The number of allowable propagation modes in a magnetic medium is reduced, if exchange ﬁelds are omitted in the equation of motion. As such, internal microwave ﬁeld amplitudes may be determined uniquely from the application of electromagnetic boundary conditions only. Electromagnetic boundary conditions include the requirement that the tangential components of ~e and ~h be continuous across the surface boundaries. Inclusion of exchange ﬁelds in the equation of motion increases the number of allowable propagation constants by two. For this general case, electromagnetic boundary conditions alone are not sufﬁcient to determine the internal microwave ﬁeld amplitudes. Hence, we must introduce additional boundary conditions besides the electromagnetic boundary conditions at the surfaces of a magneto-dielectric medium. From a mathematical point of view, we must introduce as many boundary conditions as the number of unknown internal ﬁeld parameters associatedwith eachmode of propagation in a magnetic medium. In this chapter, we introduce spin boundary conditions at the surfaces of amagneto-dielectricmedium, as the so-called additional boundary conditions (see Figure 9.1). For convenience, we choose the y-axis to be perpendicular to the pillbox. We start with the equation of motion for M (see Equations 4.19 and 5.15):
¼ ~M ~H ~M 2A M2
where ~M is the total magnetization, and M ¼ j~Mj. We rewrite the above equation and integrate over the volume of the magnetic medium:
~M ~H !