Question

The current density in a wire of radius $‘a^{\prime }$ varies with radial distance $‘r^{\prime }$ as $J=k{r}^2$ , where $k$ is a constant .

• A. The total current passing through the cross section of the wire is $I=\frac{\pi k{a}^4}{2}$
• B. The total current passing through the cross section of the wire is $I=\frac{3\pi k{a}^4}{2}$
• C. The magnetic field at a distance $r>a$ is $B=\frac{{\mu }_{0}k{a}^4}{4r}$
• D. The magnetic field at a distance $r<a$ is $B=\frac{{\mu }_{0}k{r}^3}{4}$

Answer 1

Answer: (A), (C), (D)

The correct options are
A The total current passing through the cross section of the wire is $I=\frac{\pi k{a}^4}{2}$
C The magnetic field at a distance $r>a$ is $B=\frac{{\mu }_{0}k{a}^4}{4r}$
D The magnetic field at a distance $r<a$ is $B=\frac{{\mu }_{0}k{r}^3}{4}$

Current through the wire of area $da$ is
$dI=Jda$
$\Rightarrow dI=k{r}^2\left(2\pi rdr\right)$
$\Rightarrow I=2\pi k\underset{0}{\overset{a}{\int }}{r}^{3}dr$


$I=\frac{1}{2}\pi k{a}^4$
Further, field for $r>a$ is
$B=\frac{{\mu }_{0}I}{2\pi r}=\frac{{\mu }_{0}k{a}^4}{4r}$
and field for $r<a$ is
$B=\frac{{\mu }_0\left(\frac{\pi k{r}^4}{2}\right)}{2\pi r}=\frac{1}{4}{\mu }_{0}k{r}^3$