Question

A spherical shell of radius R is uniformly charged with net charge Q and a point charge Q is placed at the centre of shell. The amount of work done by external agent to slowly increase the radius of shell to 2R is

Answer 1

-\frac{3Q^2}{16\pi\epsilon_0 R}

Answer 2

The work done by an external agent to slowly change a system's configuration is equal to the change in its total electrostatic potential energy ($\Delta U$). The total potential energy consists of the self-energy of the spherical shell and the interaction energy between the central point charge and the shell. The self-energy of a shell with charge Q and radius r is $Q^2/(8\pi\epsilon_0 r)$. The interaction energy between a point charge Q at the center and a shell of charge Q and radius r is $Q^2/(4\pi\epsilon_0 r)$. Calculate the initial total potential energy ($U_i$) with radius R and the final total potential energy ($U_f$) with radius 2R. $U_i = \frac{Q^2}{8\pi\epsilon_0 R} + \frac{Q^2}{4\pi\epsilon_0 R} = \frac{3Q^2}{8\pi\epsilon_0 R}$ $U_f = \frac{Q^2}{8\pi\epsilon_0 (2R)} + \frac{Q^2}{4\pi\epsilon_0 (2R)} = \frac{3Q^2}{16\pi\epsilon_0 R}$ The work done is $W_{ext} = U_f - U_i = \frac{3Q^2}{16\pi\epsilon_0 R} - \frac{3Q^2}{8\pi\epsilon_0 R} = -\frac{3Q^2}{16\pi\epsilon_0 R}$.