No work done moving charge along an equipotential surface.

The idea

An equipotential surface is a surface on which the electric potential is constant. Since potential difference ΔV between any two points on the surface is zero, the work done W = QΔV is also zero. This is because the electric force is always perpendicular to the equipotential surface, so the component of force along the direction of motion is zero. For example, moving a charge around a circular equipotential centred on a point charge requires no work, even though the charge experiences a force. This principle is used in understanding conductors in electrostatic equilibrium, where the surface is an equipotential, and in the design of particle accelerators where particles move along equipotentials without energy change.

What earns the marks

• State that on an equipotential surface, the potential difference between any two points is zero (ΔV = 0).
• Apply W = QΔV to show that if ΔV = 0, then W = 0, regardless of the magnitude of Q.
• Explain that the electric force is perpendicular to the equipotential surface, so the force has no component along the displacement.
• Relate this to the definition of work done: W = Fs cosθ, where θ = 90°, so cos90° = 0 and W = 0.
• Give a practical example, such as a charge moving on a spherical equipotential around a point charge, or a charge moving along the surface of a charged conductor.

Common mistakes and how to put them right

• Assuming that because a force acts on the charge, work must be done.

  Put it right: Work depends on the component of force along the displacement; perpendicular forces do no work.

• Thinking that moving along an equipotential changes the potential energy of the charge.

  Put it right: Potential energy depends on potential, which is constant, so it does not change.

• Confusing equipotential surfaces with electric field lines.

  Put it right: Equipotentials are perpendicular to field lines and represent constant potential, while field lines show the direction of force on a positive charge.

Exam tips

• When asked why no work is done, explicitly state that ΔV = 0 and that the electric force is perpendicular to the displacement.
• Use the equation W = QΔV to justify the zero work done, and mention that Q is not zero but ΔV is zero.
• If a diagram is given, identify the equipotential surface and show that the path lies along it, then explain the perpendicularity of force and motion.