A cylindrical layer of dielectric with permittivity `epsilon` is inserted into a cylindrical capacitor to fill up all the space between the electrodes. The mean radius of the electrodes equals `R`, the gap between them is equal to `d`, with `d lt lt R`. The constant voltage `V` is applied across the electrodes of the capacitor. Find the magnitude of the electric force pulling the dielectric into the capacitor.

Explanation

We known that energy of a capacitor , `U = (q^(2))/(2C)`
Hence, from `F_(x) = (del U)/(del x):|_(q =const)`, We have, `F_(x) = (q^(2))/(2) (del c)/(del x)/C^(2)` ....(1)
Now, since `d lt lt R`, the capacitance of the given capacitor can be calculated by the formula of a parallel plate capacitor. Therefore, if the dielectric is intoduced upto a depth `x` and the length of the capacitor is `l`, we have,
`C = (2pi epsilon_(0) epsilon R x)/(d) + (2pi R epsilon_(0) (l - x))/(d)` ....(2)
From (1) and (2) we get,
`F_(x) = epsilon_(0) (epsilon - 1) (pi R V^(2))/(d)`