4.4 Energy Store in parallel plate Capacitor:

The capacitance of a conductor is given by, $q = CV$.

While the capacitor is connected across a battery, charges come from the battery and get stored in the capacitor plates. But this process of energy storing is step by step only.

In the beginning, the capacitor does not have any charge or potential. 

i.e. $V = 0$ and $q = 0$.

After applying the potential from the battery, the work has to be done to transfer charges onto a conductor, against the force of repulsion from the already existing charges on it. This work is stored as a potential energy of the electric field of the conductor.

Suppose a parallel plate capacitor having capacitance $C$ to store the charge $q$ when connected across the cell of potential $V$. So, we have

$q = CV$ .......... $(i)$

When charge $dq$ is passed to the capacitor at constant $V$, then small amount of workdone is given by,

$dW = V\;dq$

$dW =$ $ \frac{q}{C}$$.dq$ .......... $(ii)$                    [∵ From equation $(i)$]

∴ The total workdone for passing the total charge $Q$ is given by,

or, $\int{dw} = $ $\int_{0}^{q} \frac{q}{C}$$.dq = $$\frac{1}{C}\int_{0}^{q}$q$.dq$

or, $ W = $ $\frac{1}{C}[\frac{q^2}{2}]_{0}^{q}$

or, $W = $ $\frac{1}{C} \frac{q^2}{2}$

But from equation $(i)$ $q = CV$, then we get

or, $W = $ $\frac{C^2V^2}{2C} = \frac{1}{2}$$CV^2$ .......... $(iii)$

This workdone is store in capacacitor is in the form of energy.

∴ $U = W = $ $\frac{1}{2}$$CV^2$ .......... $(iv)$

This is the required expression of energy store in a capacitor.

Energy Density:

It is defined as energy store in a capacitor per unit volume.

i.e. Energy density $= $ $\frac{U}{V}$

                               $ = $ $\frac{1}{2}\frac{CV^2}{A*d}$                             [∵ Volume $(V) = A * d$]

Also, $C = $ $\frac{\epsilon_0\;A}{d}$

or, Energy density $=$ $\frac{1}{2} . \frac{\epsilon_0\;A}{d} . \frac{V^2}{A * d}$

             $= $ $\frac{1}{2} \epsilon_0 (\frac{V}{d})^2$

            $ = $ $\frac{1}{2}\epsilon_0$$\;E^2$   [∵ Where $E =$ $ \frac{V}{d}$ is the electric field intensity between the plates]

If the space between the capacitor plates is filled with a dielectric having permittivity $\epsilon$, then 

Energy Density $ = $ $\frac{1}{2}\epsilon$$\;E^2$

or, Energy Density ∝ $E^2$

Thus, the energy density in any dielectric is directly proportional to the square of the electric field in that region.

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