Question

Find the current in $AB$ at time $t$. What is its limiting value as $t\to \infty$?

Answer 1

At $t=0$ capacitor offers zero resistance
$R_{net}=\frac{3R}{2}$
$\therefore i=\frac{V}{3R/2}=\frac{2V}{3R}$
$i_C=i_{AB}=\frac{1}{2}=\frac{V}{3R}$
At $t=\infty$ Capacitors offers infinite resistance. So, $i_C=0$.
$\therefore i_{battery}=i_{AB}=\frac{V}{2R}$
Now current through $AB$, increases exponentially from $\frac{V}{3R}$ to $\frac{V}{2R}$ with same time constant.
$i-t$ graph is as shown below.
$i-t$ equation correspondent to this graph is
$i=\frac{V}{3R}+\frac{V}{6R}(1-e^{-t/{\tau }_C})$