waves, then this might be surprising. A material object with more
energy goes faster, but that’s not the case for waves. The circulation
around the edge of the Amp`erian surface shown in the figure is twice
as strong for the light whose fields are doubled in strength, so the
left sides of Maxwell’s Γ equations are doubled. The right sides are
also doubled, because the derivative of twice a function is twice the
derivative of the original function. Thus if dim light moving with
a particular velocity is a solution, then so is bright light, provided
that it has the same velocity.
We can now see that all sinusoidal waves have the same velocity.
What about nonsinusoidal waves like the one in figure m? There
is a mathematical theorem, due to Fourier, that says any function
can be made by adding together sinusoidal functions. For instance,
3 sinx−7 cos 3xcan be made by adding together the functions 3 sinx
and−7 cos 3x, but Fourier proved that this can be done even for
functions, like figure m, that aren’t obviously built out of sines and
cosines in the first place. Therefore our proof that sinusoidal waves
all have the same velocity is sufficient to demonstrate that other
waves also have this same velocity.
We’re now ready to prove that this universal speed for all electro-
magnetic waves is indeedc. Since we’ve already convinced ourselves
that all such waves travel at the same speed, it’s sufficient to find
the velocity of one wave in particular. Let’s pick the wave whose
fields have magnitudesE=E ̃sin(x+vt) and
B=B ̃sin(x+vt),which is about as simple as we can get. The peak electric field
of this wave has a strengthE ̃, and the peak magnetic field isB ̃.
The sine functions go through one complete cycle asxincreases by
2 π= 6.28..., so the distance from one peak of this wave to the
next — its wavelength — is 6.28... meters. This means that it is
not a wave of visible light but rather a radio wave (its wavelength
is on the same order of magnitude as the size of a radio antenna).
That’s OK. What was glorious about Maxwell’s work was that it
unified the whole electromagnetic spectrum. Light is simple. Radio
waves aren’t fundamentally any different than light waves, x-rays,
or gamma rays.^16
The justification for puttingx+vtinside the sine functions is as
follows. As the wave travels through space, the whole pattern just
shifts over. The fields are zero atx= 0,t= 0, since the sine of
zero is zero. This zero-point of the wave pattern shifts over as time
goes by; at any timetits location is given byx+vt= 0. After one(^16) What makes them appear to be unrelated phenomena is that we experience
them through their interaction with atoms, and atoms are complicated, so they
respond to various kinds of electromagnetic waves in complicated ways.
728 Chapter 11 Electromagnetism