JEE Advanced · 2017 · 21 May 2017 · Paper 2 · Electromagnetic Induction

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?

(A)After a long time, the current through $L_{1}$ will be $\frac{V}{R} \frac{L _{2}}{L _{1} + L _{2}}$ ✓ Correct
(B)After a long time, the current through $L_{2}$ will be $\frac{V}{R} \frac{L _{1}}{L _{1} + L _{2}}$ ✓ Correct
(C)The ratio of the currents through $L_{1}$ and $L_{2}$ is fixed at all times $(t > 0)$ ✓ Correct
(D)At $t = 0$ , the current through the resistance $R$ is $\frac{V}{R}$

Answer: (A), (B), (C)

Solution

Since the inductors are in parallel and there's no mutual inductance, the voltage across both is the same at all times. This gives

L_{1} \frac{d i _{1}}{d t} = L_{2} \frac{d i _{2}}{d t}

, leading to

L_{1} i_{1} = L_{2} i_{2}

(constant ratio). After a long time, inductors act as short circuits, so

i_{1} + i_{2} = \frac{V}{R}

. Solving:

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}}

. At

t = 0

, inductors block current, so current through R is zero, not

V / R

JEE Advanced · 2017 · 21 May 2017 · Paper 2 · One or more correct options