Q.A circular coil of $100$ turns and radius $5\ \text{cm}$ carries a current of $0.5\ \text{A}$. Find the magnetic field at the centre of the coil.

Applying the Biot-Savart law to a circular arc of wire exploits a special geometric fact unique to circles: every current element along the arc is always exactly perpendicular to the radius vector drawn to the centre, and every element is the same fixed distance from the centre. This simplifies the Biot-Savart integral for the field at the centre O of an arc of radius r subtending angle $\theta$ (in radians) to $B=\dfrac{\mu_0}{4\pi}\dfrac{I}{r}\theta$.

As the special case of a COMPLETE circular loop ($\theta=2\pi$), this gives the widely-used result for the field at the centre of a full circular current loop of radius r carrying current I: $B=\dfrac{\mu_0I}{2r}$, or $B=\dfrac{\mu_0NI}{2r}$ for N turns. This centre-of-loop formula is one of the most frequently used results in the whole chapter, and is also correctly recovered as the special case z=0 of the more general on-axis field formula for a circular loop.