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

Derive an expression for energy stored in a charged capacitor.

Accepted Answer

Consider, the capacitance of the capacitor is $C$ and the constant potential difference between the capacitor plates is $V$ .

Assuming $q$ charge is present on the capacitor plates. Then, the potential difference between the plates will be: $V = \frac{q}{C}$

Now to add small amount of charge $(d q)$ is to capacitor, the work done will be: $d W = \frac{q}{C} d q$

So, to add $Q$ charge to the capacitor, the required work done will be:

$\int d W = \int_{0}^{Q} \frac{q}{C} d q \Rightarrow W = \frac{Q^{2}}{C}$

The energy stored in the capacitor will be half of the work done to the charge added.

That is, $E = \frac{1}{2} W = \frac{1}{2} \frac{Q^{2}}{C} = \frac{1}{2} C V^{2}$

Hence, the energy stored in a charged capacitor is $\frac{1}{2} C V^{2}$ .

Derive an expression for energy stored in a charged capacitor.

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