Consider a wire in the shape of a circular arc of radius of curvature R and carrying a current I. The unit vector \(\vec r\) from each current element I \(\vec {dl}\) towards
magnetic induction due to a current in a circular arc at its centre of curvature
the centre of the loop is perpendicular to I \(\vec{dl}\), i.e., the angle θ between them is 90°. The direction of the incremental magnetic induction \(\vec{dB}\) due to each current element is in the same direction, viz., perpendicular to the plane of the loop, and out of the plane of the figure for the sense of the current shown in above figure.
Since every current element is equidistant from the centre of curvature C, the magnetic field at C due to each current element in the arc by Biot-Savart’s law is
If the arc subtends an angle 0 at its centre of curvature C, the total field at C due to all the elements on the arc is
where Φ is in radian.
The magnitude of the total induction \(\vec B\) at the centre of a circular coil is, from Eq. (3),
magnetic induction due to a current in a circular loop at its centre
If a circular coil has N turns, each of radius r and carries a current I, the magnetic induction at its centre has a magnitude
B = \(\cfrac{\mu_0NI}{2R}\) ………… (6)
