What is the magnetic field at the center of an arc of current of radius R? The arc curves through an angle q. If the current is counter-clockwise the magnetic field at the center is out of the page.
The magnetic field dB from a representative piece of the wire is:
dB = (mo / 4p ) I ds ´ /r2
r = R, and ds ´ = ds
Our integral for the net field is easy:
B = ( mo I / 4pR2 ) ò ds
B = ( mo I / 4pR2 ) s
where s is the arc length, equal to Rq.
Therefore B = mo Iq / 4pR