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, as a function of the radial distance $r$ from the axis is best represented by

• A.
• B.
• C.
• D.

Answer 1

Answer: (D)

r = distance of a point from centre
For r $\le$ R/2 Using Ampere's circuital law,
$\oint B.d1orBl={\mu }_0(I_{in})$
$orB(2\pi r)orBl={\mu }_0(I_{in})orB=\frac{{\mu }_0}{2\pi }\frac{I_{in}}{r}...(i)$
Since, $I_{in}=0\Rightarrow \therefore B=0$
$For\frac{R}{2}\le r\le R{I}_{in}=[\pi r^2-\pi (\frac{R}{2}{)}^2]\sigma$
Here$\sigma$ = current per unit area.
Substituting in Eq. (i), we have
$B=\frac{{\mu }_0}{2\pi }\frac{[\pi r^2-\pi \frac{R^2}{4}]\sigma }{r}=\frac{{\mu }_0\sigma }{2r}\frac{{\mu }_0}{2\pi }(r^2-\frac{R^2}{4})$
At$r=\frac{R}{2},B=0$
At$r=R,B=\frac{3{\mu }_0\sigma R}{8}$
For e $\ge R{I}_{in}=I_{Total}=I$(say)
Therefore, substituting in Eq. (i), we have
$B=\frac{{\mu }_0}{2\pi }.\frac{I}{r}orB\propto \frac{1}{r}$