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

Electric Potential and Capacitance – Lecture 5 (University Physics II)

Overview

This lecture finishes the study of electric potential and then opens the chapter on capacitance. Building on the idea that every conservative force has an associated potential energy, it defines the electric potential $V$ as potential energy per unit charge and develops strong intuition for it through the analogy with gravitational height. From there it works out how potential behaves on and inside conductors, why charge piles up at sharp points, and how strong fields cause air to break down. The second half reverses the earlier relationship — recovering the field from the potential by differentiation ($\vec{E} = -\nabla V$) — introduces equipotential surfaces, and finally defines a capacitor and its capacitance, the charge it stores per volt.

Key Concepts

Electric potential energy and conservative forces. Every conservative force stores energy when you work against it and returns it when released. Integrating the force over distance gives the change in potential energy, so $W = U_i - U_f$: when a force does positive work, potential energy drops. For two point charges the potential energy function is $U = \dfrac{q_1 q_2}{4\pi\varepsilon_0 r}$, which is the force law $F = \dfrac{q_1 q_2}{4\pi\varepsilon_0 r^2}$ multiplied by $r$.

Electric potential. Electric potential is potential energy per unit charge:

It measures how much energy a unit charge could pick up by moving from a point — the electrical analogue of height in a gravitational field. Its unit is the volt, where $1\ \text{V} = 1\ \text{J}/\text{C}$. Only differences in potential are physical; the zero level is arbitrary, so what we measure between two points is the voltage (potential difference). For a single point charge $V = \dfrac{q}{4\pi\varepsilon_0 r}$, and for a collection of charges the potentials simply add (an integral for a continuous distribution).

Potential from the field, and the electron-volt. Just as integrating force over distance gives potential energy, integrating the field over distance gives potential: $V_A - V_B = -\int_B^A \vec{E}\cdot d\vec{l}$. The electron-volt is the energy an electron gains crossing a one-volt difference, $1\ \text{eV} = 1.602\times10^{-19}\ \text{J}$.

Conductors in electrostatic equilibrium. The field inside a conductor is everywhere zero (otherwise charges would still move). Because no work is done moving a charge through a field-free region, the potential is the same everywhere inside the conductor and equal to its surface value, $V = \dfrac{q}{4\pi\varepsilon_0 R}$ for a sphere of radius $R$. Work is required only to bring a charge from infinity up to the surface; once inside, it moves freely.

Charge density and curvature. On a conductor the surface charge density is not uniform — it is highest where the surface is most sharply curved (smallest radius of curvature). For two spheres at the same potential connected by a wire, equating $V = \dfrac{q}{4\pi\varepsilon_0 R}$ and writing $q = \sigma\,(4\pi R^2)$ gives $\sigma \propto 1/R$. This is why charge concentrates at sharp points and why a doorknob's rough edges can spark.

Dielectric breakdown. When the field at a surface reaches roughly $3\times10^6\ \text{V/m}$, the surrounding air ionizes and charge discharges into it (corona discharge, sparks, lightning). A larger radius keeps the field low for a given charge, which is why a Van de Graaff generator needs a big sphere to hold a large charge.

Recovering the field from the potential. Since integrating force gives potential energy, differentiating potential energy gives force: in one dimension $F_x = -\dfrac{dU}{dx}$, and in three dimensions $\vec{F} = -\nabla U$. The same relationship holds for fields and potentials: $\vec{E} = -\nabla V$. Differentiating is far easier than integrating, so working from the (scalar) potential is the preferred route to the (vector) field.

Equipotential surfaces. Equipotentials are surfaces of constant potential — the analogue of contour lines on a topographic map; closely spaced lines mean a steep, strong field. Field lines always meet equipotentials at right angles: moving a charge along an equipotential does no work, so the force can have no component along that surface. Because a conductor's surface is an equipotential, the field always leaves it perpendicularly.

Capacitors and capacitance. A capacitor is any pair of conductors carrying equal and opposite charge; the common case is the parallel-plate capacitor. Between two large plates with charge density $\sigma$ the field is $E = \dfrac{\sigma}{\varepsilon_0}$ and is essentially zero outside. Separating the plates stores energy in this field, so a capacitor stores energy in the field between its plates, with energy density $u = \tfrac{1}{2}\varepsilon_0 E^2$ and total stored energy $U = \tfrac{1}{2}QV$. The capacitance is the charge stored per volt,

measured in farads. Capacitance depends only on geometry — not on the charge present or the applied voltage. For an isolated sphere $C = 4\pi\varepsilon_0 R$, and for parallel plates $C = \dfrac{\varepsilon_0 A}{d}$. Given a capacitance, $Q = CV$ sets the stored charge. Capacitors charge slowly but can discharge almost instantly, which is what powers a camera flash.

Worked Example

Capacitance of a parallel-plate capacitor. Find $C$ for two plates of area $A$ separated by distance $d$, each carrying surface charge density $\pm\sigma$. Start from the definition, then substitute the plate charge $Q = \sigma A$ and the voltage $V = E\,d$, using the between-plates field $E = \sigma/\varepsilon_0$:

The charge density $\sigma$ cancels, leaving a result that depends only on the geometry: larger plates or a smaller gap give more capacitance.

Summary

• Electric potential is potential energy per unit charge, $V = U/q$, measured in volts ($1\ \text{V} = 1\ \text{J}/\text{C}$); only potential differences (voltage) are physically meaningful.
• Inside a conductor the field is zero, so the potential is constant throughout and equal to its surface value $V = \dfrac{q}{4\pi\varepsilon_0 R}$.
• Surface charge density grows where curvature is sharpest ($\sigma \propto 1/R$); once the field reaches about $3\times10^6\ \text{V/m}$, air ionizes and discharges.
• The field is the negative gradient of the potential, $\vec{E} = -\nabla V$ — differentiation undoes the integration that produced $V$ from $\vec{E}$.
• Field lines cross equipotential surfaces at right angles, because no work is done moving along an equipotential.
• A capacitor stores energy in the field between its plates; its capacitance $C = Q/V$ (in farads) depends only on geometry, with $C = \dfrac{\varepsilon_0 A}{d}$ for parallel plates.