use the capacitance formulae for capacitors in series and in parallel

Published by Patrick Mutisya · 14 days ago

Cambridge A-Level Physics 9702 – Capacitors and Capacitance

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 = \varepsilon0 \varepsilonr \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.

Suggested diagram: Cross‑section of a parallel‑plate capacitor showing plate area \$A\$, separation \$d\$, and dielectric.

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}} = C1 + C2 + C3 + \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}{C1} + \frac{1}{C2} + \frac{1}{C3} + \dots + \frac{1}{C_n}\$

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

Worked Example

  1. Three capacitors: \$C1 = 2\ \mu\text{F}\$, \$C2 = 3\ \mu\text{F}\$, \$C_3 = 6\ \mu\text{F}\$.

  2. Find \$C_{\text{eq}}\$ when:

    • All three are in parallel.

    • \$C1\$ and \$C2\$ 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 \$C1\$ and \$C2\$:

\$\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 \$C3\$:

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

Summary Table of Formulae

ConfigurationFormula 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 = \varepsilon0 \varepsilonr \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.