Question

Derive the expression for energy stored in a charged capacitor.

Answer 1

To find the electrical energy stored in a capacitor, we calculate the work required to change it. Suppose charge $Q^{\prime }$ has been deposited on the conducting plates of the capacitor, and with a constant potential difference of $V^{\prime }=\frac{Q^{\prime }}{C}$. To add more charge $d{Q}^{\prime }$, the work needed is the potential energy gained by the charge $d{Q}^{\prime }$, which is $d{Q}^{\prime }\times V^{\prime }$. Hence

$dW=d{Q}^{\prime }\times V^{\prime }=\frac{Q^{\prime }\times d{Q}^{\prime }}{C}$.

On integrating we get

$U{=}_0{\int }^{U}dW$

${=}_0{\int }^Q\frac{Q^{\prime }\times d{Q}^{\prime }}{C}$

$=\frac{Q^2}{2C}=\frac{1}{2}C{V}^2$