A parallel-plate capacitor has plates of area `A` and separation `d` and is charged to a potential difference `V`. The charging battery is then disconnected, and the plates are pulled apart until their separation is `2d`. Derive expression in terms of `A, d` and `V` for
(a) the new potential difference
(b) the initial and final stored energies, `U_i` and `U_f` and
(c) the work required to increase the separation of plates from d to `2d`.