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Electromagnetic waves
This is an
appropriate point at which to demonstrate that Maxwell's equations possess
propagating wavelike solutions. Let us start from Maxwell's equations
in free space (i.e., with no charges and no currents):
Note that these equations exhibit a nice symmetry between the electric and magnetic
fields.
There is an easy way to show that the above equations possess wavelike
solutions, and a hard way. The easy way is to assume that the solutions are
going to be wavelike beforehand. Specifically, let us search for
planewave solutions of the form:
Here, and are constant vectors, is called
the wavevector, and is the angular frequency. The frequency
in hertz, , is related to the angular frequency via
.
The frequency is conventionally defined to be positive. The quantity
is a phase difference between the electric and magnetic fields.
Actually, it is more convenient to write
where, by convention, the physical solution is the real part of the
above equations. The phase difference is absorbed into the
constant vector by allowing it to become complex. Thus,
. In general,
the vector is also complex.
A wave maximum of the electric field satisfies

(438) 
where is an integer and is some phase angle. The solution to
this equation is a set of equally spaced parallel planes
(one plane for each possible value of ), whose normals lie
in the direction of the wavevector , and
which propagate in this direction with phasevelocity

(439) 
The spacing between adjacent planes (i.e., the wavelength) is given by

(440) 
(see Fig. 35).
Figure 35:

Consider a general planewave vector field

(441) 
What is the divergence of ? This is easy to evaluate. We have
How about the curl of ? This is slightly more difficult. We have
This is easily generalized to

(444) 
We can see that vector field operations on a planewave simplify to
replacing the operator with
.
The first Maxwell equation (430) reduces to

(445) 
using the assumed electric and magnetic fields (436) and (437), and
Eq. (442). Thus, the electric field is perpendicular to the direction
of propagation of the wave. Likewise, the second Maxwell equation gives

(446) 
implying that the magnetic field is also perpendicular to the direction of
propagation. Clearly, the wavelike solutions of Maxwell's equation
are a type of transverse wave. The third Maxwell equation gives

(447) 
where use has been made of Eq. (444). Dotting this equation with
yields

(448) 
Thus, the electric and magnetic fields are mutually perpendicular. Dotting
equation (447) with yields

(449) 
Thus, the vectors , , and are mutually
perpendicular, and form a righthanded set. The final Maxwell equation
gives

(450) 
Combining this with Eq. (447) yields

(451) 
or

(452) 
where use has been made of Eq. (445). However, we know from Eq. (439) that
the phasevelocity is related to the magnitude of the wavevector and the
angular wave frequency via . Thus, we obtain

(453) 
So, we have found transverse wave solutions of the freespace Maxwell equations,
propagating at some phasevelocity , which is given by a combination of and
. The constants and
are easily measurable. The former is related to the
force acting between stationary electric charges, and the latter to the force acting between steady electric currents.
Both of these constants were fairly wellknown in Maxwell's time. Maxwell,
incidentally, was the first person to look for wavelike solutions of
his equations, and, thus, to derive Eq. (453). The modern values of
and are
Let us use these values to find the phasevelocity of ``electromagnetic
waves.'' We obtain

