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 $I_1\left(t\right)$ starts flowing through the coil. If $I_2\left(t\right)$ is the current induced in the ring. and B(t) is the magnetic field at the axis of the coil due to then as a function of time $(t>0)$, the product $I_2\left(t\right)B\left(t\right)$

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

Answer 1

The correct option is D Passes through a maximum

Using $k{1}_{,}{k}_2$ etc, as different constants.
$I_1\left(t\right)=k_1[1-e^{-t/\tau }],B\left(t\right)=k_2{I}_1\left(t\right)$
$I_2\left(t\right)=k_3\frac{dB\left(t\right)}{dt}=k_4{e}^{-t/\tau }$
$\therefore I_2\left(t\right)B\left(t\right)=k_5[1-e^{-t/\tau }]\left[e^{-t/\tau }\right]$
This quantity is zero for $t=0andt=\infty$ and positive for other value of t. It must, therefore, pass through a maximum.