What Are Equipotential Surfaces for Electric Fields and How Do They Relate to Field Lines? (HL)

Equipotential surfaces are surfaces on which every point has the same electric potential. Electric field lines always cross equipotential surfaces at right angles and point from higher potential toward lower potential.

The Reasoning and Mechanism

Electric potential, $V$, is a scalar quantity defined as the work done per unit charge in bringing a positive test charge from infinity to a point. If a charge moves between two points on the same equipotential surface, then $\Delta V=0$.

The change in electric potential energy is

$$\Delta E_{\text{p}}=q\Delta V=0.$$

Therefore, no work is required to move the charge along that surface. If the electric field had a component parallel to the surface, it would exert a force and do work along the surface, contradicting $\Delta V=0$. Consequently, the electric field must be perpendicular to the equipotential surface.

For a radial direction, the relationship is

$$E=-\frac{\Delta V}{\Delta r}.$$

The minus sign means that the electric field points in the direction of decreasing potential. Closely spaced equipotential surfaces indicate a large potential gradient and therefore a strong electric field.

For example, plates separated by $0.030\,\text{m}$ with a potential difference of $120\,\text{V}$ produce

$$E=\frac{V}{d}=\frac{120}{0.030}=4.0\times10^3\,\text{V m}^{-1}.$$

A common misconception is that equipotential surfaces follow field lines. They do not: they intersect field lines at $90^\circ$.

Exam Technique

This is Additional Higher Level (AHL) content in D.2. In an explanation, state constant potential, use $\Delta E_{\text{p}}=q\Delta V$, conclude that no work is done along the surface, and then deduce that field lines must be perpendicular.