Question

A direct current of density $j$ flows along a round uniform straight wire with cross-section radius $R$. Find the magnetic induction vector of this current at the point whose position relative to the axis of the wire is defined by a radius vector $r$. The magnetic permeability is assumed to be equal to unity throughout all the space.

Answer 1

There are two cases-

1. The point lies inside the wire.

The current enclosed in a circular loop at distance $r$ is $j\left(\pi r^2\right)$.

Hence the magnetic field at the point is given by

$⟹B_1\left(2\pi r\right)={\mu }_{0}i={\mu }_0\left(j\pi r^2\right)$ (By Ampere circuital law)

$⟹B_1=\frac{{\mu }_{0}jr}{2}$

2. The point lies outside the wire.

The current enclosed in a circular loop at distance $r$ is $j\left(\pi R^2\right)$.

Hence the magnetic field at the point is given by

$⟹B_2\left(2\pi r\right)={\mu }_{0}i={\mu }_0\left(j\pi R^2\right)$ (By Ampere Circuital law)

$⟹B_2=\frac{{\mu }_{0}j{R}^2}{2r}$