If r is the separation between the plates of the capacitor, the energy stored U is given by
U = \(\frac{Q^2}{2C} = \frac{Q^2}{2(\varepsilon_0\frac{A}{r})} = \frac{Q^2r}{2\varepsilon_0A}\) ........................(i) [∵C = \(\frac{\varepsilon_0A}{d}\) \(\)]
If the plate separation is increased to r + dr, the energy stored is given by
U' = \(\frac{Q^2(r +dr)}{2\varepsilon_0A}\) .................(ii)
Therefore, the work done to increase the plate separation by dr is
dw = U' - U = \(\frac{Q^2dr}{2\varepsilon_0A}\) ..........................(iii)
If F is the force between the plates of the capacitor, then work done dW is also given by
dW = Fdr .......................(iv)
From eq. (iii) and eq. (iv), we have
F dr = \(\frac{Q^2dr}{2\varepsilon_0A}\)
or F = \(\frac{Q^2}{2\varepsilon_0A}\)