Capacitors in series & parallel

Wire two capacitors together and they act like one. Parallel ones add; series ones add "upside-down."

Source: Serway & Jewett, Physics for Scientists and Engineers, 7th ed., §26.3.

Before you start

What you need first

Topic 12 — $C = Q/V$ and $Q = CV$.
Reading a simple circuit diagram.

What you'll be able to do

Recognise the capacitor circuit symbol.
Combine capacitors in parallel: $C_{\text{eq}} = C_1 + C_2 + \cdots$.
Combine capacitors in series: $\tfrac{1}{C_{\text{eq}}} = \tfrac{1}{C_1} + \tfrac{1}{C_2} + \cdots$.
Know what is shared in each case (voltage vs charge).

Reading the circuit

The capacitor symbol

In a circuit a capacitor is drawn as two parallel lines (the plates) with a lead on each side. A battery (a long thin line for $+$, a short thick line for $-$) drives charge onto it.

The capacitor (equal lines) and battery (long $+$, short $-$) symbols.

Side by side

Capacitors in parallel — just add

In parallel, both capacitors feel the same voltage, and their charges add. Side by side is like making the plates bigger — so the capacitances simply add:

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

Parallel: both capacitors share the battery's voltage.

📐 Worked example 1

Two capacitors in parallel

$C_1 = 3\ \mu\text{F}$ and $C_2 = 6\ \mu\text{F}$ are connected in parallel. Find the equivalent capacitance.

1Parallel — just add:

$$ C_{\text{eq}} = C_1 + C_2 = 3 + 6 = 9\ \mu\text{F} $$

The parallel total is always bigger than the largest single capacitor.

In a line

Capacitors in series — add the upside-downs

In series, the same charge sits on each capacitor, and their voltages add. In a line is like making the gap bigger — so capacitance drops. Add the reciprocals, then flip back:

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

Series: the same charge passes through each in turn.

📐 Worked example 2

Two capacitors in series

The same $C_1 = 3\ \mu\text{F}$ and $C_2 = 6\ \mu\text{F}$, now in series. Find the equivalent capacitance.

1Add the reciprocals:

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

2Flip back:

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

The series total is always smaller than the smallest single capacitor. (Don't stop at $\tfrac12$ — flip it!)

✏️ Try it yourself

Take $C_1 = 2\ \mu\text{F}$ and $C_2 = 4\ \mu\text{F}$.

(a) Find the equivalent capacitance in parallel.
(b) Find the equivalent capacitance in series.

(a) Parallel: $C_{\text{eq}} = 2 + 4 = 6\ \mu\text{F}.$ (b) Step 1. $\dfrac{1}{C_{\text{eq}}} = \dfrac{1}{2} + \dfrac{1}{4} = \dfrac{3}{4}.$ Step 2. Flip: $C_{\text{eq}} = \dfrac{4}{3} \approx 1.3\ \mu\text{F}$ (smaller than either).

Common mistakes

Mistake	Fix
Swapping the rules (adding in series).	Capacitors add in parallel; in series you add the reciprocals — opposite to resistors.
Stopping at $1/C_{\text{eq}}$ in series.	Flip it back to get $C_{\text{eq}}$.
Thinking series gives a bigger capacitance.	Series is always smaller than the smallest; parallel is bigger than the largest.
Assuming the same voltage in series.	Series shares the same charge; the voltages add. Parallel shares the voltage.

Recap — the whole topic on one screen

Connection	Rule	Shared
Parallel	$C_{\text{eq}} = C_1 + C_2 + \cdots$ (bigger)	same voltage
Series	$\dfrac{1}{C_{\text{eq}}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots$ (smaller)	same charge

Next topic

Energy stored in a capacitor

A charged capacitor holds energy, ready to release in a flash (a camera flash, a defibrillator). Next we find how much: $U = \tfrac12 CV^2$.

→ Topic 14 · Energy stored in a capacitor

Mistake	Fix
Swapping the rules (adding in series).	Capacitors add in parallel; in series you add the reciprocals — opposite to resistors.
Stopping at \(1/C_{\text{eq}}\) in series.	Flip it back to get \(C_{\text{eq}}\).
Thinking series gives a bigger capacitance.	Series is always smaller than the smallest; parallel is bigger than the largest.
Assuming the same voltage in series.	Series shares the same charge; the voltages add. Parallel shares the voltage.

Connection	Rule	Shared
Parallel	\(C_{\text{eq}} = C_1 + C_2 + \cdots\) (bigger)	same voltage
Series	\(\dfrac{1}{C_{\text{eq}}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots\) (smaller)	same charge