Question

A parellel plate capacitor with plate speration d is charged with a battery so that energy stored is `"U"`. If a plate of dielectric `"K"` and thickness d is placed betweeen the plates of capacitor and battery remains connected. The work by battery is?

Options:

1. `"KU"`

2. `"2U(K - 1)"`

3. `"U(K - 1)"`

4. `"2KU"`

Answer 1

`2U(K-1)`

Explanation:

The energy stored in a capacitor is given by `U = 1/2 CV^2` where `C` is the capacitance and `V` is the potential.

When the dielectric is introduced, the capacitance increases by a factor of `K`, but since the battery is still connected, the potential `V` does not change.

So, the charge stored in the capacitor grows from `CV` to `K times CV`. The battery must pump in this extra charge `(K-1)CV` at the constant voltage `V` - and the work it must do for this is
`(K-1)CV times V = (K-1)CV^2=2(K-1)U`

Note that this is twice the energy gained by the capacitor.