A unit positive point charge is taken slowly through an infinitesimally thin tube that is inside a charged dielectric sphere of radius R , having uniform positive charge density , as shown in the figure. The initial and final positions of the charge are marked by A and B at distances 2R and 3R respectively, from the centre of the sphere. In this process, the magnitude of the total work done on the point charge is R^2

Step-by-step solution

Since the electrostatic field is conservative, the work done in moving a charge depends only on the potential difference between the initial and final positions, regardless of the path taken. The electric potential V(r) at a distance r ( r R ) from the centre of a uniformly charged solid sphere of radius R and charge density is given by: V(r) = 1 4 ₀ Q r The total charge Q of the sphere is: Q = 4 3 R^3 Substituting Q into the potential formula, we get: V(r) = 1 4 ₀ 4 3 R^3 r = R^3 3 ₀ r The initial position A is at