Question

The inductors of two $LR$ circuits are placed next to each other, as shown in the figure. The values of the self-inductance of the inductors, resistances, mutual-inductance and applied voltages are specified in the given circuit. After both the switches are closed simultaneously, the total work done by the batteries against the induced $EMF$ in the inductors by the time the currents reach their steady state values is _____ $\text{mJ}$.

Answer 1

Mutual inductance is producing flux in same direction as self inductance.
$\Rightarrow U=\frac{1}{2}{L}_1{I}_1^2+\frac{1}{2}{L}_2{I}_2^2+M{I}_1{I}_2$
Here
$I_1=\frac{5}{5}=1\text{A}$ and
$I_2=\frac{20}{10}=2\text{A}$

$\Rightarrow U=\frac{1}{2}\times (10\times {10}^{-3})1^2+\frac{1}{2}\times (20\times {10}^{-3})\times 2^2+(5\times {10}^{-3})\times 1\times 2$

$\therefore U=55\text{mJ}$