Abattery charges capacitor a until the potential difference between the two conductors of the capacitor is v.
a second, identical capacitor, labeled b, is charged by a different battery until the potential difference of capacitor b is 2v.

how does the stored energy of capacitor b compare to that of capacitor a?

The energy stored in capacitor B is four times the energy stored in capacitor A

Explanation:

First let the write-out the formula for the energy stored in a capacitor in-terms of the capacitance(C)and the voltage(V)

For capacitor A charged to a voltage of V, The energy stored is

and for capacitor B, we have

From the energy stored in Capacitor A we can arrive at

if we substitute this value into the energy for capacitor B we arrive at

Hence we can conclude that the energy stored in capacitor B is four times the energy stored in capacitor A