Question

Two coils A and B have inductances 1.0 H and 2.0 H respectively. The resistance of each coil is $10\Omega$ Each coil is connected to an ideal battery of emf 2.0 V at t = 0.
Let $i_A$ and $i_B$ be the currents in the two cirucit at time t. Find te ration $\frac{i_A}{I_B}$ at (a) t = 100 ms, (b) T = 200 ms and (c) t = 1 s.

Answer 1

Here $L_A=1.0H;L_B=2.0H;R=10\Omega \left(a\right)t=0.1s,t_A=0.1,T_B=\frac{L}{R}=0.2i_A=i_0(1-e^{-\frac{t}{\tau }}),=\frac{2}{10}(1-e^{\frac{0.1\times 10}{1}})=0.2(1-e^{-1})=0.126424111i_B=i_O(1-e^{-\frac{t}{\tau }})=\frac{2}{10}(1-e^{\frac{0.1\times 10}{2}})=0.2(1-e^{-\frac{1}{2}})=0.078693\frac{i_A}{i_B}=\frac{0.126411}{0.78693}=1.6\left(b\right)t=200ms=0.2s{l}_A=0.2(1-e^{-0.2\times 10.1})=0.2\times 0.864664716=0.1729329943l_B=0.2(1-e^{-0.2\times 10.2})=0.2\times 0.632120=0.126424111\therefore \frac{i_A}{i_B}=\frac{0.172932343}{0.126424111}=1.36=1.4\left(c\right)t=1s,i_A=0.2(1-e-1\times 10.1)=0.2-0.9999549i_B=0.2(1-e^{-1\times 10.2})=0.2\times 0.99326=0.19865241\therefore \frac{i_A}{i_B}=\frac{0.19999092}{0.19999092}\approx 1.0$