Physics 5. Part IV: Electricity and Magnetism 5.3. Capacitance 5.3.3. Series and Parallel Capacitors

5.3.3. Series and Parallel Capacitors

Table of Contents

Combining Capacitors

In many circuits, more than one capacitor is used. When capacitors are connected together, the total effect of the combination depends on how they are arranged. The two most important arrangements are series and parallel. Learning these combinations lets you replace several capacitors with one equivalent capacitor, which simplifies circuit analysis.

The idea of an equivalent capacitor is simple. A group of capacitors can often be replaced by a single capacitor that stores the same total charge for the same applied voltage.

The equivalent capacitance, $C_{\text{eq}}$, is the capacitance of a single capacitor that behaves like the whole combination.

Capacitors in Parallel

When capacitors are connected in parallel, each capacitor is attached across the same two points in the circuit. This means that every capacitor has the same voltage across it.

Rendering scientific sketch…

Capacitors in parallel

Because the voltage is the same across all capacitors in parallel, we write

$$
V = V_1 = V_2 = V_3 = \cdots
$$

The total charge stored by the combination is the sum of the charges on each capacitor:

$$
Q_{\text{total}} = Q_1 + Q_2 + Q_3 + \cdots
$$

Using $Q = CV$ for each capacitor,

$$
Q_{\text{total}} = C_1V + C_2V + C_3V + \cdots
$$

Factor out the common voltage:

$$
Q_{\text{total}} = (C_1 + C_2 + C_3 + \cdots)V
$$

If we define the equivalent capacitor by

$$
Q_{\text{total}} = C_{\text{eq}}V
$$

then

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

For capacitors in parallel,
$$
C_{\text{eq}} = C_1 + C_2 + C_3 + \cdots
$$
The voltage is the same across each capacitor.

This result makes physical sense. Putting capacitors in parallel increases the total area available to store charge, so the equivalent capacitance becomes larger than any individual capacitance.

Example of a Parallel Combination

Suppose two capacitors, $2\,\mu\text{F}$ and $5\,\mu\text{F}$, are connected in parallel. Then

$$
C_{\text{eq}} = 2\,\mu\text{F} + 5\,\mu\text{F} = 7\,\mu\text{F}
$$

So the pair behaves like one $7\,\mu\text{F}$ capacitor.

Capacitors in Series

When capacitors are connected in series, they are placed one after another along a single path. The same charge passes through each capacitor during charging, so each capacitor ends up with the same magnitude of charge.

Rendering scientific sketch…

Capacitors in series

For capacitors in series, the charge on each capacitor is the same:

$$
Q = Q_1 = Q_2 = Q_3 = \cdots
$$

The total voltage across the series combination is the sum of the individual voltages:

$$
V_{\text{total}} = V_1 + V_2 + V_3 + \cdots
$$

Using $V = Q/C$ for each capacitor,

$$
V_{\text{total}} = \frac{Q}{C_1} + \frac{Q}{C_2} + \frac{Q}{C_3} + \cdots
$$

Factor out $Q$:

$$
V_{\text{total}} = Q\left(\frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots\right)
$$

If the equivalent capacitor satisfies

$$
V_{\text{total}} = \frac{Q}{C_{\text{eq}}}
$$

then

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

For capacitors in series,
$$
\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \cdots
$$
The charge is the same on each capacitor.

For two capacitors only, this can be written as

$$
C_{\text{eq}} = \frac{C_1C_2}{C_1 + C_2}
$$

In series, the equivalent capacitance is always smaller than the smallest individual capacitor.

Why Series Capacitance Gets Smaller

A series arrangement effectively increases the separation involved in storing charge, making the system less able to store charge for a given voltage. That is why the equivalent capacitance decreases.

Example of a Series Combination

Suppose two capacitors, $3\,\mu\text{F}$ and $6\,\mu\text{F}$, are connected in series. Then

$$
\frac{1}{C_{\text{eq}}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2}
$$

with capacitances measured in $\mu\text{F}$. Therefore,

$$
C_{\text{eq}} = 2\,\mu\text{F}
$$

So the pair behaves like a single $2\,\mu\text{F}$ capacitor.

Comparing Series and Parallel

The key differences are shown below.

Arrangement	Same for each capacitor	Adds directly	Equivalent capacitance
Parallel	Voltage	Charge	$C_{\text{eq}} = C_1 + C_2 + \cdots$
Series	Charge	Voltage	$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$

A quick memory rule is this. In parallel, capacitances add. In series, reciprocals add.

Mixed Combinations

Some circuits contain both series and parallel parts. In such cases, simplify one part at a time. Replace a simple series or parallel group by its equivalent capacitance, then continue until only one equivalent capacitor remains.

For example, if $C_2$ and $C_3$ are in parallel, first find

$$
C_{23} = C_2 + C_3
$$

If that result is then in series with $C_1$, use

$$
\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_{23}}
$$

This step by step approach is the standard method for more complicated capacitor networks.

A Simple Mixed Example

Suppose $C_1 = 4\,\mu\text{F}$ is in series with a parallel pair $C_2 = 2\,\mu\text{F}$ and $C_3 = 2\,\mu\text{F}$.

First combine the parallel capacitors:

$$
C_{23} = 2 + 2 = 4\,\mu\text{F}
$$

Now the circuit becomes two $4\,\mu\text{F}$ capacitors in series:

$$
C_{\text{eq}} = \frac{4 \times 4}{4 + 4} = \frac{16}{8} = 2\,\mu\text{F}
$$

So the full combination has equivalent capacitance

$$
C_{\text{eq}} = 2\,\mu\text{F}
$$

Common Mistakes

A common mistake is to use the resistor rules by memory without checking whether the circuit element is a capacitor. Capacitors combine in the opposite way from resistors in many basic cases. Another common mistake is forgetting what stays the same. In parallel, voltage is the same. In series, charge is the same.

It is also important to identify the arrangement correctly. Two capacitors are in parallel only if both ends connect to the same two nodes. They are in series only if they are connected one after another with no branching point between them.

Final Check of Results

You can often tell whether an answer is reasonable without detailed calculation.

If capacitors are in parallel, the equivalent capacitance should be larger than each single capacitor.

If capacitors are in series, the equivalent capacitance should be smaller than the smallest one.

Reasonableness check:
For parallel, $C_{\text{eq}}$ must increase.
For series, $C_{\text{eq}}$ must decrease.
If your result does not follow this pattern, recheck the circuit.

Summary

Series and parallel combinations let us replace many capacitors by one equivalent capacitor. In parallel, capacitances add directly because each capacitor has the same voltage. In series, reciprocals add because each capacitor carries the same charge and the voltages add. These rules are essential for simplifying capacitor networks in electric circuits.

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