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 starts flowing through the coil. If $I_1$ is the current induced in the ring, and $B$ is the magnetic field at the axis of the coil due to $I_2$, 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

The correct option is D Passes through a maximum

Using $k_1,k_2$ etc, as different constants.
$I_1\left(t\right)=k_1[1-e^{\frac{-t}{\tau }}],B\left(t\right)=k_2{I}_1\left(t\right)$
$I_2\left(t\right)=k_3\frac{dB\left(t\right)}{dt}=k_4{e}^{\frac{-t}{\tau }}$
$\therefore l_2\left(t\right)B\left(t\right)=k_5[1-e^{\frac{-t}{\tau }}]\left[e^{\frac{-t}{\tau }}\right]$
This quantity is zero for t=0 and $t=\infty$ and positive for other value of t. It must , therefore, pass through a maximum.