Question

An equilateral triangular loop of wire of side $\ell$ carries a current i. The magnetic field produced at the circumcentre of the loop is :

• A. $\frac{{\mu }_0}{4\pi }\frac{3\sqrt{3}i}{\ell }$
• B. $\frac{{\mu }_0}{4\pi }\frac{9i}{\ell }$
• C. $\frac{{\mu }_0}{4\pi }\frac{18i}{\ell }$
• D. $\frac{{\mu }_0}{4\pi }\frac{6i}{\ell }$

Answer 1

The correct option is C $\frac{{\mu }_0}{4\pi }\frac{18i}{\ell }$

If $l$ is the side of the triangle, the distance of circumcenter from each of the sides of the triangle is $r=\frac{l\sqrt{3}}{6}$. The magnetic induction due to each of the sides of the triangle carrying a current i is $\frac{{\mu }_0}{4\pi }\frac{i}{r}(\sin {60}^{\circ }+\sin {60}^{\circ })=\frac{{\mu }_0}{4\pi }\frac{6i}{\ell }$. Since the direction of magnetic field in each case is the same, three times this would be the total magnetic induction.
Therefore total magnetic induction = $=\frac{{\mu }_0}{4\pi }\frac{18i}{\ell }$