Question

In the figure shown, the magnetic field induction at 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. $\frac{{\mu }_0}{4\pi }\frac{i}{r}(\pi -2)$

Answer 1

Answer: (B)

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 $\frac{{\mu }_0\pi i}{4\pi r}$.

$\therefore B=B_a+B_b+B_c$
$=\frac{{\mu }_0}{4\pi }\frac{i}{r}+\frac{{\mu }_0}{4\pi }\frac{\pi i}{r}+\frac{{\mu }_0}{4\pi -}\frac{i}{x}$
$=\frac{{\mu }_0}{4\pi }\frac{i}{r}(\pi +2)$ out of the page