Question

Derive an expression for energy stored in a capacitor. In which form energy is stored?

Answer 1

Hint: Energy is stored in the form of potential energy. A capacitor is a device consisting of two dielectric plates in which electrical energy is being stored in an electric field. Here we use the idea of energy stored in the capacitor.

Complete step by step answer:
A capacitor is a device consisting of two dielectric plates in which electrical energy is being stored in an electric field.
Now let us calculate the energy stored in a capacitor,
The capacitance of the capacitor is given as c and V be the potential energy between the dielectric plates
The charge on each plate is +q and -q on the other. Let the charging of capacitors be a gradual process. Now, the charge on the capacitor is q every time.
Therefore, the potential difference = \[V=\dfrac{q}{c}\] .
The work done in giving an additional charge to the capacitor is \[\partial W=\dfrac{q}{C}\times \partial q\]
Therefore, the total work done when giving a total charge Q to the capacitor will be,\[W=\int{\partial W}\]
Providing the limits from 0 to +Q
\[W=\int\limits_{0}^{Q}{\dfrac{Q}{C}}\]
$W=\dfrac{1}{C}\times \left[ \dfrac{{{Q}^{2}}}{2} \right]_{0}^{Q}$
After performing the integration, $W=\dfrac{{{Q}^{2}}}{2C}$$Q=CV$
As we all know, the work done in a capacitor is stored as energy there.
Therefore, energy stored in capacitor=E
And $Q=CV$
We can write that,
\[E=\dfrac{{{Q}^{2}}}{2C}=\dfrac{C{{V}^{2}}}{2}=\dfrac{QV}{2}\]
In a capacitor, energy is stored in the form of potential energy. Therefore the answer for the question is \[E=\dfrac{{{Q}^{2}}}{2C}\]
Where Q is the total charge stored in the capacitor and C is the capacitance of the capacitor.

Note:
The limits given to the integration should be taken care of. It is from 0 to Q not from 0 to q. The energy stored in a capacitor is a scalar quantity. Hence the answer for the question is \[E=\dfrac{{{Q}^{2}}}{2C}\]