Physics - Magnetism Question with Solution | TestHub

The magnetic field at the centre of the loop is the superposition of the fields due to the straight wire and the circular loop.

Magnetic field at the centre due to the infinitely long straight wire:
$B_1=\frac{{\mu }_{0}I}{2\pi r}$
By the right-hand rule, since the current is flowing to the right, this field is directed outwards from the plane of the paper.

Magnetic field at the centre due to the circular loop:
$B_2=\frac{{\mu }_{0}I}{2r}$
Since the current in the loop is clockwise, this field is directed inwards into the plane of the paper by the right-hand rule.

Since the two fields are in opposite directions, the net magnetic field is their difference:
$B=B_2-B_1$
$B=\frac{{\mu }_{0}I}{2r}-\frac{{\mu }_{0}I}{2\pi r}$
$B=\frac{{\mu }_{0}I}{2r}\left(1-\frac{1}{\pi }\right)$
$B=\frac{{\mu }_{0}I}{2\pi r}(\pi -1)$

Multiplying the numerator and denominator by $2$ gives:
$B=\frac{{\mu }_0}{4\pi }\times \frac{2I}{r}(\pi -1)$

Answer: $\frac{{\mu }_{\mathrm{o}}}{4\pi }\times \frac{2\mathrm{I}}{\mathrm{R}}(\pi -1)$