A wire loop formed by joining two semi-circular wires of...

A wire loop formed by joining two semi-circular wires of radii $a_{1}$ and $a_{2}$ carries a current $i$ as shown in the adjoining figure. The magnetic induction at the centre $O$ is (a) $\frac{\mu_{o} i}{4}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right)$ (b) $\frac{\mu_{\theta} i}{4}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)$ (c) $\frac{\mu_{o} i}{2}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right)$ (d) $\frac{\mu_{e} i}{2}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)$

A wire loop formed by joining two semi-circular wires of radii $a_{1}$ and $a_{2}$ carries a current $i$ as shown in the adjoining figure. The magnetic induction at the centre $O$ is
(a) $\frac{\mu_{o} i}{4}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right)$
(b) $\frac{\mu_{\theta} i}{4}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)$
(c) $\frac{\mu_{o} i}{2}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right)$
(d) $\frac{\mu_{e} i}{2}\left(\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)$

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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 allows the calculation of the magnetic field produced at a point by a small segment of current-carrying wire by integrating the contributions of all such segments over the whole current distribution.

Magnetic Field Contribution from a Circular Arc

When a current flows through a circular arc, the magnetic field at the center of the arc can be computed by integrating the Biot?Savart Law over the angle subtended by the arc. For a full circle, the field is given by (??i)/(2R); for a semi?circular arc, this result is halved, showing the inverse dependence on the arc’s radius and a proportionality to the current.

Superposition Principle

The superposition principle in magnetostatics states that the net magnetic field created by multiple sources is the vector sum of the individual fields generated by each source independently. In problems involving composite geometries, such as a loop formed by joining arcs of different radii, the fields due to each arc can be calculated separately and then added to obtain the total magnetic field at a point of interest.

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