Question

Figure 29-32 shows four circular Amperian loops (a, b, c, d) and, in cross section, four long circular conductors (the shaded regions), all of which are concentric. Three of the conductors are hollow cylinders; the central conductor is a solid cylinder. The currents in the conductors are, from smallest radius to largest radius, 4 A out of the page, 9 A into the page, 5 A out of the page, and 3 A into the page. Rank the Amperian loops according to the magnitude of$\mathbf{\oint }\overrightarrow{\mathbf{B}}\mathbf{.}\mathit{d}\overrightarrow{\mathbf{s}}$around each, greatest first.

Short answer

The ranking of the Amperian loops according to the magnitude of $\oint \overrightarrow{B}.d\overrightarrow{s}$around each is $b>a>d>c$.

Step-by-step solution

Step 1: Given

• Figure showing four circular Amperian loops.
• The current is uniform across the wire’s circular cross-section.

Determining the concept.

Using Ampere’s law, Rank theloops according to the magnitude offrom the corresponding values of the magnitude of enclosed currents.

Formulae are as follows:

$\oint \overrightarrow{B}.d\overrightarrow{s}={\mu }_0{i}_{enc}$

Where,

$\overrightarrow{B}=$is the magnetic field,

role="math" localid="1663001431872" $d\overrightarrow{s}=$is the infinitesimal segment of the integration path,

${\mu }_0=$is the empty's permeability,

$i=$is the enclosed electric current by the path.

Determining the ranking of Amperian loops according to the magnitude  ∮B→.ds→.

According to Ampere’s law,

$\oint \overrightarrow{B}.d\overrightarrow{s}={\mu }_0{i}_{enc}$

So,

$\left|\oint \overrightarrow{B}.d\overrightarrow{s}\right|=\left|{\mu }_0{i}_{enc}\right|$

From the given figure, it interprets that,

role="math" localid="1663001760102" $$\begin{gathered} \left|i_{enc,a}\right|=4\text{A} \\ \left|i_{enc,b}\right|=5\text{A} \\ \left|i_{enc,c}\right|=0\text{A} \\ \left|i_{enc,d}\right|=3\text{A} \end{gathered}$$

Hence, the ranking of Amperian loops according to the magnitude of $\oint \overrightarrow{B}.d\overrightarrow{s}$around each is $b>a>d>c$.

Ampere’s law gives the relation between magnetic flux and current enclosed by a loop.