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

Explanation

Energy stored in the capacitor: 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 ...(1)
where `V=Q/C`
The total work done to charge a capacitor is
`W=int_0^Q Q/C dQ =Q^2/(2C)` ...(2)
this work done is stored as electrostatic potential energy `(U_E)` in the capacitor .
`U_E=Q^2/(2C) =1/2 CV^2(therefore Q=CV)` ...(3)
where Q = CV is used. This stored energy is thus directly proportional to the capacitance of the capacitor and the square of the voltage between the plates of the capacitor. But where is this energy stored in the capacitor? To understand this question, the equation (3) is rewritten as follows using the results
`C=(epsilon_0A)/d` and V=Ed
`U_E=1/2 ((epsilon_0A)/d)(Ed)^2=1/2 epsilon_0 (Ad)E^2` ....(4)
where Ad = volume of the space between the capacitor plates. The energy stored per unit volume of space is defined as energy density `u_E=U/"Volume"` . From equation (4) we get
`u_E=1/2epsilonE^2` ...(5)
From equation (5), we infer that the energy is stored in the electric field existing between the plates of the capacitor. Once the capacitor is allowed to discharge, the energy is retrieved.