Equivalent capacitance

Also called Total capacitance, C_eq

Equivalent capacitance is the single capacitance that could replace a group of capacitors without changing what the rest of the circuit sees. Capacitors in parallel add directly, while capacitors in series combine as reciprocals, which is the reverse of the rule for resistors.

Both CEDs file this under RC circuits rather than under the capacitor topic, at 11.8.A.1: a collection of capacitors may be analyzed as though it were a single capacitor with an equivalent capacitance $C_{\text{eq}}$ . Both equation sheets print the pair:

\frac{1}{C_{\text{eq},s}} = \sum_i \frac{1}{C_i} \qquad C_{\text{eq},p} = \sum_i C_i

Read them next to the resistor rules and the swap is unmistakable. Series capacitors combine like parallel resistors, and parallel capacitors combine like series resistors. Memorising four formulas in a row is how students end up choosing the wrong one; deriving them from what is shared is how you stop.

In series the shared quantity is charge. The CED gives the reason at 11.8.A.2: as a result of conservation of charge, each capacitor in series must have the same magnitude of charge on each plate. The potential differences then add, and since $\Delta V = Q/C$ for each, the reciprocals add. The CED states the consequence outright at 11.8.A.1.ii: the equivalent capacitance of a set of capacitors in series is less than the capacitance of the smallest one.

In parallel the shared quantity is potential difference. Each capacitor sits across the same $\Delta V$ , so the stored charges add, and $C = Q/\Delta V$ makes the capacitances add.

Contrast the two with equivalent resistance, where series shares current and parallel shares voltage. Same principle, opposite arithmetic, because capacitance is a charge-per-volt while resistance is a volt-per-amp.

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