Irodov Problem 3.41

Without loss of generality, let us choose the direction of the dipole moment as the z-axis. The external uniform electric field is thus also aligned along the z-axis.

Consider a sphere of radius R with its center at the origin. From Problem 3.39 we know that the potential at any point on this sphere in the absence of the electric field is given by,





In other words the potential difference between any two points (x1,y1,z1) and (x2,y2,z2) in the absence of the external electric field is given by,







Since the external electric field is aligned with the electric dipole moment, it is along the z-axis. The additional potential difference induced due to the electric field between points is given by E(z1-z2). The net potential difference between at any two points on this sphere are thus given by,







On an equipotential surface the potential difference between any two points is zero. Hence, for this spherical surface to be equipotential,