Question

A very long wire of cross section radius a carry current such that its current density is $J=J_{0}r$. $J_0$ is constant and $r$ is distance from axis of wire. Magnetic field as a function of $r$ is best represented by

• A.
• B.
• C.
• D.

Answer 1

The correct option is A


$dI=JdA=J_{0}r2\pi rdr$
$I=2\pi J_0{\int }_0^r{r}^{2}dr$ $[0<r<a]$
$I=\frac{2\pi J_0}{3}{r}^3$
$\int Bdl={\mu }_{0}I\Rightarrow b\times 2\pi r={\mu }_0\times \frac{2\pi J_0{r}^3}{3}\Rightarrow B\alpha r^2$
For $r>a$ $\Rightarrow B\times 2\pi r={\mu }_0{I}_0\Rightarrow B\alpha \frac{1}{r}$