c
(c) Initial energy of the system
\({U_i} = \frac{1}{2}C{V_1}^2 + \frac{1}{2}C{V_2}^2\)
When the capacitors are joined, common potential \(V = \frac{{C{V_1} + C{V_2}}}{{2C}} = \frac{{{V_1} + {V_2}}}{2}\)
Final energy of the system
\({U_f} = \frac{1}{2}(2C){V^2} = \frac{1}{2}2C\,{\left( {\frac{{{V_1} + {V_2}}}{2}} \right)^2} = \frac{1}{4}C{({V_1} + {V_2})^2}\)
Decrease in energy = \({U_i} - {U_f} = \frac{1}{4}C{({V_1} - {V_2})^2}\)