A capacitor of capacitance C which is initially uncharged is connected with a battery. Find out heat dissipated in the circuit during the process of charging.

 

Let potential at point A is 0, so at B also 0 and at C and D it is $\epsilon$. finally, charge on the capacitor

$Q_c=\epsilon C$

$U_i=0$

$U_f=\frac{1}{2}C{V}^2=\frac{1}{2}C{\epsilon }^2$

work done by battery = $\int Pdt$

$\mathrm{W}=\int \epsilon \text{ idt }=\epsilon \int \mathrm{idt}=\epsilon ·\mathrm{Q}=\epsilon .\epsilon \mathrm{C}={\epsilon }^2\mathrm{C}$

(Now onwards remember that w.d. by battery = $\epsilon$Q if Q has flown out of the cell from high potential and w.d. on battery is $\epsilon$Q if Q has flown into the cell through high potential)

$\text{ Heat produced }=W-\left(U_f-U_i\right)={\epsilon }^{2}C-\frac{1}{2}{\epsilon }^{2}C=\frac{C{\epsilon }^2}{2}$