Question

A current carrying hollow wire has inner radius $\frac{R}{2}$ and outer radius $R$ .Which of the following plots correctly represent variation of magnetic field with respect to distance from its axis.

• A.
• B.
• C.
• D.

Answer 1

The correct option is D


Let, $r$ be the distance of a point from centre.

$\text{Case I:}$ For $r\le \frac{R}{2}$

Using Ampere's circuital law, we have

$\oint \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_0{i}_{net}$

$\Rightarrow Bl={\mu }_0\left(I_{en}\right)$

$\Rightarrow B\left(2\pi r\right)={\mu }_0\left(I_{en}\right)$

$\Rightarrow B=\frac{{\mu }_0}{2\pi }\frac{I_{en}}{r}....\left(1\right)$

$\Rightarrow B=0\because I_{en}=0$

For , $\frac{R}{2}\le r\le R$,

$I_{en}=[\pi r^2-\pi {\left(\frac{R}{2}\right)}^2]\sigma$

Here, $\sigma =$ current passing through a unit area

Substituting, in equation $\left(1\right)$, we have

$B=\frac{{\mu }_0}{2\pi }\frac{[\pi r^2-\pi {\frac{R}{4}}^2]\sigma }{r}=\frac{{\mu }_0\sigma }{2r}(r^2-\frac{R^2}{4})$

At, $r=\frac{R}{2},B=0$

At, $r=R,B=\frac{3{\mu }_0\sigma R}{8}$

$\text{Case II:}$ For $r\ge R$

$I_{in}=I_{Total}=I$

Therefore, substituting in equation $\left(1\right)$, we have

$B=\frac{{\mu }_0}{2\pi }.\frac{I}{r}$

$\therefore B\propto \frac{1}{r}$

Combining the $\text{Cases I}$ and $\text{II}$, we find that option (d) is the correct answer.