Question

Figure 32-21 shows, in two situations, an electric field vector $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$and an induced magnetic field line. In each, is the magnitude $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$increasing or decreasing?

Short answer

In both situations,$\stackrel{\rightharpoonup }{\mathrm{E}}$ is decreasing.

Step-by-step solution

Given

Figure 32-21.

Determining the concept

From Maxwell's law of induction, the relation between magnetic flux induced due to electric field and enclosed current. Then using the right-hand thumb rule, find the directions of the enclosed current in both situations. From this, conclude whether the magnitude$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$is increasing or decreasing.

The formula is as follows:

role="math" localid="1663041558546" width="200" height="46">∮B⇀.ds⇀=μ0ε0dfEdt=μ0ienc

Determining in each situation, whether the magnitude E⇀ is increasing or decreasing

Maxwell’s law of induction gives

$\oint \stackrel{\rightharpoonup }{B}.d\stackrel{\rightharpoonup }{s}={\mu }_0{\epsilon }_0\frac{d{f}_E}{dt}={\mu }_0{i}_{enc}$

This implies that it $\stackrel{\rightharpoonup }{E}$ is increasing, i_enc is in the direction of the electric field, and vice versa. According to the right-hand thumb rule, i_enc is upward in situation (a) and downward in the situation (b).

Hence, in both situations,$\stackrel{\rightharpoonup }{E}$is decreasing.