j/Example 23. The incident
and reflected waves are drawn
offset from each other for clar-
ity, but are actually on top of
each other so that their fields
superpose.
wave terminology, we say that the wave is transverse, not longitudi-
nal.) The wave pattern in figure i is impossible, because it diverges
from the middle. For virtually any choice of Gaussian surface, the
magnetic and electric fluxes would be nonzero, contradicting the
equations ΦB= 0 and ΦE= 0.^12
Reflection example 23
The wave in figure j hits a silvered mirror. The metal is a good
conductor, so it has constant voltage throughout, and the electric
field equals zero inside it: the wave doesn’t penetrate and is 100%
reflected. If the electric field is to be zero at the surface as well,
the reflected wave must have its electric field inverted (p. 376), so
that the incident and reflected fields cancel there.
But the magnetic field of the reflected wave isnotinverted. This is
because the reflected wave has to have the correct right-handed
relationship between the fields and the direction of propagation.Polarization
Two electromagnetic waves traveling in the same direction through
space can differ by having their electric and magnetic fields in dif-
ferent directions, a property of the wave called its polarization.The speed of light
What is the velocity of the waves described by Maxwell’s equa-
tions? Maxwell convinced himself that light was an electromagnetic
wave partly because his equations predicted waves moving at the ve-
locity of light,c. The only velocity that appears in the equations is
c, so this is fairly plausible, although a real calculation is required in
order to prove that the velocity of the waves isn’t something like 2c
orc/π— or zero, which is alsocmultiplied by a constant! The fol-
lowing discussion, leading up to a proof that electromagnetic waves
travel atc, is meant to be understandable even if you’re reading this
book out of order, and haven’t yet learned much about waves. As
always with proofs in this book, the reason to read it isn’t to con-
vince yourself that it’s true, but rather to build your intuition. The
style will be visual. In all the following figures, the wave patterns
are moving across the page (let’s say to the right), and it usually
doesn’t matter whether you imagine them as representing the wave’s
magnetic field or its electric field, because Maxwell’s equations in
a vacuum have the same form for both fields. Whichever field we
imagine the figures as representing, the other field is coming in and
out of the page.
The velocity of the waves is not zero. If the wave pattern was(^12) Even if the fields can’t be parallel to the direction of propagation, one might
wonder whether they could form some angle other than 90 degrees with it. No.
One proof is given on page 730. A alternative argument, which is simpler but
more esoteric, is that if there was such a pattern, then there would be some other
frame of reference in which it would look like figure i.
726 Chapter 11 Electromagnetism