Question

Answer the following question.
Obtain the expression for the energy stored in a capacitor connected across a dc battery. Hence define energy density of the capacitor

Answer 1

Energy Stored in a Charged Capacitor:
The energy of a charged capacitor is measured by the total work done in charging the capacitor to a given potential.
Let us assume that initially both the plates are uncharged. Now, we have to repeatedly remove small positive charges from one plate and transfer them to the other plate.
Let
q →Total quantity of charge transferred
V →Potential difference between the two plates
Then,
q = CV
Now, when an additional small charge dq is transferred from the negative plate to the positive plate, the small work done is given by,

`dW = Vdq = q/"C" dq`

The total work done in transferring charge Q is given by,
`W = int_0^Q q/"C"dq = 1/"C"int_0^Qqdq = 1/"C"[q^2/2]_0^Q`

`W = Q^2/(2"C")`

This work done is stored as electrostatic potential energy U in the capacitor.
`U = Q^2/(2"C")`

Hence energy stored in the capacitor = `1/2 Q^2/"C" = (Asigma)^2/2 xx d/(epsilon_oA)`
The surface charge density `sigma` is related to the electric field `E` between the plates, `E` = `sigma/epsilon_o`
So, energy stored in the capacitor = `1/2epsilon_oE^2 xx Ad`
Here, Ad is volume between the plates of capacitor.
We define energy density as energy stored per unit volume of space.
Energy density of electric field = `U = 1/2epsilon_oE^2`