Question

A current of i amp flows in a loop having circular arc of radius r subtending an angle $\theta$ as shown in the figure. The magnetic field at the centre of the circle is:

• A. $\frac{{\mu }_{0}i\theta }{4\pi r}$
• B. $\left(\frac{{\mu }_{0}i}{4r}\right)\sin \theta$
• C. $\left(\frac{2{\mu }_{0}i}{2r}\right)\sin \theta$
• D. $\left(\frac{{\mu }_{0}i}{4\pi r}\right)\sin \theta$

Answer 1

The correct option is A $\frac{{\mu }_{0}i\theta }{4\pi r}$

The wire AOB will not contribute in magnetic field at 0 as it is along the length of the wire.
For the rest of the wire,
Magnetic field at centre if $2\pi$ angle is covered $B=\frac{{\mu }_{o}i}{2r}$
Magnetic field at the centre if $\theta$ radian is covered $B=\frac{{\mu }_{o}i\times \theta }{2r\times 2\pi }=\frac{{\mu }_{o}i\theta }{4\pi r}$