Magnetic Field Equations in Magnetized Regions Carrying Steady Current

Abstract

We proceed in essentially the same way as in the electric case and first formally define a single magnetic dipole moment analytically from our present magnetic apparatus. To this end we consider a closed current loop in which a steady electric current I flows as shown in Figure 25. In all of our closed loop expressions for B and A, the loop integral

$$ \oint_{c} {f({r_{1}})} Ids = I\oint_{c} {f({r_{1}})} ds, $$

(7.1.1)

exists. The application of a well-known integral theorem¹⁸ yields

$$ I\oint_{c} f ds = I\oint_{s} {nx} \nabla fdS. $$

(7.1.2)