Idea that no work is done when moving along an equipotential surface.

The idea

An equipotential surface is a surface on which the gravitational potential V is the same at every point. If a mass m moves from one point on the surface to another, the potential difference ∆V is zero, so the change in potential energy m∆V is zero. The gravitational force acts perpendicular to the equipotential surface, while the displacement along the surface is tangential, so the angle between force and displacement is 90°. Since work done is force × displacement × cos θ and cos 90° = 0, the gravitational field does no work on the mass. This is why no energy is transferred when a satellite moves in a circular orbit, because its path lies along an equipotential. It also explains why equipotential surfaces are always perpendicular to field lines.

What earns the marks

• Define an equipotential surface as a surface where the gravitational potential V is constant, so ∆V = 0 between any two points on it.
• Use ∆W = m∆V to show that if ∆V = 0 then ∆W = 0, so no work is done by or against the gravitational field.
• Explain that the gravitational force is perpendicular to the equipotential surface, so the component of force along the displacement is zero.
• Apply the work formula W = Fs cos θ and state that θ = 90° for motion along an equipotential, giving cos 90° = 0.
• Link the zero work to a circular orbit, where the gravitational force provides the centripetal force and the satellite's speed remains constant.
• Describe how equipotential surfaces are drawn perpendicular to field lines, with closer spacing where the field is stronger.

Common mistakes and how to put them right

• Claiming that no force acts on a mass on an equipotential surface.

  Put it right by stating that the gravitational force still acts, but it is perpendicular to the displacement.

• Thinking that no work is done because the mass is stationary.

  Put it right by explaining that the mass may move at constant speed along the surface while the field does zero work.

• Assuming potential energy is zero on an equipotential.

  Put it right by stating that potential energy mV is constant but not necessarily zero, since V is measured relative to infinity.

• Confusing equipotential surfaces with surfaces of constant field strength.

  Put it right by noting that equipotentials join points of equal potential, while field strength depends on the spacing of equipotentials.

Exam tips

• In explanations, explicitly state that ∆V = 0 on an equipotential and substitute into ∆W = m∆V to justify zero work.
• Use the perpendicularity argument with a clear diagram showing the force arrow at 90° to the equipotential surface and the displacement along it.
• When discussing circular orbits, link constant speed to zero work done by gravity, avoiding any claim that gravity is absent.
• Define equipotential surfaces in words before using them in calculations or explanations, so the meaning of ∆V = 0 is unambiguous.