Question

Figure 32-23 shows a face-on view of one of the two square plates of a parallel-plate capacitor, as well as four loops that are located between the plates. The capacitor is being discharged. (a) Neglecting fringing of the magnetic field, rank the loops according to the magnitude of$\mathbf{\oint }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{·}\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{s}}$along them, greatest first. (b) Along which loop, if any, is the angle between the directions of $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{s}}$constant (so that their dot product can easily be evaluated)? (c) Along which loop, if any, is B constant (so that B can be brought in front of the integral sign in Eq. 32-3)?

Short answer

1. The ranking of loops according to the magnitude of $\oint \stackrel{\rightharpoonup }{\mathrm{B}}·d\stackrel{\rightharpoonup }{\mathrm{s}}$ along them is a = b > c > d.
2. There is no such loop along which the angle between the directions of $\stackrel{\rightharpoonup }{B}andd\stackrel{\rightharpoonup }{s}$is constant.
3. There is no such loop along which B is constant.

Step-by-step solution

Given

Figure 32-23.

Determining the concept

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

Formulae are as follows:

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

Where,$\stackrel{\rightharpoonup }{\mathrm{B}}$is the magnetic field.

(a) Determining the ranking of loops according to the magnitude of ∮B⇀·ds⇀ along them.

Maxwell’s law of induction gives,

$\oint \stackrel{\rightharpoonup }{\mathrm{B}}·d\stackrel{\rightharpoonup }{\mathrm{s}}={\mu }_0{i}_{enc}$

This implies that$\stackrel{\rightharpoonup }{\mathrm{B}}·d\stackrel{\rightharpoonup }{\mathrm{s}}$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⇀·ds⇀ along them is a = b > c > d,

(b) Determining the loop along which the angle between the directions of B⇀ and ds⇀ is constant.

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

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

(c) Determining the loop along which B is constant

There is no such loop along which B is constant.

The loops according to the magnitude of $\oint \stackrel{\rightharpoonup }{\mathrm{B}}·d\stackrel{\rightharpoonup }{\mathrm{s}}$from the area enclosed by them.

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