Question

A long straight cylindrical wire of radius $R$ carries a current distributed uniformly over its cross section. Find the locations where the magnetic field is maximum and minimum.

Answer 1

Let total current through the wire be $I$.

To get the magnetic field intensity we will be applying Ampere's Circuit as law.

Case $-1$

For $r.R$

$\int B.dl={\mu }_{0}I$

we will take the loop in the from circle of radius $r$

$\int B.dl={\mu }_{0}I$

$B.2\pi r={\mu }_{0}I$

$B=\frac{{\mu }_{0}I}{2\pi r}$

Case$-2$

$r=R$

We will take ampreres loop as a circle of radius $R$

$\int B.dl={\mu }_{0}IB.2\pi R={\mu }_{0}I$

$B=\frac{{\mu }_{0}I}{2\pi R}$

Case $-3$

$r<R$

Here we eill take an ampere loop of radius $r$

current through the loop $=\frac{I}{\pi R^2}\times \pi r^2$

$=\frac{I{r}^2}{R^2}$

$\int B.dl={\mu }_{0}i$

$B.2\pi r={\mu }_0\frac{I{r}^2}{R^2}B=\frac{{\mu }_{0}Ir}{2\pi R^2}$

Magnetic Field will be maximum at $R$

Magnetic Field will be minimum at $r=0$ at the centre

(If infinete distance is not considered)