Question

A long cylindrical conductor of radius $R$ carries current such that current density $J\left(r\right)$ varies with distance $‘r^{\prime }$ from central axis of cylinder. If magnetic field at a distance $r(r<R)$ from axis is $B\left(r\right)=2r^3$. Then current density $J\left(r\right)$ will be given as

• A. $J\left(r\right)=\frac{2r^2}{{\mu }_0}$
• B. $J\left(r\right)=\frac{8r^2}{{\mu }_0}$
• C. $J\left(r\right)=\frac{r^2}{{\mu }_0}$
• D. $J\left(r\right)=\frac{4r^2}{{\mu }_0}$

Answer 1

The correct option is B $J\left(r\right)=\frac{8r^2}{{\mu }_0}$

$\int B.dl={\mu }_0{l}_{\left(in\right)})$
$l_{\left(in\right)}=\int 2\pi rdr.J\left(r\right)={\mu }_{0}2\pi {\int }_0^{r}J\left(r\right)rdr$

$\int B.dl=2r^3.2\pi r={\mu }_{0}2\pi \int J\left(r\right)rdr$

$\Rightarrow 2r^4={\mu }_0\int J\left(r\right)rdr$

Differentiating both side

$2\times 4r^{3}dr={\mu }_{0}J\left(r\right).rdr$

$\Rightarrow J\left(r\right)=\frac{8r^2}{{\mu }_0}$