Name: Energy Density in a Capacitor
Uploaded: 2019-07-19
Duration: 3 min 1 s
Description: The energy in the capacitor is stored in the form of the electric field. Let us derive a mathematical formula to calculate this energy in terms of the electr...

Question

Calculate the energy of a charged conductor of capacitance

C

raised to a potential

V

. How does this energy depend on charge on the conductor?

Accepted Answer

Let the charge on a plate of the conductor be $Q$ and the plates to be of area $A$ , placed at distance $d$ from each other. The potential between the plates is given to be $V$ . The charge stored is directly proportional to the potential difference between its plates.

$∴ Q \propto V \Rightarrow \frac{Q}{V} = (constant)$

This constant is called the capacitance of the capacitor. For the given capacitor, the capacitance is $C$ .

$∴ \frac{Q}{V} = C \Rightarrow Q = C V . . . . (1)$

We know the formula for determining the capacitance of a capacitor is given as:

$C = \frac{ε_{0} A}{d} . . . . (2)$

Now, the force between the plates is given by:

$F = \frac{Q^{2}}{2 ε_{0} A} . . . . (3)$

The work done in bringing the plates at a distance $d$ with respect to each other is:

$W = F d \Rightarrow W = (\frac{Q^{2}}{2 ε_{0} A}) \times d (but d = \frac{ε_{0} A}{C} from (3)) \Rightarrow W = (\frac{Q^{2}}{2 ε_{0} A}) \times \frac{ε_{0} A}{C} \Rightarrow W = \frac{Q^{2}}{2 C}$

This work is stored in the capacitor in the form of a potential energy.

$∴ U = \frac{Q^{2}}{2 C}$

From the relation of potential energy of capacitor given above, It is clear that this energy is directly proportional to the square of the charge stored on the conductor.

Calculate the energy of a charged conductor of capacitance $C$ raised to a potential $V$ . How does this energy depend on charge on the conductor?

Important Questions on Capacitance and Capacitors

Learn and Prepare for any exam you want !

Calculate the energy of a charged conductor of capacitance C raised to a potential V. How does this energy depend on charge on the conductor?

Important Questions on Capacitance and Capacitors

Learn and Prepare for any exam you want !

Calculate the energy of a charged conductor of capacitance $C$ raised to a potential $V$ . How does this energy depend on charge on the conductor?