What Are Equipotential Surfaces? (HL)

Equipotential surfaces are surfaces on which every point has the same gravitational potential. Moving a mass along one of these surfaces causes no change in gravitational potential energy, so the gravitational field does no work.

The Reasoning

Gravitational potential $V_g$ is the work done per unit mass in bringing a small test mass from infinity to a point. It is a scalar measured in $\text{J kg}^{-1}$, with zero defined at infinity.

For an isolated spherical mass $M$,

$$V_g=-\frac{GM}{r}.$$

Therefore, every point at the same radial distance $r$ has the same potential. The equipotential surfaces are concentric spheres around the mass; on a two-dimensional diagram, their cross-sections appear as concentric circles.

The relationship between field strength and potential gradient is

$$g=-\frac{\Delta V_g}{\Delta r}.$$

The negative sign means the gravitational field points toward decreasing potential. Near a mass, the potential becomes more negative and changes rapidly with distance, so equally spaced potential values produce surfaces that are closer together.

For example, moving a $2.0\,\text{kg}$ mass between two points on the same surface gives

$$\Delta E_p=m\Delta V_g=(2.0)(0)=0\,\text{J}.$$

A common misconception is that an equipotential surface has zero gravitational field. This is incorrect: the potential is constant only along the surface, while it changes in the perpendicular direction.

Exam Technique

This is Additional Higher Level content in D.1 Gravitational fields. In an exam sketch, show equipotential surfaces perpendicular to field lines and closer together where the field is stronger. For an Explain question, link $\Delta V_g=0$ directly to $\Delta E_p=0$ and therefore zero work along the surface.