Question

Figure 29-30 shows four circular Amperian loops (a, b, c, d) concentric with a wire whose current is directed out of the page. The current is uniform across the wire’s circular cross section (the shaded region). Rank the loops according to the magnitude of $\mathbf{\oint }\overrightarrow{\mathit{B}}\mathit{.}\overrightarrow{\mathit{d}\mathit{s}}$around each, greatest first.

Short answer

The ranking of the loops according to the magnitude of$\mathbf{\oint }\overrightarrow{\mathit{B}}\mathit{.}\overrightarrow{\mathit{d}\mathit{s}}$around each is,

.$c=0>0>a$

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 ofaround each.

The formula is as follows:

$\mathbf{\oint }\overrightarrow{\mathit{B}}\mathbf{.}\overrightarrow{\mathit{d}\mathit{s}}\mathbf{=}{\mathit{\mu }}_{\mathbf{0}}{\mathit{i}}_{\mathbf{e}\mathbf{n}\mathbf{c}}$

Where, $\overrightarrow{\mathbf{B}}$is the magnetic field,

$\overrightarrow{\mathit{d}\mathit{s}}$is the infinitesimal segment of the integration path,

${\mathit{\mu }}_{\mathbf{0}}$is the empty's permeability,
localid="1662742639512" ${\mathit{i}}_{\mathbf{enc}}$is the enclosed electric current by the path.

Determining the rank of the loops according to the magnitude of  around each.

According to Ampere’s law,

$\mathbf{\oint }\overrightarrow{\mathit{B}}\mathbf{.}\overrightarrow{\mathit{d}\mathit{s}}\mathbf{=}{\mathit{\mu }}_{\mathbf{0}}{\mathit{i}}_{\mathbf{enc}}$

From the given figure, it can interpret that,

$i_{enc,c}=i_{enc,d}$

Also,

$i_{enc,b}>i_{enc,a}$

And,

$i_{enc,c}>i_{enc,b}$

(Since the area enclosed by loop b is greater than that by loop a and the area enclosed by loop c is greater than that by b.)

Hence, the ranking of the loops according to the magnitude ofeach is,$c=0>0>a$

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