91Ó°ÊÓ

Figure 32-23.

FromMaxwell’s law of induction, the relation between $\mathbf{âˆ\circledR }\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}\mathbf{·}\mathit{d}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{s}}$andtheenclosed current can be found. Comparingthearea oftheplate enclosed bytheloop, find the ranking of loops according to the magnitude of$\mathbf{âˆ\circledR }\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}\mathbf{·}\mathit{d}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{s}}$along them. Then from the type of the capacitor, predict the loop along which the angle between the directions of$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{B}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{d}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{s}}$is constant and the loop along which Bis constant.

Formulae are as follows:

role="math" localid="1663046482332" width="119" height="37">∮B⇶Ä·ds⇶Ä=μ0ienc

Where,$\stackrel{â‡\P Ä}{\mathrm{B}}$is the magnetic field.

Maxwell’s law of induction gives,

$âˆ\circledR \stackrel{â‡\P Ä}{\mathrm{B}}·d\stackrel{â‡\P Ä}{\mathrm{s}}={\mathrm{μ}}_0{i}_{enc}$

This implies that$\stackrel{â‡\P Ä}{\mathrm{B}}·d\stackrel{â‡\P Ä}{\mathrm{s}}$depends on i_enc.

From the given figure, infer that loops a and b enclose the same current, since they enclose the entire square plate.

Sincethearea enclosed by loop c is greater than that enclosed by loop d, the current enclosed by loop c is greater than that by d.

Therefore, the ranking of loops according to the magnitude of role="math" localid="1663046936053" width="59" height="37">∮B⇶Ä·dsâ‡¶Ä along them is a = b > c > d,

Since the capacitor is a parallel plate capacitor and not a circular plate capacitor, the angle between the directions of $\stackrel{â‡\P Ä}{B}andd\stackrel{â‡\P Ä}{s}$is not constant.

Therefore, there is no such loop along which the angle between the directions of $\stackrel{â‡\P Ä}{B}andd\stackrel{â‡\P Ä}{s}$is constant.

There is no such loop along which B is constant.

The loops according to the magnitude of $âˆ\circledR \stackrel{â‡\P Ä}{\mathrm{B}}·d\stackrel{â‡\P Ä}{\mathrm{s}}$from the area enclosed by them.

In a circular plate capacitor, the angle between the directions of $\stackrel{â‡\P Ä}{B}andd\stackrel{â‡\P Ä}{s}$is constant.