Question

In the following circuit, the switch is closed at $t=0.$ Find the currents $i_1,i_2,i_3$ and $d{i}_3/dt$ at $t=0$ and at $t=\infty$ . Initially, all currents are zero.

Answer 1

At $t=0$, inductor will act as open circuit and no current will flow in that branch.

Hence, $i_1=i_2=\frac{E}{R+R}=\frac{E}{2R}$

And, $i_3=0$

At $t=\infty$, inductor will act as closed circuit and will offer no resistance.

Hence, $i_1=\frac{E}{R+R/2}=\frac{2E}{3R}$

And, $i_2=i_3=\frac{i_1}{2}=\frac{E}{3R}$

Current through inductor is given by-

$i=i_o(1-e^{-t/\tau })$

Here, $i_o=\frac{E}{3R}=$max current

And, $\tau =\frac{L}{R}$

Hence, $i_3=\frac{E}{3R}(1-e^{-Rt/L})$

$⟹\frac{d{i}_3}{dt}=\frac{E}{3L}{e}^{-Rt/L}$

At, $t=0$ $\frac{d{i}_3}{dt}=\frac{E}{3L}$

At, $t=\infty$ $\frac{d{i}_3}{dt}=0$