(456) 
Of course, we immediately recognize this as the velocity of light. Maxwell also made
this connection back in the 1870's. He conjectured that light, whose nature had
previously been unknown, was a form of electromagnetic radiation. This was
a remarkable
prediction. After all, Maxwell's equations were derived from the results of benchtop
laboratory experiments, involving charges, batteries, coils, and currents, which apparently
had nothing
whatsoever to do with light.
Maxwell was able to make another remarkable prediction. The wavelength of
light was wellknown in the late nineteenth century from studies of diffraction
through slits, etc.
Visible light actually occupies a surprisingly
narrow wavelength range. The shortest wavelength blue light which is visible
has microns (one micron is meters).
The longest wavelength red light which is visible has
microns. However, there is nothing in our analysis which suggests that
this particular range of wavelengths is special. Electromagnetic waves
can have any wavelength.
Maxwell concluded that visible light was a small part of a vast spectrum of
previously undiscovered
types of electromagnetic radiation. Since Maxwell's time, virtually all of the
nonvisible parts of the electromagnetic spectrum have been observed.
Table 1 gives a brief guide to the electromagnetic spectrum.
Electromagnetic waves are of particular importance because they
are our only source of information regarding the universe around us.
Radio waves and microwaves (which are comparatively
hard to scatter) have provided much of
our knowledge about the centre of our own galaxy. This is completely unobservable
in visible light, which is strongly scattered by interstellar gas and dust
lying in the galactic plane.
For the same reason, the spiral arms of our galaxy can only be mapped out using radio waves.
Infrared radiation is useful for detecting
protostars, which are not yet hot enough to emit visible radiation.
Of course, visible radiation is still the mainstay of astronomy.
Satellite based ultraviolet observations have yielded invaluable insights into
the structure and distribution of distant galaxies. Finally, Xray and ray
astronomy usually concentrates on exotic objects in the Galaxy, such as pulsars
and supernova remnants.
Table 1:
The electromagnetic spectrum
Radiation Type 
Wavelength Range () 
Gamma Rays 

XRays 
 
Ultraviolet 
 
Visible 
 
Infrared 
 
Microwave 
 
TVFM 
 
Radio 


Equations (445), (447), and the relation , imply that

(457) 
Thus, the magnetic field associated with an electromagnetic wave is smaller
in magnitude than the electric field by a factor . Consider
a free charge interacting with an electromagnetic wave. The force exerted on the
charge is given by the Lorentz formula

(458) 
The ratio of the electric and magnetic forces is

(459) 
So, unless the charge is relativistic, the electric force greatly exceeds the
magnetic force. Clearly, in most terrestrial situations electromagnetic waves are
an essentially electric phenomenon (as far as their interaction with matter goes).
For this reason, electromagnetic waves are usually characterized by their wavevector
(which specifies the direction of propagation and the wavelength) and
the plane of polarization (i.e., the plane of oscillation) of the associated electric
field. For a given wavevector , the electric field can have any direction in
the plane normal to . However, there are only two independent
directions in a plane (i.e., we can only define two linearly independent
vectors in a plane). This implies that there are only two independent polarizations
of an electromagnetic wave, once its direction of propagation is
specified.
Let us now derive the velocity of light from Maxwell's equation the hard way.
Suppose that we take the curl of the fourth Maxwell equation, Eq. (433). We obtain

(460) 
Here, we have used the fact that
. The third Maxwell equation,
Eq. (432), yields

(461) 
where use has been made of Eq. (456). A similar equation can obtained for the electric field
by taking the curl of Eq. (432):

(462) 
We have found that electric and magnetic fields both satisfy equations of the
form

(463) 
in free space. As is easily verified, the
most general solution to this equation (with a positive frequency) is
where , , and are
onedimensional
scalar functions. Looking along the direction of the wavevector, so that
,
we find that
The component of this solution is shown schematically in Fig. 36. It clearly propagates in
with velocity .
If we look along a direction which is perpendicular to then
, and there is no propagation.
Thus, the components
of
are arbitrarily shaped pulses which propagate, without changing shape, along
the direction of
with velocity .
These pulses can be related to the sinusoidal planewave solutions which we found earlier
by Fourier transformation. Thus, any arbitrary shaped pulse propagating in the direction of
with velocity can be broken down into lots of sinusoidal oscillations propagating
in the same direction with the same velocity.
Figure 36:

The operator

(470) 
is called the d'Alembertian. It is the fourdimensional equivalent of the Laplacian. Recall that
the Laplacian is invariant under rotational transformation. The d'Alembertian goes one better
than this, since it is both rotationally invariant and Lorentz invariant.
The d'Alembertian is conventionally denoted . Thus, electromagnetic waves in free space
satisfy the wave equations
When written in terms of the vector and scalar potentials, Maxwell's equations
reduce to
These are clearly driven wave equations. Our next task is to find the solutions to these equations.
Next: Green's functions
Up: Timedependent Maxwell's equations
Previous: Potential formulation
Richard Fitzpatrick
20060202