Question

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 $I_{2}(t) B(t)$

• A. Increases with time
• B. Decreases with time
• C. Does not vary with time
• D. Passes through a maximum

Answer 1

Answer: (D)

Using $k_{1}, k_{2}$ etc. as different constants, we have
$I_{1}(t)=k_{1}\left[1-e^{-t / \tau}\right], B(t)=k_{2} I_{1}(t)$
$I_{2}(t)=k_{3} \frac{d B(t)}{d t}=k_{4} e^{-t / \tau}$
$I_{2}(t) B(t)=k_{5}\left[1-e^{-t / \tau}\right]\left[e^{-t / \tau}\right]$
This quantity is zero for $t=0$ and $t=\propto$ and positive for other value of $t$. It must, therefore, pass through a maximum.