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
$I_1(t)=k_1\left[1-e^{-t/\tau }\right],B(t)=k_2{I}_1(t)$
$I_2(t)=k_3\frac{dB(t)}{dt}=k_4{e}^{-t/\tau }$
$I_2(t)B(t)=k_5\left[1-e^{-t/\tau }\right]\left[e^{-t/\tau }\right]$
This quantity is zero for $t=0$ and $t=\propto$ and positive for other value of $t$. It must, therefore, pass through a maximum.