Multiplying these last two equations by each other, we get
c^2 B ̃E ̃=v^2 E ̃B ̃
c^2 =v^2
v=±c.
This is the desired result. (The plus or minus sign shows that the
wave can travel in either direction.)
As a byproduct of this calculation, we can find the relationship
between the strengths of the electric and magnetic fields in an elec-
tromagnetic wave. If, instead of multiplying the equationsc^2 B ̃=vE ̃
andE ̃=vB ̃, we divide them, we can easily show thatE ̃=cB ̃.o/The electromagnetic
spectrum.
Figure o shows the complete spectrum of light waves. The wave-
lengthλ(number of meters per cycle) and frequencyf (number of
cycles per second) are related by the equationc=fλ. Maxwell’s
equations predict that all light waves have the same structure, re-
gardless of wavelength and frequency, so even though radio and x-
rays, for example, hadn’t been discovered, Maxwell predicted that
such waves would have to exist. Maxwell’s 1865 prediction passed an
important test in 1888, when Heinrich Hertz published the results
of experiments in which he showed that radio waves could be ma-
nipulated in the same ways as visible light waves. Hertz showed, for
example, that radio waves could be reflected from a flat surface, and
that the directions of the reflected and incoming waves were related
in the same way as with light waves, forming equal angles with the
surface. Likewise, light waves can be focused with a curved, dish-
shaped mirror, and Hertz demonstrated the same thing with radio
waves using a metal dish.Momentum of light waves
A light wave consists of electric and magnetic fields, and fields
contain energy. Thus a light wave carries energy with it when it trav-
els from one place to another. If a material object has kinetic energy
and moves from one place to another, it must also have momentum,
so it is logical to ask whether light waves have momentum as well.
It can be proved based on relativity^17 that it does, and that the(^17) See problem 11 on p. 460, or example 24 on p. 438.
730 Chapter 11 Electromagnetism