A point charge Q is fixed at the centre O of a square ABCD. M, N and P are the midpoints of the sides AB, CD and BC respectively. A small test charge q is carried slowly between each pair of points listed below, along a path that does not pass through O.
(A) From corner A to corner B
(B) From corner A to corner C
(C) From corner A to the point M
(D) From the point M to the point N
(E) From the point M to the point P
In how many of the above cases is the work done by the electric field of Q zero?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4
The work done by the field of Q in carrying q from one point to another is \( q(V_{\mathrm{initial}}-V_{\mathrm{final}}) \), whatever the path. The equipotential surfaces of a point charge are spheres centred on it. So the work is zero whenever the two points are at the same distance from O. For a square of side a, every corner is at \( a/\sqrt{2} \) from O and the midpoint of every side is at \( a/2 \).
The moves in which the work done is zero are:
(A) A to B, two corners
(B) A to C, two corners
(D) M to N, two midpoints
(E) M to P, two midpoints
The move in which the work done is not zero is:
(C) A to M — the corner A is at \( a/\sqrt{2} \) from O but M is at \( a/2 \), so the two points are at different potentials.
Total = 4
NCERT (2026-27):
Equipotential Surfaces – “An equipotential surface is a surface with a constant value of potential at all points on the surface. For a single charge q, the potential is given by …”
\( \displaystyle V=\frac{1}{4\pi\varepsilon_0}\,\frac{q}{r} \)
“This shows that V is a constant if r is constant. Thus, equipotential surfaces of a single point charge are concentric spherical surfaces centred at the charge.”
Equipotential Surfaces – “… there is no potential difference between any two points on the surface and no work is required to move a test charge on the surface. …”