Question

Suppose that the current density in a wire of radius a varies with r according to $J=K{r}^2$, where K is a constant and r is the distance from the axis of the wire. Find the magnetic field at a point distance r from the axis when (a) r < a and (b) r > a

Answer 1

Choose a circular path centred on the axis of the conductor and apply Ampere's law.
(a) To find the current passing through the area enclosed by the path Integrate
$dl=JdA=\left(K{r}^2\right)\left(2\pi rdr\right)$
$\Rightarrow I=\int dl=K{\int }_0^{r}2\pi r^{3}dr=\frac{K\pi r^4}{2}$
since $\int \overrightarrow{B}.dl={\mu }_{0}I$
$\Rightarrow B2\pi r={\mu }_0.\frac{\pi K{r}^4}{2}\Rightarrow B=\frac{{\mu }_{0}K{r}^3}{4}$
(b) If r > a, then net current through the Amperian loop is
$I={\int }_0^{a}K{r}^{2}2\pi dr=\frac{\pi K{a}^4}{2}$
$\Rightarrow B=\frac{{\mu }_{0}K{a}^4}{4r}$