a
(a) In case of a charged conducting sphere
\({V_{{\rm{inside}}}} = {V_{{\rm{centre }}}} = {V_{{\rm{surface}}}} = \frac{1}{{4\pi {\varepsilon _o}}}.\frac{q}{R}\), \({V_{{\rm{outside}}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{q}{r}\)
If \(a\) and \(b\) are the radii of sphere and spherical shell respectively, then potential at their surface will be
\({V_{{\rm{sphere }}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{Q}{a}\) and \({V_{{\rm{shell}}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{Q}{b}\)
\(\therefore \)\(V = {V_{{\rm{sphere }}}} - {V_{{\rm{shell}}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\left[ {\frac{Q}{a} - \frac{Q}{b}} \right]\)
Now when the shell is given charge \((-3Q)\), then the potential will be
\(V{'_{{\rm{sphere}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{Q}{a} + \frac{{( - 3Q)}}{b}} \right],\)\(V{'_{{\rm{shell}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{Q}{b} + \frac{{( - 3Q)}}{b}} \right]\)
\(\therefore \)\(V{'_{{\rm{sphere }}}} - V{'_{{\rm{shell}}}} = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{Q}{a} - \frac{Q}{b}} \right] = V\)