Question

If the current density in a linear conductor of radius $a$ varies with $r$ according to relation $J=k{r}^2$, where $k$ is a constant and $r$ is the distance of a point from the axis of conductor, find the magnetic field induction at a point of distance $r$ from the axis when $r<a$. Assume relative permeability of the conductor to be unity.

• A. $\frac{{\mu }_{0}k\pi a^4}{4r}$
• B. $\frac{{\mu }_{0}k{r}^3}{4}$
• C. $\frac{{\mu }_{0}k\pi a^4}{2r}$
• D. $\frac{{\mu }_{0}k{r}^3}{2}$

Answer 1

Answer: (B)

$\frac{{\mu }_{0}k{r}^3}{4}$

Magnetic field at any point on Ampere's loop can be due to all currents passing through inside or outside the loop. But net contribution in the left hand side will come form inside current only.

For $r<a$ current passing through within the cylinder of radius $r$ is given by
${\int }_0^{r}JdA={\int }_0^{r}k{r}^{2}2\pi rdr=2\pi k{\int }_0^r{r}^{3}dr=k\pi r^4/2$
Now using Ampere's law:
$B\times 2\pi r={\mu }_{0}I={\mu }_{0}k\pi r^4/2$

$\Rightarrow B=\frac{{\mu }_{0}k{r}^3}{4}$