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

D

Explanation

The equation of `I_(1)(t), I_(2)(t)` and `B(t)` will take the following forms
`I_(1)(t) = K_(1)(1-e^(-k2t)) rarr` current growth in `L-R` circuit
`B = (t) = K_(3)(1-e^(-k 2t)) rarr B(t) prop I_(1)(t)`
`I_(2)(t) = K_(4)e^(-k2t)`
`[ I_(2)(t) = (e_(2))/(R ) and e_(2) prop = -M (dI_(1))/(dt)]`
Therefore, the product
`I_(2)(t) B(t) = K_(5)e^(-k 2t) (1-e^(-k2t))`
The value of this product zero at `t = 0` and `t = oo`. Therefore, the product will pass through a maximum value.