Question

A current of i amp flows in the loop having circular arc r metre, subtending an angle  as shown in the figure. The magnetic field at centre O of circle is

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

Answer 1

The Correct Option is A

Explanation:
Magnetic field due to the current carrying arc at the centre of curvature $B=\frac{{\mu }_{0}i}{4\pi r}\theta$

Answer 2

_{Biot-Savart Law}

Biot-Savart law is used to determine the strength of the magnetic field at any point due to a current-carrying conductor.

Consider a very small current element of length dl of a conductor carrying current I, the strength of magnetic field dB at distance r from the element is found to be

\[dB=\frac{\mu_0}{4\pi}\frac{I\,dl\,sin\theta}{r^2}\]

Where

• µ₀ is absolute permeability
• θ is the angle between dl and r

Using Biot-Savart law, the magnetic field at the center of a current carrying loop of radius R is given by

\[B=\frac{\mu_0}{4\pi}\frac{2\pi I}{R}\]

This equation can be rewritten as

\[B=\frac{\mu_0}{4\pi}\frac{I}{R}(2\pi)\]

Where 2π is the angle subtended by the circular at its center.

Thus, if θ will be the angle subtended by an arc of radius R, its magnetic field at the center is given by

\[B=\frac{\mu_0}{4\pi}\frac{I}{R}(θ)\]