use the capacitance formulae for capacitors in series and in parallel

Capacitors and Capacitance

A capacitor stores electric charge $Q$ on two conductors separated by an insulating material (dielectric). The ability of a capacitor to store charge is quantified by its capacitance $C$, defined as

$$C = \frac{Q}{V}$$

where $V$ is the potential difference between the plates.

Capacitance of a Parallel‑Plate Capacitor

For an ideal parallel‑plate capacitor filled with a dielectric of relative permittivity $\varepsilon_r$:

$$C = \varepsilon_0 \varepsilon_r \frac{A}{d}$$

$\varepsilon_0 = 8.85\times10^{-12}\ \text{F m}^{-1}$ is the permittivity of free space, $A$ is the plate area, and $d$ is the separation.

Combining Capacitors

In circuits, capacitors are often connected in series or in parallel. The equivalent capacitance $C_{\text{eq}}$ can be found using specific formulae.

Capacitors in Parallel

When capacitors are connected such that each experiences the same voltage, their total capacitance is the sum of the individual capacitances:

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

Charge distributes according to each capacitor’s value, but the voltage across each is identical.

Capacitors in Series

When the same charge $Q$ passes through each capacitor, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances:

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

The voltage across the series combination is the sum of the individual voltages.

Worked Example

1. Three capacitors: $C_1 = 2\ \mu\text{F}$, $C_2 = 3\ \mu\text{F}$, $C_3 = 6\ \mu\text{F}$.
2. Find $C_{\text{eq}}$ when:
   • All three are in parallel.
   • $C_1$ and $C_2$ are in series, and this combination is in parallel with $C_3$.

Solution

Parallel combination:

$$C_{\text{eq,\,parallel}} = 2 + 3 + 6 = 11\ \mu\text{F}$$

Series of $C_1$ and $C_2$:

$$\frac{1}{C_{12}} = \frac{1}{2} + \frac{1}{3} = \frac{3+2}{6} = \frac{5}{6}$$ $$C_{12} = \frac{6}{5} = 1.2\ \mu\text{F}$$

Now $C_{12}$ in parallel with $C_3$:

$$C_{\text{eq}} = C_{12} + C_3 = 1.2 + 6 = 7.2\ \mu\text{F}$$

Summary Table of Formulae

Configuration	Formula for $C_{\text{eq}}$	Key Points
Parallel	$C_{\text{eq}} = \displaystyle\sum_{i=1}^{n} C_i$	Same voltage across each capacitor; charges add.
Series	$\displaystyle\frac{1}{C_{\text{eq}}} = \sum_{i=1}^{n} \frac{1}{C_i}$	Same charge on each capacitor; voltages add.
Parallel‑Plate (ideal)	$C = \varepsilon_0 \varepsilon_r \dfrac{A}{d}$	Increase area $A$ → larger $C$; increase separation $d$ → smaller $C$.

Common Mistakes to Avoid

• Adding voltages for capacitors in parallel – the voltage is the same across each.
• Adding charges for capacitors in series – the charge on each is identical.
• Confusing the reciprocal rule for series with the direct sum rule for parallel.
• Neglecting the effect of a dielectric ($\varepsilon_r$) when calculating $C$ for real capacitors.

Practice Questions

1. Two capacitors, $4\ \mu\text{F}$ and $6\ \mu\text{F}$, are connected in series across a $12\ \text{V}$ battery. Calculate the charge on each capacitor and the voltage across each.
2. A $10\ \mu\text{F}$ capacitor is placed in parallel with a combination of three $5\ \mu\text{F}$ capacitors in series. Find the total capacitance.
3. For a parallel‑plate capacitor with plate area $0.02\ \text{m}^2$, plate separation $1.5\ \text{mm}$, and air as the dielectric ($\varepsilon_r \approx 1$), compute the capacitance.