17.1.Plane Monochromatic Waves in Nonconducting Media
Differentiating with respect to the time yields the phase velocity,
Vp =
Vp = c =
17.2.Polarization
There more to be said about the complex vector amplitudes and . In fact, we have not yet discussed fully what we mean by a complex vector. Two obvious meanings suggest themselves: A complex quantity whose real and imaginary parts are real vectors.
= Er + iEi
or a vector whose components (with respect to real basis vectors) are complex scalars,
= pp + ss + uu
17.3.Energy Density and Flux
We have freely used complex expressions for the E and B fields, with the understanding that the actual physical quantities are given by the real parts of the complex quantity. The mathematical justification of this prosedure is that the Maxwell equations are linear equations that are satisfied separately by the real and imaginary parts of a complex solution. The expressions
17.4.Plane Monochromatic Waves in Conducting Media
However, the physical interpretation of the solution is considerably more difficult here. There is another approach that does not simplify the physical interpretation, but it does provide some additional insight into electromagnetic waves, in conducting media in particular.
Here we used K, the complex dielectric constant, defined as
K = K +
It is convenient to define the complex index of refraction n by analogy to what was done earlier for the case of real index of refraction and real dielectric constant:
A2 = K
17.8.Spherical Waves
As an example of a more difficult wave problem, where in fact it is not easy to find even the elementary waves, we consider the wave equation in spherical coordinates. The wave equation for the electric field in a vacuum is
For monochromatic waves, the equation for the spatial portion becomes
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