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 maximum

Answer

B

Explanation

The magnetic field at the centre of the coil.
`B()=(mu_0)n(I_1)`
As the current increases, B will also increases with time till it reaches a maximum value (when the current become steady).
The induced emf in the ring
`e=-(d phi)/(dt)``=-(d)/(dt)((vec B)(vecA))=-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)`
where `(I_1)-(I_(max))(1-(e^(-t//tau))`
The relevant graphs are
.