Question

The work done in increasing the potential of a capacitor from $V$ volts to $2V$ volts is $W$. If the work done in increasing the potential of the same capacitor from $2V$ volts to $4V$ volts will be $nW$, then find the value of $n$

Answer 1

The energy of the capacitor $E=\frac{1}{2}C{V}^2$ work done in increasing the potential of the capacitor from $V$ to $2V$ is
$W=\frac{1}{2}C\left[\right(2V{)}^2-\left(V{)}^2\right]=\frac{1}{2}C[4V^2-V^2]W=\frac{1}{2}C(3V{)}^2=\frac{3V^{2}C}{2}....(1)$
from $2V$ to $4V$ work done is -
$W^{\prime }=\frac{1}{2}C\left[\right(4V{)}^2-\left(2V{)}^2\right]=\frac{1}{2}C[16V^2-4V^2]$
$W^{\prime }=\frac{1}{2}C\left(12V^2\right).......\left(2\right)$
From equation (1) and (2)
$\frac{W^{\prime }}{W}=\frac{12V^{2}C}{2}\times \frac{2}{3V^{2}C}$
$W^{\prime }=4W$