A source of constant voltage is connected to a resistance and two ideal inductors and through a switch as shown. There is no mutual inductance between the two inductors. The switch is initially open. At , the switch is closed and current begins to f

Question

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 [2017]

Smart Achievers · Accepted Answer

(1, 2, 3)

At $R$ inductors $L_{1}$ and $L_{2}$ will offer infinite resistance, hence current through circuit is zero.

After a long time, the current through the resistor is constant.

I will divide into two parts through $L_{1}$ and $L_{2}$ , which are in parallel.

$∴$ $I_{1} L_{1} = I_{2} L_{2}$ $(I = I_{1} + I_{2})$

$I_{1} = \frac{V}{R} (\frac{L_{2}}{L_{1} + L_{2}})$ and $I_{2} = \frac{V}{R} (\frac{L_{1}}{L_{1} + L_{2}})$

Also, the ratio of currents through $L_{1}$ and $L_{2}$ is fixed at all times.

At $t = 0,$ $I \approx 0$

Solution

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

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

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

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

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Solution

1 After a long time, the current through L1 will be VRL2L1+L2

2 After a long time, the current through L2 will be VRL1L1+L2

3 The ratio of the currents through L1 and L2 is fixed at all times (t>0)

4 At t=0, the current through the resistance R is VR

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

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

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

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