Question

By evaluation $\int iRdt,$ show that when a capacitor is a charged by connecting in to a battery through a resistor, the energy dissipate as heat equals the energy stored in the capacitor.

Answer 1

For capacitive circuit:

$i=i_0{e}^{-t/RC}$

Form given condiation'

$\int i^{2}Rdt=\int i_0^{2}R{e}^{-2t/RC}dt=i_0^{2}R\int e^{-2t/RC}dt$

$=i_0^{2}R(-RC/2)e^{-2t/RC}=\frac{1}{2}C{i}_0^2{R}^2{e}^{-2t/RC}=\frac{1}{2}C{V}^2$ (Proved).