Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Work done by electric field on moving a positive charge on an equipotential surface is always zero.
Reason (R) : Electric lines of forces are always perpendicular to equipotential surfaces.
In the light of the above statements, choose the most appropriate answer from the options given below :

We need to evaluate two statements about equipotential surfaces and electric fields.

Assertion (A): Work done by electric field on moving a positive charge on an equipotential surface is always zero. This statement is true. The work done by the electric field in moving a charge $$q$$ from point 1 to point 2 is: $$W = q(V_1 - V_2)$$. On an equipotential surface, all points have the same potential, so $$V_1 = V_2$$, which gives: $$W = q(V_1 - V_2) = q \times 0 = 0$$. This holds true regardless of the path taken along the equipotential surface.

Reason (R): Electric lines of forces are always perpendicular to equipotential surfaces. This statement is true. The electric field $$\vec{E}$$ is related to the potential by $$\vec{E} = -\nabla V$$, which means the electric field points in the direction of the steepest decrease in potential. Since the potential is constant along an equipotential surface ($$dV = 0$$), the gradient of $$V$$ (and hence $$\vec{E}$$) must be perpendicular to the surface.

Because the electric field lines are perpendicular to the equipotential surface, the component of the electric force along the surface is zero. Therefore, when a charge moves along the surface, the force and displacement are perpendicular, and the work done ($$W = \vec{F} \cdot \vec{ds} = F \cdot ds \cdot \cos 90° = 0$$) is zero. The perpendicularity directly explains why the work is zero, so (R) is the correct explanation of (A).

The correct answer is Option 4: Both (A) and (R) are correct and (R) is the correct explanation of (A).