Question

A straight conductor carrying current $\mathbf{i}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{A}$splits into identical semicircular arcs as shown in Figure. What is the magnetic field at the center C of the resulting circular loop?

Short answer

The magnetic field at the center is$B=0$.

Step-by-step solution

Given Information

Angle made by arc$=\mathrm{\pi }$

Determining the formula for the magnetic field

Formula:

Magnetic field at center of circular arc is given by:

${\mathbf{B}}_{\mathbf{C}}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}}{\mathbf{4}\mathbf{\pi }}\mathbf{\times }\frac{\mathbf{i}\varphi }{\mathbf{R}}$

Here, $\varphi$is the angle of arc and R is the radius of the arc.

Calculating the magnetic field at the center C of the resulting circular loop 

Magnetic field at center due to arc is given by:

$B=\frac{{\mu }_0}{4\pi }\times \frac{i\varphi }{R}$

Fields due to both arcs are opposite to each other.

Angle due to semicircular arcs $=\mathrm{\pi }$,

Net field due to both semicircular arcs is

$B=\frac{{\mu }_0}{4\pi }\times \frac{i\pi }{R}+\frac{{\mu }_0}{4\pi }\times \frac{i\left(-\pi \right)}{R}$

$B=0$