k/Red and blue light travel
at the same speed.l/Bright and dim light travel
at the same speed.m/A nonsinusoidal wave.standing still in space, then the right sides of the Γ equations would
be zero, because there would be no change in the field over time
at a particular point. But the left sides are not zero, so this is
impossible.^13
The velocity of the waves is a fixed number for a given wave pat-
tern.Consider a typical sinusoidal wave of visible light, with a dis-
tance of half a micrometer from one peak to the next peak. Suppose
this wave pattern provides a valid solution to Maxwell’s equations
when it is moving with a certain velocity. We then know, for in-
stance, that therecannotbe a valid solution to Maxwell’s equations
in which the same wave pattern moves with double that velocity.
The time derivatives on the right sides of Maxwell’s equations for
ΓE and ΓB would be twice as big, since an observer at a certain
point in space would see the wave pattern sweeping past at twice
the rate. But the left sides would be the same, so the equations
wouldn’t equate.
The velocity is the same for all wave patterns. In other words,
it isn’t 0.878cfor one wave pattern, and 1.067cfor some other pat-
tern. This is surprising, since, for example, water waves with dif-
ferent shapes do travel at different speeds. Similarly, even though
we speak of “the speed of sound,” sound waves do travel at slightly
different speeds depending on their pitch and loudness, although the
differences are small unless you’re talking about cannon blasts or ex-
tremely high frequency ultrasound. To see how Maxwell’s equations
give a consistent velocity, consider figure k. Along the right and
left edges of the same Amp`erian surface, the more compressed wave
pattern of blue light has twice as strong a field, so the circulations
on the left sides of Maxwell’s equations are twice as large.^14 To
satisfy Maxwell’s equations, the time derivatives of the fields must
also be twice as large for the blue light. But this is true only if the
blue light’s wave pattern is moving to the right at thesamespeed as
the red light’s: if the blue light pattern is sweeping over an observer
with a given velocity, then the time between peaks is half as much,
like the clicking of the wheels on a train whose cars are half the
length.^15
We can also check that bright and dim light, as shown in figure
l, have the same velocity. If you haven’t yet learned much about
(^13) A young Einstein worried about what would happen if you rode a motorcycle
alongside a light wave, traveling at the speed of light. Would the light wave have
a zero velocity in this frame of reference? The only solution lies in the theory of
relativity, one of whose consequences is that a material object like a student or
a motorcycle cannot move at the speed of light.
(^14) Actually, this is only exactly true of the rectangular strip is made infinitesi-
mally thin.
(^15) You may know already that different colors of light have different speeds
when they pass through a material substance, such as the glass or water. This
is not in contradiction with what I’m saying here, since this whole analysis is for
light in a vacuum.
Section 11.6 Maxwell’s equations 727