Electromagnetic Boundary Problems

1.4    DUALITY; EQUIVALENCE; SURFACE SOURCES

1.4.1    DUALITY AND MAGNETIC SOURCES

Next, we look at the idea of duality. Our fields are required to satisfy the modified Maxwell’s equations (1.61) with magnetic as well as electric sources present (corresponding to some choice of definitions for B, D, and H as in the previous section). Boundary conditions for the fields when magnetic surface currents and charges are present must also be modified, but we defer discussion of this topic to the next section. In a linear isotropic medium, these equations become

$\begin{array}{lll}\hfill \nabla \cdot \left(ϵ\mathbf{E}\right)& =\hfill & \rho \hfill \\ \hfill \nabla \cdot \left(\mu \mathbf{H}\right)& =\hfill & {\rho }_m\hfill \\ \hfill \nabla \times \mathbf{H}& =\hfill & \mathbf{J}+\frac{\partial \left(ϵ\mathbf{E}\right)}{\partial t}\hfill \\ \hfill \nabla \times \mathbf{E}& =\hfill & -\mathbf{M}-\frac{\partial \left(\mu \mathbf{H}\right)}{\partial t}\hfill \end{array}$

(1.87)

If E and H satisfy (1.87) for some given ϵ, µ and source densities, then the dual fields

$\begin{array}{ccc}{E}^{\prime }=-\mathbf{H}& \text{and}& H^{\prime }=\mathbf{E}\end{array}$

will obey a dual set of Maxwell’s ...