Question

A long straight conductor, carrying a current $i$, is bent to form an almost complete circular loop of radius $r$ as shown. The magnetic field at the centre of the loop :

• A. has magnitude $\frac{{\mu }_{0}i}{r}(1-\frac{1}{\pi })$
• B. has magnitude $\frac{{\mu }_{0}i}{r}(1+\frac{1}{\pi })$
• C. has magnitude $\frac{{\mu }_{0}i}{2r}(1-\frac{1}{\pi })$
• D. has magnitude $\frac{{\mu }_{0}i}{2r}(1+\frac{1}{\pi })$

Answer 1

The correct option is C has magnitude $\frac{{\mu }_{0}i}{2r}(1-\frac{1}{\pi })$

The field at the center of the loop due to a straight part is,

$B=\frac{{\mu }_{0}I}{2\pi r}$

This field is directed into the paper.

The field due to the loop is,

$B^{\prime }=\frac{{\mu }_{0}I}{2r}$

This field is directed out of the paper.

The magnitude of the field is

$B^{\prime }-B=\frac{{\mu }_{0}I}{2r}(1-\frac{1}{\pi })$