In the circuit shown, the switch `S` is shifted to position `2` from position `1` at `t=0`, having been in position `1` for a long time.Find the current in the circuit as a function of time.

Answer

A, B, C

Explanation

Let `I_(0)` be the initial current in the steady state condition before shifting the swith. `therefore I_(0)=epsilon/R`
Let `I'_(0)` be the current in the circuit at time `t=0`
Since the flux associated with inductors will be same just before and after the shifting of switch, (otherwise the induced `emf` would be infinite) `phi=I_(0)L=I'_(0)(L+2L)`
or `I'_(0)=I_(0)/3=r/(3R)`
If `I` is the instantaneous current in the circuit at time `t`, then `epsilon-3L(dI)/(dt)-IR=0 rArr 3L(dI)/(dt)epsilon-IR or underset(I'_(0))overset(I)int(dI)/(epsilon-IR)=2/(3L)underset(0)overset(t)intdt`
`rArr In((epsilon-IR)/(epsilon-I'_(0)R))=(-Rt)/(3L)` or `(epsilon-IR)/(epsilon-epsilon/(3R)R)=e^((-2Rt)/(3L))`
or `epsilon-IR=(2epsilon)/3xxe^((-2Rt)/(3L))` or `I=epsilon/R(1-2/3xxe^((-2Rt)/(3L)))`