Consideration of the relationship between the magnetic field produced by a magnetic shell and that produced by a current loop requires the evaluation of the line integral ɸH̅ • dl̅, where a portion of the path must cross the dipole layer. The correct line integral to be evaluated must be ɸB̅ • dl̅. The evaluation of this integral leads immediately to the correct Maxwell equation in ∆ x H̅. Gaussian units are used.
McCloud, H. E.
"Logical Transition from Magnetic Poles to Current Loops,"
Journal of the Arkansas Academy of Science: Vol. 27, Article 10.
Available at: https://scholarworks.uark.edu/jaas/vol27/iss1/10