Question

Show that the plates of a parallel plate capacitor attract each other with a force given by

F = \(\frac{Q^2}{2\varepsilon_0A}\)

The symbols have their usual meanings.

Answer 1

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 = \(\frac{Q^2}{2\varepsilon_0A}\)