Question

A current I flows in an infinity long wire with cross section in the from of a semicircular ring of radius R the magnitude of the magnetic induction along its axis is :-

• A. $\frac{{\mu }_{0}I}{2\pi R}$
• B. $\frac{{\mu }_{0}I}{4\pi R}$
• C. $\frac{{\mu }_{0}I}{{\pi }^{2}R}$
• D. $\frac{{\mu }_{0}I}{2{\pi }^{2}R}$

Answer 1

The correct option is C $\frac{{\mu }_{0}I}{{\pi }^{2}R}$

Let semicircular ring of radius $R$ as shown in figure

An elementary length $dl$ cut for finding magnetic field,

So,

$dl=Rd\theta$

So, elementary current along $dl$ is $di=\frac{i}{\pi R}\times Rd\theta =\frac{id\theta }{\pi }$.

Now, you can see that elementary part of length is perpendicular upon $dB$.

so,

$dB=\frac{{\mu }_{0}di}{2\pi R}$

$=\frac{{\mu }_{0}di}{2{\pi }^{2}R}$

Now, magnetic filed along axis $B=\underset{0}{\overset{\pi }{\int }}db.\sin \theta d\theta =2dB$

Now, put $dB$ in here,

So, $B=\frac{{\mu }_{o}i}{{\pi }^{2}R}$

Hence,

Option $C$ is the correct answer.