Imagine a point charge sitting in the middle of infinite space,
its electric field lines radiating straight out to infinity.
Suddenly, the charge starts to oscillate, back and forth.
What happens to its field lines?
And if you are a million kilometers away from the charge, when do you
find out that it has begun to oscillate?
The answers to these questions are provided by Maxwell's equations,
and by the idea of electromagnetic waves.
Let's answer the second question first.
The news that the charge has started to oscillate travels outward from
the charge at the speed of light.
Hence, you find out about the oscillation after a time
.
This is pretty quick, considering the distance, but it is
not instantaneous.
Now for the answer to the first question: when the charge starts to oscillate, its field
lines no longer radiate straight out.
Instead, Maxwell's equations predict that the field lines snake out through space
in the shape of sinusoidal waves, with the wave crests moving
outward at the speed
If we substitute the numbers into this formula we get
the well-known speed of light. When he discovered this connection, Maxwell drew the only reasonable conclusion: light must be an electromagnetic wave of the type produced by oscillating charges. In fact, all electromagnetic waves are produced by oscillating charges. Charges oscillate back and forth in the large antennas of television and radio stations, charges oscillate in the klystrons of radar installations and microwave ovens; charges oscillate in the atoms that radiate in the infrared, visible, ultraviolet, and x-ray parts of the electromagnetic spectrum, and finally, charges oscillate in the nuclei that radiate gamma rays.
Maxwell's equations tell us more about these waves than just their speed.
They also tell us that the magnitudes of the electric and magnetic fields
in these waves always satisfy the relation
This means that even if the electric field amplitude is quite large, the magnetic field will be quite small because c is so large. For this reason we almost always forget about the magnetic field when thinking about what these waves do. For instance, when your car radio antenna picks up a radio station, it is responding only to the electric field of the incoming radio waves.
Maxwell's equations also tell us that these waves are transverse, meaning
that both 
and 
are perpendicular to the direction the
waves are traveling.
In addition, and are perpendicular to each
other, with their cross product pointing in the traveling direction
of the waves:
As an example, let's write down the wave functions for an electromagnetic
wave of frequency 
and wavenumber k traveling along the
x-axis with an electric field of amplitude 
pointing in the
y-direction.
Note the appearance of x in the argument of the sine function,
indicating that the wave travels along the x-axis.
Also note that
points in the x-direction, as it should.
As usual, and k are related to each other through the
wave speed, as discussed in Chapter 16:
