Question

An arc of a circle of radius $R$ subtends an angle $\frac{\pi }{2}$ at the centre. It carries a current $i$. The magnetic field at the centre will :

• A. $\frac{{\mu }_{0}i}{2R}$
• B. $\frac{{\mu }_{0}i}{8R}$
• C. $\frac{{\mu }_{0}i}{4R}$
• D. $\frac{{\mu }_{0}i}{5R}$

Answer 1

The correct option is B $\frac{{\mu }_{0}i}{8R}$

Magnetic field due to current carring arc

$dB=\frac{{\mu }_o}{4\pi }\frac{idl\sin 90}{R^2}$

$=\frac{{\mu }_o}{4\pi }\cdot \frac{idl}{R^2}$

$N_{Net}=\int dB=\int \frac{{\mu }_o}{4\pi }\frac{idl}{R^2}$ $\{\theta =\frac{l}{R}\}$

$B_{Net}=\frac{{\mu }_o}{4\pi }\frac{i}{R^2}\int dl$

$=\frac{{\mu }_o}{4\pi }\frac{i}{R^2}\left(R\theta \right)$

$B_{net}=\frac{{\mu }_o}{4\pi }\cdot \frac{i\theta }{R}$

if $\theta =\frac{\pi }{2}$

$B_{Net}=\frac{{\mu }_o}{4\pi }\cdot \frac{i\times \pi /2}{R}$

$=\frac{{\mu }_{o}i}{8R}$.