A coil of wire having inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time t=0, so that a time-dependent current `1_(1)(t)` starts following through the coil. If `I_(2)(t)` is the current induced in the ring, and B (t) is the magnetic field at the axis of the coil due to `I_(1)(t)` then as a function of time `(tgt0)`, the product `I_(2)(t)B(t)`

Options

A. increases with time
B. decreases with time
C. does not vary with time
D. passes through a mximum

Answer

D

Explanation

The magnetic field at the centre of the is `B(t) = mu_(0)nI_(1)`.
As the current increases, `B` will also increase with time till it reaches a maximum value (when the current becomes steady).
The induced emf in the ring
`e = (d phi)/(dt) = (d)/(dt) (vec(B)* vec(A))`
`= A (d)/(dt) (mu_(0)nI_(1))`
`:.` The induced current in the ring
`I_(2)(t) = (|e|)/(R ) = (mu_(0)nA)/(R ) (dI_(1))/(dt)`
`I_(2)B prop i_(1) (dI_(1))/(dt)`
`rArr I_(2) B = KI_(0) [1 - e^(-t//tau)]((I_(0))/(tau)e^(-t//tau))`
`= (KI_(0)^(2))/(tau) (e^(-t//tau)) [1 - e^(-t//tau))`
at `t = 0` and `t = oo`, `I_(2)B = 0`