The magnetic induction at the centre O of the current loop...

The magnetic induction at the centre $O$ of the current loop shown in the adjoining diagram is (a) $\frac{\mu_{o} i}{4 \pi}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right) \theta$ (b) $\frac{\mu_{a} i}{2 \pi}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right) \theta$ (c) $\frac{\mu_{o} i}{4 \pi}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right) \theta$ (d) $\frac{\mu_{o} i}{2 \pi} \cdot \frac{\theta}{a_{2}}$

The magnetic induction at the centre $O$ of the current loop shown in the adjoining diagram is
(a) $\frac{\mu_{o} i}{4 \pi}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right) \theta$
(b) $\frac{\mu_{a} i}{2 \pi}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right) \theta$
(c) $\frac{\mu_{o} i}{4 \pi}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right) \theta$
(d) $\frac{\mu_{o} i}{2 \pi} \cdot \frac{\theta}{a_{2}}$

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Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in electromagnetism that relates magnetic fields to the currents which are their sources. It provides a method to compute the magnetic field produced by a small current element by integrating over the entire current distribution. This law is essential for calculating magnetic fields of various geometries, including wires, loops, and arcs, and serves as the basis for deriving specific formulas in cases of symmetry.

Magnetic Field of a Current-Carrying Arc

A current-carrying arc produces a magnetic field at its center that depends on the current, the subtended angle of the arc, and the radius of the arc. The contribution to the magnetic field from an elemental segment of the arc is integrated over the angle to yield a formula involving the total angle in radians and the inverse of the radius. This concept is crucial for understanding magnetic induction in configurations where the current path is not a complete loop but rather spans a specified angular segment.

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