Week 14

Basic Facts

Complex Notation: Because linear operators do not mix the real and imaginary
parts of a complex number we can represent a real sinusoidal wave as either the
real or the imaginary part of a complex wave. That is, we can represent the plane wave
by the imaginary part of the complex plane wave
and the real plane wave
by the real part of the same complex wave.

Non-Uniform Plane Waves: When an electromagnetic wave enters a conductor
there is a conduction current term that must be added to the wave equation as well
as the displacement current term. This additional term is a dissipative or friction-
like term that represents energy lost to heat in the conductor. Because of this term,
the plane wave decays exponentially as travels through the conductor with a decay
length d called the Skin Depth. In a good conductor the conduction current is
much larger than the displacement current and the decay is very rapid. To an
excellent approximation we find that

Reflection and Refraction: When a plane wave crosses a surface from a medium of
refractive index n₁ to a medium of index n₂ a portion of the energy is transmitted and
a portion is reflected. We define the direction of the waves by the angles between the
wave crests and the surface or, equivalently, by the angles between the rays and the
normals to the surface as shown below.