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, $|\overrightarrow{B}|$ 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, $\oint \overrightarrow{B}\cdot \overrightarrow{\mathrm{dl}}={\mu }_0{I}_{inside}$
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 $\frac{R}{2}<r<R,$
$B\times 2\pi r={\mu }_0\left(\frac{I}{\pi R^2-\pi {\left(\frac{R}{2}\right)}^2}\times \pi \left\{r^2-{\left(\frac{R}{2}\right)}^2\right\}\right)\Rightarrow B=\frac{{\mu }_{0}I}{2\pi r}\left[\frac{r^2-{\left(\frac{R}{2}\right)}^2}{R^2-{\left(\frac{R}{2}\right)}^2}\right]$