b
(b) After inserting the dielectric slab
New capacitance \(C' = K.C = \frac{{K{\varepsilon _0}A}}{d}\)
New potential difference \(V' = \frac{V}{K}\)
New charge \(Q' = C'V' = \frac{{{\varepsilon _0}AV}}{d}\)
New electric field \(E' = \frac{{V'}}{d} = \frac{V}{{Kd}}\)
Work done \((W) =\) Final energy \(-\) Initial energy
W \( = \frac{1}{2}C'V{'^{\,2}} - \frac{1}{2}C{V^2}\) \( = \frac{1}{2}(KC)\,{\left( {\frac{V}{K}} \right)^2} - \frac{1}{2}C{V^2}\)
\( = \frac{1}{2}C{V^2}\left( {\frac{1}{K} - 1} \right) = - \frac{1}{2}C{V^2}\left( {1 - \frac{1}{K}} \right)\)
\( = - \frac{{{\varepsilon _0}A{V^2}}}{{2d}}\left( {1 - \frac{1}{K}} \right)\) so \(|W|\, = \frac{{{\varepsilon _0}A{V^2}}}{{2d}}\left( {1 - \frac{1}{K}} \right)\).