Q3Q - Fundamentals Of Physics

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?

Step-by-Step Solution

Answer

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

Step 1: Given

Figure 32-21.

Step 2: 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: 

width="200" height="46" style="max-width: none; vertical-align: -15px;" $\oint \stackrel{\rightharpoonup }{B}.d\stackrel{\rightharpoonup }{s}={\mu }_0{\epsilon }_0\frac{d{f}_E}{dt}={\mu }_0{i}_{enc}$

Step 3: 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.