Equipotential surfaces — why work done moving charge along them is zero

Question

What are equipotential surfaces? Prove that the work done in moving a charge along an equipotential surface is zero. Why are equipotential surfaces always perpendicular to electric field lines?

(NCERT Class 12 — Electrostatic Potential)

Solution — Step by Step

An equipotential surface is a surface where every point has the same electric potential. If you move along this surface, the potential does not change.

Examples:

For a point charge: concentric spheres centred on the charge
For a uniform field: planes perpendicular to the field
For a dipole: complex 3D shapes (not spheres)

The work done in moving charge $q$ from point A to point B is:

W = q(V_A - V_B)

On an equipotential surface, $V_A = V_B$ (same potential everywhere on the surface).

W = q(V_A - V_A) = q \times 0 = \mathbf{0}

No work is done moving a charge along an equipotential surface, regardless of the path taken on that surface.

The work done by the electric field when moving charge $q$ through displacement $d\vec{l}$ is:

dW = q\vec{E} \cdot d\vec{l} = qE\,dl\cos\theta

On an equipotential surface, $dW = 0$ for any displacement along the surface. This means:

qE\,dl\cos\theta = 0

Since $q \neq 0$ , $E \neq 0$ , and $dl \neq 0$ , we must have $\cos\theta = 0$ , i.e., $\theta = 90°$ .

Therefore, $\vec{E}$ is perpendicular to the equipotential surface at every point.

Why This Works

Electric potential is like “height” in a gravitational analogy. An equipotential surface is like a flat floor — moving along it involves no change in height, so no work against gravity. Moving perpendicular to it (changing floors) requires work.

The electric field always points from higher to lower potential, perpendicular to the equipotential surface. This is analogous to how gravity points straight down, perpendicular to the level ground.

Two equipotential surfaces can never intersect. If they did, the intersection line would have two different potentials — a contradiction.

Alternative Method

Using the gradient relationship: $\vec{E} = -\nabla V$ . The gradient of $V$ is perpendicular to the surface $V = \text{constant}$ (by definition of gradient — it points in the direction of maximum change). So $\vec{E}$ is perpendicular to the equipotential surface. This is the calculus-based proof preferred at the JEE level.

For drawing equipotential surfaces in CBSE board exams: draw them as dashed lines perpendicular to the electric field lines. Label each with a potential value (higher potential closer to the positive charge). Spacing between surfaces indicates field strength — closer spacing means stronger field.

Common Mistake

Students sometimes draw equipotential surfaces parallel to field lines instead of perpendicular. Remember: if you walk along a field line, the potential changes (you are going “downhill”). An equipotential surface must cut across the field lines at right angles, like contour lines on a map cutting across the slope direction.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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