ENERGY STORED IN A CHARGED CAPACITOR (ELECTROSTATIC ENERGY IN A CAPACITOR)
Derive an expression for the energy stored in a charged capacitor. In what form the energy is stored in a capacitor?
The process of charging a capacitor is equivalent to that of transferring charge from one plate to the other plate of the capacitor. At any stage of the charging, there is a potential difference between the plates of the capacitor.
Therefore, some work must be done to transfer charge from one plate to another plate of the capacitor. This work done is stored as electrostatic potential energy in the capacitor.
Let at any instant, a charge $q$ be on the capacitor. Then potential difference between the plates of the capacitor is given by $V = q/C$.
If extra charge $dq$ is transferred to the capacitor, then work done to do so is given by
$dW = V dq = \frac{q}{C} dq \quad \dots (i)$
If the final charge on the capacitor is $Q$, then the total work done is given by
$W = \int dW = \int_0^Q \frac{q}{C} dq = \frac{1}{C} \int_0^Q q \, dq$
$ = \frac{1}{C} \left[ \frac{q^2}{2} \right]_0^Q$
$= \frac{1}{C} \left[ \frac{Q^2}{2} \right] = \frac{Q^2}{2C} \quad \dots (ii)$
This work done is stored as the electrostatic potential energy (U) of the capacitor. Hence, eqn. (ii) can be written as
$U = \frac{1}{2} \frac{Q^2}{C} \quad \dots (iii) $
which is the expression for the energy stored in the capacitor.
Other Forms of eqn. (iii)}
We know that, $Q = CV$
Hence eqn. (iii) becomes
$U = \frac{1}{2} \frac{(CV)^2}{C} = \frac{1}{2} CV^2 \quad \dots (iv)$
Also,
$C = \frac{Q}{V}$
eqn. (iii) becomes
$U = \frac{1}{2} \frac{Q^2}{Q/V} = \frac{1}{2} QV \quad \dots (v)$
Thus, energy stored in a capacitor is given by
$U = \frac{1}{2} CV^2 = \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} QV \quad \dots (vi)$
The electrostatic potential energy of a capacitor is stored in the form of electric or electrostatic field between the plates of the capacitor.
Derive an expression for energy density in the case of a parallel plate capacitor. Give S.I. unit and dimensional formula of energy density.
Energy Density: Energy stored per unit volume of the space between the plates of the capacitor is known as energy density.
Consider a parallel plate capacitor of capacitance $C$. Let $A$ be the area of each plate and $d$ be the distance between the two plates of the capacitor. When the capacitor is charged to voltage $V$, the energy stored in the capacitor is given by
$U = \frac{1}{2} CV^2 \quad \dots (i)$
But capacitance of a parallel plate capacitor is given by
$C = \frac{\epsilon_0 A}{d} \quad \dots (ii)$
and potential difference, $V = Ed \dots (iii)$
Using eqn. (ii) and (iii) in eqn. (i), we get
$U = \frac{1}{2} \epsilon_0 E^2 Ad$
Since volume of the capacitor $= Ad$
$\therefore \text{Energy stored per unit volume} = \frac{U}{Ad} = \frac{1}{2} \epsilon_0 E^2 \quad \dots (iv)$
But energy stored per unit volume of a capacitor is known as energy density $(U_d)$.
$\therefore \text{energy density}, \quad U_d = \frac{1}{2} \epsilon_0 E^2 = \frac{1}{2} \epsilon_0 \vec{E} \cdot \vec{E} \quad \dots (v)$
S.I. unit of energy density is J m$^{-3}$.
Dimensional formula of energy density $= \frac{[\text{energy}]}{[\text{volume}]} = \frac{[ML^2 T^{-2}]}{[L^3]} = [ML^{-1} T^{-2}]$