19.4 Combinations of Capacitors

Capacitors can be connected in two fundamental ways: series and parallel. Each arrangement produces a different equivalent capacitance and has distinct practical applications.

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Series Combination

When capacitors are connected end-to-end so that the same current path passes through each one, they are said to be in series.

Key Property

The charge $Q$ on every capacitor in a series combination is identical: $Q_1=Q_2=Q_3=\cdots =Q$

This occurs because of electrostatic induction: the charge displaced from one plate must reside on the adjacent plate of the next capacitor.

Voltage Distribution

The total voltage $V$ supplied by the source equals the sum of the individual voltages: $V=V_1+V_2+V_3+\cdots$

Since $V=Q/C$ for each capacitor: $\frac{Q}{C_{eq}}=\frac{Q}{C_1}+\frac{Q}{C_2}+\frac{Q}{C_3}+\cdots$

Dividing through by $Q$:

$\boxed{\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\cdots }$

Important Result

The equivalent capacitance $C_{eq}$ in series is always less than the smallest individual capacitance. Physically, series connection increases the effective plate separation, reducing capacitance.

Worked Example

Find $C_{eq}$ for $C_1=3\mu F$ and $C_2=6\mu F$ in series: $\frac{1}{C_{eq}}=\frac{1}{3}+\frac{1}{6}=\frac{2}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}$ $C_{eq}=2\mu F$

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Parallel Combination

When capacitors are connected so that both plates of each capacitor are connected to the same two nodes, they are in parallel.

Key Property

The potential difference $V$ across every capacitor in a parallel combination is identical and equal to the source voltage: $V_1=V_2=V_3=\cdots =V$

Physically, parallel connection increases the total plate area available for storing charge while keeping the plate separation constant.

Charge Distribution

The total charge supplied by the source is shared among the capacitors: $Q=Q_1+Q_2+Q_3+\cdots$

Since $Q=CV$ for each capacitor: $C_{eq}V=C_{1}V+C_{2}V+C_{3}V+\cdots$

Dividing through by $V$:

$\boxed{C_{eq}=C_1+C_2+C_3+\cdots }$

Important Result

The equivalent capacitance in parallel is always greater than the largest individual capacitance.

Worked Example

Find $C_{eq}$ for $C_1=3\mu F$ and $C_2=6\mu F$ in parallel: $C_{eq}=3+6=9\mu F$

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Comparison: Series vs Parallel

Energy Comparison

Since $U=\frac{1}{2}{C}_{eq}{V}^2$, for the same source voltage $V$, the parallel combination stores more energy because $C_{eq}$ is larger.

For $n$ identical capacitors each of capacitance $C$:

• $C_{parallel}=nC$
• $C_{series}=C/n$
• Ratio of energies: $\frac{U_{parallel}}{U_{series}}=\frac{nC}{C/n}=n^2$