Question

An infinite uniform current carrying wire is kept along $z$-axis, carrying current $I_0$ in the direction of the positive $z$-axis. $OABCDEFG$ represents a circle (where all the points are equally spaced), whose centre at point $(4m,0m)$ and radius $4m$ as shown in the figure. If $\underset{\text{DEF}}{\int }\overrightarrow{B}.\overrightarrow{dl}=\frac{{\mu }_0{I}_0}{k}$ in S.I. unit, then the value of $k$ is

Answer 1

Let $\underset{\text{DEF}}{\int }\overrightarrow{B}.\overrightarrow{dl}=\frac{{\mu }_0{I}_0}{k}=x$
Draw a circle assuming $OF$ as radius and $O$ as centre

$\underset{\text{MDEFM}}{\oint }\overrightarrow{B}.d\overrightarrow{l}={\int }_M^D\overrightarrow{B}.d\overrightarrow{l}+\underset{\text{DEF}}{\int }\overrightarrow{B}.\overrightarrow{dl}+{\int }_F^M\overrightarrow{B}.d\overrightarrow{l}=0$
$\Rightarrow 0+x-{\mu }_0{I}_0\frac{{45}^{\circ }}{{360}^{\circ }}=0$
$\Rightarrow x=\frac{{\mu }_0{I}_0}{8}$