The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Capacitors and Capacitance – Lecture 6 (University Physics II)

Overview

This lecture introduces the capacitor — any two conductors separated by an insulator — and the quantity that characterizes it, capacitance. After a short recap of how to compute electric potential by choosing a sensible reference point, the focus turns to how capacitors store energy in the electric field between their plates, why capacitance is fixed entirely by geometry and the material between the conductors, and how capacitors combine in series and in parallel. The lecture closes with a rigorous derivation of the stored energy and a first, qualitative look at dielectrics.

Key Concepts

Electric potential and the choice of reference (recap)

Only differences in potential matter, so you are free to call any convenient point $V = 0$ — just as with gravitational height.
Between two parallel plates the field is uniform, so the potential a height $y$ above the negative plate is $V = Ey$ , and the potential energy of a charge is $U = qEy$ (directly analogous to $U = mgy$ ).
For an infinite line charge, $V \propto \ln r$ , so you cannot place the zero at infinity ( $\ln\infty$ diverges) or on the axis ( $\ln 0$ diverges); pick the finite surface of the rod instead.
For a uniformly charged ring, every point on the ring is equidistant from a point on the axis, so it acts like a point charge: $V = \dfrac{Q}{4\pi\varepsilon_0\sqrt{a^2 + x^2}}$ .

Capacitor and the field between plates

A capacitor is two conductors separated by an insulator (even vacuum). Unless stated otherwise, "capacitor" means a parallel-plate capacitor, and the plates carry equal and opposite charge.
The field between parallel plates is uniform: $E = \dfrac{\sigma}{\varepsilon_0}$ , where $\sigma = Q/A$ is the surface charge density. Enlarging the plates at fixed $\sigma$ does not change $E$ .

Capacitance

Definition: $C = \dfrac{Q}{V}$ , measured in farads (1 F = 1 C/V), named for Faraday.
Capacitance is a property of geometry and material only — not of how much charge or voltage happens to be present. A store-bought capacitor has its rated capacitance even when uncharged.
Representative geometries:
- Isolated conducting sphere: $C = 4\pi\varepsilon_0 R$ .
- Parallel plates: $C = \dfrac{\varepsilon_0 A}{d}$ — larger area or smaller gap gives more capacitance.
- Spherical capacitor: $C = 4\pi\varepsilon_0\dfrac{R_a R_b}{R_b - R_a}$ .
- Cylindrical capacitor of length $L$ : $C = \dfrac{2\pi\varepsilon_0 L}{\ln(R_b/R_a)}$ .
The farad is enormous: a 1 F parallel-plate capacitor with a 1 mm gap would need plates of roughly $10^{8}\ \text{m}^2$ (about 6 miles on a side). Real capacitors are typically micro- ( $\mu$ F), nano-, or picofarads.

Energy storage

Charging a capacitor builds an electric field, and that field stores energy. Equivalent expressions:

\begin{aligned} U &= \tfrac{1}{2}QV = \tfrac{1}{2}CV^2 = \frac{Q^2}{2C} \end{aligned}

Energy density of any electric field (not just in capacitors): $u = \tfrac{1}{2}\varepsilon_0 E^2$ . Doubling the field quadruples the stored energy per unit volume.

Combining capacitors

Series: each capacitor carries the same charge; the equivalent capacitance is smaller than any single one: $\dfrac{1}{C_\text{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots$
Parallel: each capacitor has the same voltage; capacitances simply add: $C_\text{eq} = C_1 + C_2 + \cdots$
Redistributing charge between capacitors (e.g. closing a switch onto an uncharged capacitor) lowers the total stored energy; the difference is dissipated as heat in the connecting wires.

Dielectrics (qualitative preview)

Inserting a nonconducting dielectric between the plates polarizes its molecules, creating an internal field opposite to the plates' field and partially canceling it.
The weaker net field lets more charge sit at the same voltage, so the dielectric increases the capacitance. It also lets the plates sit closer without arcing — air breaks down (ionizes) near $3\times10^{6}\ \text{V/m}$ . A good dielectric is easily polarized yet a poor conductor.

Worked Example

A rolled-up capacitor has a total plate area $A = 2\ \text{m}^2$ and plate separation $d = 5\ \text{mm}$ , connected across $V = 10\ \text{kV}$ . Find its capacitance, the stored charge, and the field between the plates. State each rule symbolically, then substitute:

\begin{aligned} C &= \frac{\varepsilon_0 A}{d} = \frac{(8.85\times10^{-12})(2)}{5\times10^{-3}} \approx 3.54\times10^{-9}\ \text{F} = 3.54\ \text{nF} \\ Q &= CV = (3.54\times10^{-9})(1.0\times10^{4}) \approx 3.5\times10^{-5}\ \text{C} \\ E &= \frac{V}{d} = \frac{1.0\times10^{4}}{5\times10^{-3}} = 2.0\times10^{6}\ \text{V/m} \end{aligned}

The field is safely below the $\sim 3\times10^{6}\ \text{V/m}$ breakdown of air.

Summary

A capacitor is two conductors separated by an insulator; capacitance is defined as $C = Q/V$ and measured in farads.
Capacitance depends only on geometry and the dielectric material — for parallel plates, $C = \varepsilon_0 A/d$ (bigger area, smaller gap ⇒ larger $C$ ).
Stored energy can be written $U = \tfrac{1}{2}QV = \tfrac{1}{2}CV^2 = Q^2/2C$ , and the field itself holds energy density $u = \tfrac{1}{2}\varepsilon_0 E^2$ .
In series, capacitors share the same charge and combine reciprocally (equivalent capacitance drops); in parallel, they share the same voltage and simply add.
Letting charge redistribute between capacitors loses stored energy to heat in the wires.
A dielectric polarizes to cancel part of the internal field, raising capacitance and allowing more charge at the same voltage; air itself ionizes near $3\times10^{6}\ \text{V/m}$ .