Question

A current i flows along a thin wire shaped as shown in figure. radius of the curved part of the wire is r. The field at the centre O of the coil is:

• A. $\frac{{\mu }_0}{4}.\frac{i}{r}$
• B. $\frac{m{u}_0}{2\pi r}$
• C. $\frac{{\mu }_{0}i}{2\pi r}$
• D. $\frac{{\mu }_{0}i}{8\pi r}(3\pi +4)$

Answer 1

The correct option is D $\frac{{\mu }_{0}i}{8\pi r}(3\pi +4)$

Here,

Magnetic field due to line current $AB$,

$B_1=\frac{{\mu }_{0}I}{4\pi R/\sqrt{2}}\times (\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}})$

$B_1=\sqrt{2}\times \sqrt{2}\frac{{\mu }_{0}I}{4\pi R}=\frac{{\mu }_{0}I}{2\pi R}$ (in)

Magnetic field due to arc,

$B_2=\frac{{\mu }_{0}I}{4\pi R}\times \frac{3\pi }{2}$ (in)

Net Magnetic Field $B$,

$B=B_1+B_2=\frac{{\mu }_{0}I}{8\pi R}(4+3\pi )$

Option $\text{D}$ is the correct answer