Question

Obtain an expression for magnetic field due to infinitely long straight current carrying conductor by using Amperes law.

Answer 1

Ampere's law equation
$\oint \overrightarrow{B}.\overrightarrow{d\ell }=\mu \sum i$
Here $\overrightarrow{B}.\overrightarrow{\ell }=\mu \sum i$
here $\overrightarrow{d\mu }$ length element perpendicular to the plane of paper
$\oint \overrightarrow{B}.\overrightarrow{d\ell }={\mu }_o\sum i$
B $\to$ magnetic field which is also perpendicular to plane of paper & align with length elements
$\oint \overrightarrow{B}.\overrightarrow{d\ell }={\mu }_o\sum i$
$B.\oint d\ell ={\mu }_{o}i$
$B.2\pi ={\mu }_{o}i$
$B=\frac{{\mu }_{o}i}{2\pi r}$