Starting with the differential expression

for the magnetic induction at the point P with coordinate x produced by an increment of current I dl at x', show explicitly that for a closed loop carrying a current I the magnetic induction at P is

B = ?₀I/4???

where ? is the solid angle subtended by the loop at the point P. This corresponds to a magnetic scalar potential, ?? = ? ?₀I?/4?. The sign convention for the solid angle is that ? is positive if the point P views the "inner" side of the surface spanning the loop, that is, if a unit normal n to the surface is defined by the direction of current flow via the right-hand rule, ? is positive if n points away from the point P, and negative otherwise. This is the same convention as in Section 1.6 for the electric dipole layer.

- ' x'* dl' x d 4 4 -