In this chapter we explore the propagation of the simplest wave solutions of Maxwell s equations. We begin with the one-dimensional case and generalize to a planar wave in vector coordinates. We shall find that many properties of plane waves are identical to the flow of voltages and currents along a transmission line. We derive equations for wave reflection and refraction into various media. Finally, we introduce the Poynting vector to facilitate calculations involving the power flow of an electromagnetic field.

14.1 Maxwell s Equations in Source-Free Regions

One-dimensional waves

In a source-free region ? = 0 and J = 0, which simplifies Maxwell s equations to

Assume that E and H are uniform in the x ? y plane so that and . For this case the ? E simplifies to

Similarly, writing out the curl of H in rectangular coordinates

Time variation in the direction is zero. Thus the fields are entirely transverse to the direction of propagation. We call such fields TEM waves.

Polarized TEM fields

For simplicity, assume E _y = 0. We say the field is polarized in the -direction. This implies that H _x = 0 and H _y ? 0

We finally have it, a one-dimensional wave equation

Notice the similarity between this equation and the wave equation we derived for voltages and currents along a transmission line (see for instance Eq. (9.12)). As before, the wave velocity is and the general...