Question

State Ampere's circuital law. Use it to find magnetic field due to an infinity long thin current carrying wire.

Answer 1

Ampere's circuital law states that the line integral of the magnetic field $\overrightarrow{B}$ around any closed circuit is equal to ${\mu }_0$ (permeability constant) times the total current $I$ passing through this closed circuit.

Mathematically,

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

Consider the figure given above.

Let the current through the long straight wire is $I$ and here from the wire at distance $r$ at point $P$ we have to find the magnetic field .

Taking the circular path as the Ampere's circle then we get that $dl$ is parallel to $B$ at every place applying the cork screw rule .

Therefore , here applying the Ampere's circuital law we get ,

$\oint \overrightarrow{B}.\overrightarrow{dl}=\oint Bdlcos{0}^0=B\oint dl=B\times 2\pi r$

Now here,

$B2\pi r={\mu }_{0}I$

$⟹B=\frac{{\mu }_{0}I}{2\pi r}=\frac{{\mu }_0}{4\pi }\frac{2I}{r}$