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
${\int }_{DEF}\overrightarrow{B}\cdot d\overrightarrow{\ell }=\frac{{\mu }_0{I}_0}{k}$ in $S.I$ unit, then the value of $k$ is:

• A. $4$
• B. $8$
• C. $3$
• D. $6$

Answer 1

The correct option is B $8$

We will draw a circle with current carrying wire at center of radius $OF$

${\oint }_{MF}B.dl={\mu }_0{I}_0\times \frac{\pi }{4\times 2\pi }=\frac{+{\mu }_0{I}_0}{8}$

in loop $MDEFM$

${\oint }_{MDEFM}B.dl={\oint }_{MD}B.dl+{\int }_{DEF}B.dl+{\int }_{FM}B.dl=0$ (as no current)

$=o+x-\frac{{\mu }_0{I}_0}{8}=0$

as $\overline{B}\perp DM$

$x=\frac{{\mu }_0{I}_0}{8}=\frac{{\mu }_0{I}_0}{K}$

$K=8$