19.6 Capacitors in Series and Parallel

Capacitors can be connected in series or parallel, and each configuration behaves differently in terms of charge, voltage, and total capacitance. Knowing how to tell them apart and calculate equivalent capacitance is essential for simplifying circuits down to a single effective capacitor.

Capacitor Configurations and Equivalent Capacitance

Capacitance formulas for configurations

Series configuration

In a series connection, capacitors are linked end-to-end so there's only one path for charge to flow. Think of it like a single-lane road: the same amount of charge must pass through every capacitor. That means charge is the same on each capacitor, but the voltage divides among them.

The formula for equivalent capacitance in series:

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$

The equivalent capacitance is always less than the smallest individual capacitor. For example, two 10 µF capacitors in series give $C_{eq} = 5 \text{ µF}$.

Since each capacitor holds the same charge $Q$, you can find the voltage across any individual capacitor using $V = Q / C$. The individual voltages add up to the total applied voltage.

Parallel configuration

In a parallel connection, every capacitor's positive plate connects together and every negative plate connects together. This means voltage is the same across each capacitor, but the charge divides among them (each capacitor stores a different amount of charge based on its capacitance).

The formula for equivalent capacitance in parallel:

$C_{eq} = C_1 + C_2 + ... + C_n$

The equivalent capacitance is always greater than the largest individual capacitor. For example, two 10 µF capacitors in parallel give $C_{eq} = 20 \text{ µF}$.

The total charge stored equals the sum of the charges on each capacitor: if $C_1$ holds 10 µC and $C_2$ holds 20 µC, the combination stores 30 µC total.

Series vs parallel capacitor connections

Here's a quick comparison to keep the key differences straight:

This pattern is the opposite of what happens with resistors, which is a common source of confusion. For capacitors, series reduces the total capacitance and parallel increases it.

A useful special case: for two capacitors in series, you can use the shortcut formula instead of dealing with reciprocals:

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

Equivalent capacitance in combined circuits

Most real circuits mix series and parallel connections. To find the single equivalent capacitance for the whole circuit, work from the inside out:

1. Identify the innermost series or parallel group. Look for capacitors that are purely in series or purely in parallel with each other.

2. Replace that group with its equivalent capacitance using the appropriate formula:

   • Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$
   • Parallel: $C_{eq} = C_1 + C_2 + ... + C_n$

3. Redraw the circuit with the equivalent capacitor in place of the group.

4. Repeat until the entire circuit reduces to a single capacitor.

For example, suppose $C_1 = 4 \text{ µF}$ and $C_2 = 12 \text{ µF}$ are in series, and that combination is in parallel with $C_3 = 7 \text{ µF}$. First, find the series equivalent: $C_{12} = \frac{4 \times 12}{4 + 12} = 3 \text{ µF}$. Then add in parallel: $C_{eq} = 3 + 7 = 10 \text{ µF}$.

Once you have the total $C_{eq}$, you can work backwards through the circuit to find the charge and voltage on each individual capacitor, using $Q = CV$ and the rules that series capacitors share charge while parallel capacitors share voltage.

Capacitor Properties and Electric Field

• Capacitance measures a capacitor's ability to store electric charge for a given voltage, defined by $C = Q / V$.
• The electric field between the plates is what actually stores the energy. For a parallel-plate capacitor, the field is uniform and points from the positive plate to the negative plate.
• Inserting a dielectric material (an insulator like glass or plastic) between the plates increases the capacitance. The dielectric reduces the effective electric field inside the capacitor, which allows the plates to hold more charge at the same voltage.