Chapter 22.2, Problem 1aTH

On the two diagrams below, indicate the direction of the induced current in the loop at each of the instants shown. If the current is zero, state that explicitly. Explain how you determined your answers.

To determine

The direction of the induced current in the loop at each instant.

Explanation of Solution

Introduction:

Write the expression to calculate the magnetic flux.

  $\begin{array}{c}\varphi =BS\cos \theta \\ =BS\cos \omega t\end{array}$

Here, $\varphi$ is the magnetic flux, $B$ is the magnetic field, $S$ is the surface area, $\theta$ is the angle, $\omega$ is the angular speed and $t$ is the time.

Write the expression to calculate induced emf.

  $e=-\frac{d\varphi }{dt}$

Here, $e$ is the emf.

Case 1:

Consider time 1.

Calculate the magnetic flux.

  $\begin{array}{c}\varphi =BS\cos 90°\\ =BS\left(0\right)\\ =0\end{array}$

The magnetic flux will decrease as the angle reaches to $90°$ .

Calculate the induced emf.

  $\begin{array}{c}e=-\frac{d\varphi }{dt}\\ =\frac{d\left(BS\cos \left(\omega t\right)\right)}{dt}\\ =BS\omega \sin \left(\omega t\right)\end{array}$

The emf will increase which leads to the increase of current. The coil will produce a magnetic field in the same direction. According to Lenz’s law the direction of current will be anticlockwise when seen from the left side of the loop.

Case 2:

Calculate the magnetic flux.

  $\begin{array}{c}\varphi =BS\cos 0°\\ =BS\left(1\right)\\ =BS\end{array}$

The magnetic flux will decrease as the angle reaches to $90°$ .

Calculate the induced emf.

  $\begin{array}{c}e=-\frac{d\varphi }{dt}\\ =\frac{d\left(BS\cos \theta \right)}{dt}\\ =0\end{array}$

The emf is zero.

Calculate the current.

  $\begin{array}{c}I=\frac{e}{R}\\ =\frac{0}{R}\\ =0\end{array}$

For the time 2, the emf induced will be zero which will lead to zero current in the loop.

Conclusion:

Therefore, for time 1 the current will be in anticlockwise direction and for time 2 the current is zero.