the same (equal to the battery voltage $V$). The total charge stored in the combination is the sum of the charges on each capacitor.

• Let the charge on $C_1$ be $Q_1 = C_1 V$, on $C_2$ be $Q_2 = C_2 V$, and so on.
• Total charge: $Q = Q_1 + Q_2 + \dots + Q_n = (C_1 + C_2 + \dots + C_n)V$.

If the effective capacitance of the combination is $C_p$, then by definition $Q = C_p V$. Comparing the two expressions for $Q$:

When to use: This formula applies only when all capacitors are connected in parallel (same voltage across each).

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2. Capacitors in Series

When capacitors are connected in series, they are connected end-to-end in a single line. The left plate of the first capacitor is connected to the positive terminal of the battery, and the right plate of the last capacitor is connected to the negative terminal.

Key idea: The charge on each capacitor is the same (equal to the charge $Q$ supplied by the battery). The total voltage across the combination is the sum of the voltages across each capacitor.

• Let the voltage across $C_1$ be $V_1 = Q/C_1$, across $C_2$ be $V_2 = Q/C_2$, and so on. …