The current in Fig. E29.18 obeys the equation $\mathit{I}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{I}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{-}\mathbf{b}\mathbf{t}}$, where b > 0. Find the direction (clockwise or counter clockwise) of the current induced in the round coil for t > 0.

We have a circular loop that is close to a long wire carrying a current as shown in the following figure. The current is given by $I\left(t\right)=I_0{e}^{-bt}$.

According to this function, the current is decreasing as time passes. We need to find the direction of the current in the loop. The magnetic field due to a very long wire is inversely proportional to the distance from the wire and directly proportional to the current since the distance from the wire is constant, so only the current plays the role in our case. Therefore, when the current decreases, the magnetic field also decreases and hence, the magnetic flux through the loops decreases.

So, according to the Lenz’s law, the induced current in the loop must flow such that to oppose the external magnetic field, to oppose the decrease in the magnetic field, the induced magnetic field must point out of the page.

Hence, the induced current must flow counter-clockwise.