Sol. When switch S is closed, battery forces a current in the circuit which increases gradually. Due to increase in current, emf is induced in both the solenoids. Since, two solenoids are in parallel with each other, therefore, emf in the two solenoids is always equal to each other, though these emfs vary with time.

If at an instant t current through solenoids L 1 and L 2 be i 1 and i 2 respectively, then magnitude of induced emf in the solenoids will be

| e 1 | = L and | e 2 | = L 2 , respectively.

But | e 1 | = | e 2 |

∴ L 1 = L 2

di 2 = di 1

Integrating i 2 = i 1 ….. (1)

According to Len’z law, induced emfs will oppose increase of current in respective solenoids. Hence, at time t, current and polarities of induced emfs in the circuit will be as shown in Fig.

Applying Kirchhoff’s voltage law on outer mesh,

+ | e 1 | = (i 1 + i 2 )R – E = 0

∴ L 1 + R – E = 0.

or L 1 =

or = ….. (2)

But at t = 0, i 1 = 0

Integrating equation (2), =

i 1 =

Substituting this value in equation (1),

i 2 =