Capacitors in Series

In series, same charge Q on each capacitor, voltages add. Equivalent capacitance is always less than the smallest individual capacitance.

Charge constraint

In series, the capacitors are connected end-to-end with no branch points between them. The inner plates form isolated conductors. By charge conservation and induction, every capacitor carries the same charge $Q$.

Voltage constraint

The total voltage $V$ across the series combination equals the sum of individual voltages:

$$V = V_1 + V_2 + \cdots + V_n = \frac{Q}{C_1} + \frac{Q}{C_2} + \cdots + \frac{Q}{C_n} = Q\sum_{i=1}^n \frac{1}{C_i}$$

Equivalent capacitance

$$C_{eq} = \frac{Q}{V} = \frac{1}{\displaystyle\sum_{i=1}^n \frac{1}{C_i}}$$ $$\boxed{\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n}}$$

Key property

$C_{eq} < C_{min}$ — series combination is always less capacitive than the smallest individual capacitor. This is the opposite of resistors in series.

For two capacitors:

$$C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$$