Question

An infinitely long hollow conducting cylinder with inner radius $R/2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $\left|\right.\overset{ \rightarrow }{B}\left|\right.$ as a function of the radial distance $r$ from the axis is best represented by :-

• A.
• B.
• C.
• D.

Answer 1

Answer: (D)

From Ampere's law, $\oint\overset{ \rightarrow }{B}\cdot \overset{ \rightarrow }{d l}=\mu _{0}I_{i n s i d e}$
For $r < \frac{R}{2},B\times 2\pi r=\left(\mu \right)_{0}\left(0\right)\Rightarrow B=0$
For $r>R,B\times 2\pi r=\mu _{0}I\Rightarrow B=\frac{\mu _{0} I}{2 \pi r} \propto \frac{1}{r}$
For $\frac{R}{2} < r < R,$
$B \times 2 \pi r=\mu_{0}\left(\frac{I}{\pi R ^{2}-\pi\left(\frac{ R }{2}\right)^{2}} \times \pi\left\{ r ^{2}-\left(\frac{ R }{2}\right)^{2}\right\}\right) \Rightarrow B =\frac{\mu_{0} I }{2 \pi r }\left[\frac{ r ^{2}-\left(\frac{ R }{2}\right)^{2}}{ R ^{2}-\left(\frac{ R }{2}\right)^{2}}\right]$