Material boundaries cause discontinuities in the electric and magnetic fields. The effects can be found by considering either vanishing small pillboxes or loops that span the boundary between the two regions. By using the integral form of Maxwell’s equations on these differential structures, the integrals reduce to simple algebraic expressions. These arguments can be found in most electromagnetic texts and we give only the results. Conversely, we will discover that artificial boundaries such as shadow and reflection boundaries used in geometric optics (ray optics) cannot cause a discontinuity in the fields because they are not material boundaries. The idea that the fields remain continuous across the boundary leads to the necessity of adding terms to extend ray optics methods. We discuss these ideas when considering the uniform theory of diffraction (UTD) method used with ray optics.

Suppose that we have a locally plane boundary in space described by a point and a unit normal vector ˆn that points from region 1 to region 2. We compute the tangential fields from the vector (cross) product of the fields and the normal vector. The fields can be discontinuous at the interface between the two regions if surface magnetic MS or electric current JS densities exist on the surface.

  (1a,b)

The normal components of the fields change due to the differing dielectric and magnetic properties of the materials and the charges induced on the surface:

  (2)

with ρS and τS given as electric and magnetic surface charge densities, respectively. Perfect dielectric and magnetic materials can have no currents, which reduces Eq. (1) to

  (3)

Equation (3) means that the tangential fields are continuous across the boundary.

These boundary conditions are used in the method of moment analyses to determine currents. The method applies the boundary condition in integral equations to determine the coefficients of the expansion of currents in the sum of basis functions. The currents described as these sums do not satisfy the boundary conditions at all points but do when integrated over a region. This method leads to approximations that will converge as more terms are included in the expansions.

When doing analysis we find two types of surfaces convenient.We use these surfaces to reduce analysis effort by using planes of symmetry. The first one is the perfect electric conductor (PEC). A PEC surface causes the fields to vanish inside and to have electric currents induced on it:

  (4a,b)

A PEC surface is also called an electric wall. The second surface is the perfect magnetic conductor (PMC) and is a hypothetical surface. Whereas good conductors approximate PEC, there are no PMC materials. The PMC has no internal fields like the PEC and forces the tangential magnetic field to be zero:

  (5)

A PMC surface supports the hypothetical magnetic current density MS. We find that the magnetic wall (PMC) concept simplifies analysis.

FIGURE 1 Ground-plane images.

We use images of currents to include material boundaries in analysis. Figure 1 illustrates ground-plane images. When we analyze radiation from currents in the presence of a boundary, we include the actual antenna and its image to compute the fields. The figure shows an infinite ground plane, but a finite ground-plane image can be used in the angular region where a reflected wave occurs in the finite plane. We consider this idea further when discussing geometric optics. We can use images in dielectric boundaries provided that we calculate the polarization sensitive reflection coefficients to adjust the magnitude and phase of the image.