Physics · Electric Fields · A2 Level

Electric Potential

Moving a charge through an electric field generally takes work. This lesson introduces electric potential — the quantity that measures how much work is needed to bring a charge to a point — along with the equipotential surfaces that map out where a charge can move freely, without any work being done at all.

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Electric potential and potential energy

Moving a charge through an electric field generally takes work — pushing a positive charge closer to another positive charge, against the repulsive force, requires energy input. This is captured by electric potential energy; the quantity used to measure it per unit charge is electric potential, $V$, defined at a point as the work done per unit positive charge bringing a test charge from infinity (zero potential) to that point.

Potential is a scalar (unlike field strength), measured in volts (V), where $1\,\text{V} = 1\,\text{J C}^{-1}$. The work done moving a charge $Q$ between two points at different potentials follows directly:

$W = Q\,\Delta V$

where $\Delta V$ is the potential difference between start and end points — the same equation used to define the volt at AS for circuits, now applied to motion through a field.

Field strength and potential are related, not the same

Field strength $E$ is force per unit charge — how hard the field pushes. Potential $V$ is potential energy per unit charge — how much work is needed to bring a charge there from infinity. They're connected (field strength is the rate at which potential changes with distance) but answer different questions, with different units (N C⁻¹, equivalently V m⁻¹, for $E$; volts for $V$). A region can have large potential but zero field strength, or vice versa.

How potential varies with distance

Around an isolated point charge $Q$, the potential at a distance $r$ from it is:

$V = \dfrac{Q}{4\pi\varepsilon_0 r}$

where $\varepsilon_0$ is the permittivity of free space. For a positive charge, $V$ is positive everywhere and falls away with distance: greatest right next to the charge, dropping quickly at first, then more gradually, creeping toward zero as $r\to\infty$ but never quite reaching it at any finite distance. For a negative charge, the same shape appears reflected below zero, since $Q$ is negative.

Animated graph of electric potential V against distance from a point charge r. The curve starts high and steep close to the charge, then falls away steeply at first and flattens out, approaching but never reaching zero as r increases. — $V$ against $r$ for an isolated positive point charge. Potential is greatest close to the charge, falls away quickly at first, then flattens out, approaching zero at large $r$ but never reaching it — the same fall-off shape traced by the equipotentials below, now viewed as a single curve.

Finding $|\Delta V|$ graphically: area under an E–r graph

Field strength and potential are linked by more than a shared source charge: since $V=\dfrac{Q}{4\pi\varepsilon_0 r}$ decreases as $r$ increases, its rate of change is negative, and it's this rate of change (with a sign flip) that gives the field strength: $E=-\dfrac{dV}{dr}$. Rearranging and integrating both sides between two radii $r_1$ and $r_2$ turns this into a statement about area:

$\displaystyle\int_{r_1}^{r_2} E\,dr = -\big(V(r_2)-V(r_1)\big) = V(r_1)-V(r_2)$

The integral on the left is exactly the area under a graph of $E$ against $r$, between $r_1$ and $r_2$. So the shaded area under an $E$–$r$ graph, between any two values of $r$, gives the size of the change in potential between those points, $|\Delta V|$ — directly analogous to how the area under a force–extension graph gives work done, or the area under a velocity–time graph gives displacement. The magnitude sign matters here: unlike the gravitational case, $E$ (always plotted as a positive quantity) and the sign of $\Delta V$ point the same way in size but not necessarily in sign, so the area gives $|\Delta V|$ rather than a signed $\Delta V$.

Graph of electric field strength E against distance from a point charge r, decreasing steeply then flattening out. The region under the curve between two marked distances r1 and r2 is shaded grey and labelled area equals modulus delta V, showing that this shaded area equals the size of the change in electric potential between r1 and r2. — The shaded area under an $E$–$r$ graph, between any two distances $r_1$ and $r_2$, equals the size of the change in electric potential $|\Delta V|$ between those points — a useful graphical way to find $|\Delta V|$ directly from a sketch or plotted graph of $E$, without needing $Q$ or $r$ explicitly.

Equipotentials

An equipotential surface (or line, in 2D) joins points in a field sharing the same potential. Around an isolated point charge, equipotentials are concentric spheres (circles in cross-section) centred on the charge, since every point at radius $r$ has the same potential. Between parallel plates, equipotentials are flat planes running parallel to the plates.

A positive point charge with field lines radiating outward, overlaid with three dashed concentric circles representing equipotential lines at increasing radii, each crossing every field line at a right angle, with the circles spaced more widely apart the further out they are. — The solid lines are the field, radiating outward; the dashed circles are equipotentials, always perpendicular to the field lines. Note how the circles are bunched close together near the charge and spread further apart with distance — potential falls steeply at first, then more gradually, mirroring how the field itself weakens with distance.

Two properties of equipotentials are common exam points:

No work is done moving a charge along an equipotential. Every point on the surface is at the same potential, so $\Delta V=0$ and $W=Q\Delta V=0$ regardless of path. Work is only done moving between different equipotentials — i.e. in the direction of the field.
Field lines always cross equipotentials at right angles. If a field line had any component along an equipotential, moving a charge along it would do work with no change in potential — a contradiction. So field lines and equipotentials are always mutually perpendicular, just as contour lines on a map are perpendicular to the direction of steepest slope.

Key takeaways

Electric potential $V$ is the work done per unit charge bringing a test charge from infinity to a point; work done moving a charge $Q$ through a potential difference $\Delta V$ is $W=Q\Delta V$.
Around a point charge, $V=\dfrac{Q}{4\pi\varepsilon_0 r}$ — positive and falling toward zero with distance for a positive charge, negative and rising toward zero for a negative charge.
Field strength and potential are related but distinct: one measures force per unit charge, the other measures energy per unit charge.
Equipotentials join points of equal potential (concentric spheres for a point charge, parallel planes between plates); no work is done moving along one, and field lines always cross them at right angles.

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Electric Field Strength

Coulomb's Law