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A source of constant voltage V is connected to a resistance R and two ideal inductors $L_{1}$ and $L_{2}$ through a switch S as shown. There is no mutual inductance between the two inductors. The switch S is initially open. At t=0, the switch is closed and current begins to flow. Which of the following options is/are correct?

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After a long time, the current through $L_{1}$ will be $\frac{V}{R} \frac{L_{2}}{L_{1} + L_{2}}$

After a long time, the current through $L_{2}$ will be $\frac{V}{R} \frac{L_{1}}{L_{1} + L_{2}}$

The ratio of the currents through $L_{1}$ and $L_{2}$ is fixed at all times $(t > 0)$

At t=0, the current through the resistance R is $\frac{V}{R}$

answer is A, B, C.

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Detailed Solution

$After a long time, t = \infty, inductor behaves as conducting wire$

$\begin{array}{l} i = \frac{V}{R} and L_{1} i_{1} = L_{2} i_{2} and i = i_{1} + i_{2} \\ \Rightarrow i_{1} = \frac{L_{2}}{L_{1} + L_{2}} i = \frac{V}{R} \frac{L_{2}}{L_{1} + L_{2}} \\ and i_{2} = \frac{V}{R} \frac{L_{1}}{L_{1} + L_{2}} \end{array}$

$At t = 0, inductor behaves like infinte resistance, so no current pass through resistor.$

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