Question

An infinity long conductor carrying a current i with a semiconductor loop in the x-y plane and two straight parts. one parallel to x-axis and another coinciding with z-axis. What is the magnetic field at the centre P of the semicircular loop?

Answer 1

The magnetic field at P due to the straight part of conductor parallel to x-axis is
${\overrightarrow{B}}_1=\frac{{\mu }_0}{4\pi }.\frac{i}{\left(d/2\right)}.\hat{k}$ .......(1)
The magnetic field at the point P due to the semicircular loop on x-y plane is given by
${\overrightarrow{B}}_2=\frac{{\mu }_0}{4\pi }.\frac{\pi i}{\left(d/2\right)}(+.\hat{k})$ .......(2)
The magnetic field at P due to the straight part coinciding with z-axis is given by,
${\overrightarrow{B}}_2=\frac{{\mu }_0}{4\pi }.\frac{i}{\left(d/2\right)}.(-\hat{i})$ .......(3)
The total magnetic field at P is given by
$\overrightarrow{B}={\overrightarrow{B}}_1+{\overrightarrow{B}}_2+{\overrightarrow{B}}_3=-\frac{{\mu }_0}{4\pi }.\frac{2i}{d}(1+\pi )\hat{k}-\frac{2i}{d}\hat{i}=\frac{{\mu }_0}{4\pi }.\frac{2l}{d}\left[\right(1+\pi )\hat{k}-\hat{i}]$