Question

A current $I$ flows in an infinitely long wire with corss section in the form of a semi-circular ring of raidus $R$. The magnitude of the magnetic induction along its axis is

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

Answer 1

Answer: (B)

$B=\int dB\sin \theta$
$B=\int \frac{{\mu }_{0}di}{2\pi R}\sin \theta$
$=\frac{{\mu }_0}{2\pi R}\left(\frac{i}{\pi R}\right)R\underset{0}{\overset{\pi }{\int }}\sin \theta d\theta$
$=\frac{{\mu }_{0}i}{{\pi }^{2}R}$