Obtain the expression for the energy stored in a parallel plate capacitor.

Explanation

Capacitor not only stores the charge but also it stores energy. When a battery is connected to the capacitor, electrons of total charge Q are transferred from one plate to the other plate. To transfer the charge, work is done by the battery. This work done is stored as electrostatic potential energy in the capacitor.
To transfer an infinitesimal charge dQ for a potential difference V, the work done is given by
dW = VdQ
where, `V = Q/C`
The total wor done to charge a capacitor is
`W = int_0^Q Q/CdQ = Q^2/(2C)`
This work done is stored as electrostatic potential energy `(U_E)` in the capacitor
`U_E = Q^2/(2C) = 1/2 CV^2` `( :.Q = CV)`
where Q = CV is used. This sotred energy is thus directly proportional to the capacitance of the capacitor and the square of the voltage between the plates of the capacitor.
`C = (epsilon_0A)/d` and V = Ed
`U_E = 1/2((epsilon_0A)/d)(Ed)^2`
`1/2epsilon_0(Ad)E^2`
where Ad = volume of the spacw between the capacitor plates. The energy stored per unit volume of space is defined as energy density `U_E = U/("Volume")` From above equation we get
`U_E = 1/2epsilon_0E^2`