question_answer

A coil of wire having finite 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 \[{{I}_{1}}(t)\] starts flowing 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 (t > 0), the product I2 (t) B(t) [IIT-JEE (Screening) 2000]

A)            Increases with time          

B)            Decreases with time

C)            Does not vary with time  

D)            Passes through a maximum

Correct Answer: D

Solution :

                   Using k1, k2 etc, as different constants.                    \[{{I}_{1}}(t)={{k}_{1}}[1-{{e}^{-t/\tau }}],\ B(t)={{k}_{2}}{{I}_{1}}(t)\]                    \[{{I}_{2}}(t)={{k}_{3}}\frac{dB(t)}{dt}={{k}_{4}}{{e}^{-t/\tau }}\]                    \[\therefore {{I}_{2}}(t)\ B(t)={{k}_{5}}[1-{{e}^{-t/\tau }}][{{e}^{-t/\tau }}]\]                    This quantity is zero for \[t=0\] and \[t=\infty \] and positive for other value of t. It must, therefore, pass through a maximum.