Question

In the following circuit (Fig.)the switch is closed at `t = 0`. Intially, there is no current in inductor. Find out the equation of current in the inductor coil as s function of time.

Answer 1

At any time `t, - epsilon + i_(1) R - (i - i_(1)) R = 0`
`- epsilon + 2i_(1) R - iR = 0`
`i_(1) = (iR + epsilon)/(2R)`

Now
`- epsilon + i_(1)R + iR + L(di)/(dt) = 0`
`- epsilon+ ((iR + epsilon)/(2)) + iR + L(di)/(dt) = 0`
`(epsilon)/(2) + (3iR)/(2) = - L(di)/(dt)`
`((epsilon + 3iR)/(2)) dt = - L di rArr - int_(0)^(t) (dt)/(2L) = int_(0)^(i) (di)/(-epsilon + 3iR)`
`(t)/(2L) = (1)/(3R)` In `((- varepsilon + 3iR)/(-varepsilon)) rArr -In ((- varepsilon + 3iR)/(-varepsilon)) = (3Rt)/(2L)`
`i = + (varepsilon)/(3R) (1 - e^(-(3Rt)/(2L)))`