Physics - Magnetism Question with Solution | TestHub

The wire can be divided into three parts: a lower semi-infinite straight wire, a semicircular arc, and an upper semi-infinite straight wire.

The magnetic field at point O due to the lower semi-infinite straight wire is given by $B_1=\frac{{\mu }_{0}i}{4\pi R}$. Applying the right-hand rule, the current is towards the left and point O is above the wire, so the magnetic field is directed into the page.

The magnetic field at point O due to the semicircular arc is given by $B_2=\frac{{\mu }_{0}i}{4R}=\frac{\pi {\mu }_{0}i}{4\pi R}$. The current is in the clockwise direction, so the magnetic field is directed into the page.

The magnetic field at point O due to the upper semi-infinite straight wire is given by $B_3=\frac{{\mu }_{0}i}{4\pi R}$. The current is towards the right and point O is below the wire, so the magnetic field is directed into the page.

Since all three magnetic field components are in the same direction, the total magnetic field at O is their sum:

$B_{net}=B_1+B_2+B_3$

$B_{net}=\frac{{\mu }_{0}i}{4\pi R}+\frac{\pi {\mu }_{0}i}{4\pi R}+\frac{{\mu }_{0}i}{4\pi R}$

$B_{net}=\frac{{\mu }_{0}i}{4\pi R}(1+\pi +1)$

$B_{net}=\frac{{\mu }_{0}i}{4\pi R}(2+\pi )$