Question

A current $I$ flows in an infinitely long wire with cross section in the form of a semicircular ring of radius $R$. The magnitude of the magnetic induction along its axis is

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

Answer 1

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

Let semicircular ring of radius R as shown in the figure,

An elementary length dl cut for finding magnetic field .so, dl = Rdθ

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, $dB=\frac{\mu ₀I}{2\pi ²R}$

Now, magnetic field along axis $B={\int }_0^{\pi }dBsin\theta d\theta =2dB$

Now, put dB in here,

So,$B=\frac{\mu ₀I}{\pi ²R}$

Hence the correct option is D.