Question

In the figure shown, the magnetic field induction as the point $O$ will be

• A. $\frac{{\mu }_{0}i}{2\pi r}$
• B. $\left(\frac{{\mu }_0}{4\pi }\right)\left(\frac{i}{r}\right)(\pi +2)$
• C. $\left(\frac{{\mu }_0}{4\pi }\right)\left(\frac{i}{r}\right)(\pi +1)$
• D. $\frac{{\mu }_{0}i}{4\pi r}(\pi -2)$

Answer 1

The correct option is B $\left(\frac{{\mu }_0}{4\pi }\right)\left(\frac{i}{r}\right)(\pi +2)$

Field due to a straight wire of infinite length is $\frac{{\mu }_{0}i}{4\pi r}$ if the point is on a line perpendicular to its length while at the centre of a semicircular coil is $\frac{{\mu }_0\pi i}{4\pi r}$
$\therefore B=B_a+B_b+B_c$
$=\frac{{\mu }_0}{4\pi }\frac{i}{r}+\frac{{\mu }_0}{4\pi }\frac{\pi i}{r}+\frac{{\mu }_0}{4\pi }\frac{i}{r}$
$=\frac{{\mu }_{0}i}{4\pi r}(\pi +2)$ out of the phase