Magnetic Field Produced by a Current in a Circular Arc of a Wire

Having worked out the field of a straight wire (Section 10.10.1), the next natural shape to consider is a CIRCULAR arc of wire -- and, as a special case, a complete circular loop, which turns out to be considerably simpler to handle than the straight-wire case, on account of a special geometric fact unique to circles.

Consider a circular arc $AB$ of a wire, of radius $r$, centred at a point $O$, subtending an angle $\theta$ (measured in RADIANS) at that centre, and carrying a steady current $I$ (Fig. 10.18). Take a current element $d\vec{l}$ anywhere along this arc. Because every point on a circle is, by the very definition of a circle, always exactly the SAME distance $r$ from the centre, and because the tangent direction to a circle at any point is always exactly PERPENDICULAR to the radius drawn to that point, the current element $d\vec{l}$ (which points along the local tangent direction of the wire) is ALWAYS perpendicular to the radius vector $\vec{r}$ drawn from that element to the centre $O$ -- for EVERY element along the arc, without exception. This means the angle between $d\vec{l}$ and $\vec{r}$ in the Biot-Savart law is always exactly $90^\circ$, so $\sin\theta=1$ throughout, and moreover $r$ itself stays constant (equal to the arc's own radius) as the integration proceeds along the arc. Both of these simplifications together mean the general Biot-Savart integral collapses to something very simple:

(using $\int dl = r\theta$, the standard relation between arc length, radius, and the SUBTENDED ANGLE in radians), directed, throughout the arc, into (or out of) the plane of the page according to the right-hand rule applied to the current's direction of flow.

As an important special case, letting the arc become a COMPLETE circle -- that is, $\theta=2\pi$ radians, the full angle around a circle -- gives the field at the CENTRE of a full circular current loop of radius $r$, carrying current $I$: …