One of the hallmarks of scientific thought of the last 20 centuries is the derivation of the fundamental connection between electric and magnetic fields that leads to the notion of the electromagnetic wave.
This derivation, first done by James Clerk Maxwell around 1865, laid the foundation for the fundamental discoveries of radio that have lead to the basis for remote communication throughout the world. While the mathematics needed to rigorously derive the results of Maxwell starting from Maxwell's equations are generally beyond what would normally be expected for someone taking a freshman course in electricity and magnetism, Maxwell's approach can be done in a "physics" sense that more or less mimics the actual derivation of the wave equation for electricity and magnetism. This approach begins by first thinking about the characteristics of electric and magnetic fields for cases where both must always be produced. Essentially any situation which causes static charges to move will create a situation in which both electric and magnetic fields are present. Using Maxwell's equations, we can then look at the constraints on the form these fields must take.
We begin with the description of an oscillating electric dipole. The picture here is one of a dipole in which we allow the electric charges to change their relative positions. For our purposes, we wish to have both the charges move such that their positions with respect to time are defined by sinusoidal functions (let's say a cosine to be definite although sine would do just as well). In this case, the charges oscillate back and forth as though each were connected to a (separate) spring which had the middle of the dipole as its equilibrium position. The charges therefore go from one end of the dipole to the other as shown in the Java animation (see figure 12.1) if you click the "Start the animation" button. Their positions are independently specified by cos(t/T + f) where T is the period of the motion and f is, of course, the phase specifying the initial position of the charges.
The animation shows two views, one from the side and one from the top, above the dipole. These two views allow us to see both the electric field (in the side view) and the magnetic field (top view) where, in both views, the other field is perpendicular to the screen. Note that the magnetic field exists because, as the charges oscillate, they constitute a current and every current creates a magnetic field. According to Ampere's Law, this field is perpendicular to the plane containing the current and is seen to have equal magnitude for a circular path of given radius around the current as shown in the top view of the animation.
Let's focus on the electric field for now. A static picture of the dipole at any given time would appear as shown in figure 12.2a.)
When the charges are in their reversed position, we see the field as shown in figure 12.2b.) These two views come about with period T. Therefore, if we try to find what the field looks like at intermediate points in time, that is, between the two views, we are faced with the situation shown in figure 12.3.
Since we don't assume the change can propagate instantly once the new position is reached, the "blank" area represents what has to happen to the fields in the meantime. Since Faraday has explained that E field lines can't cross and that we need to have them be continuous (except when they end on charges), then a plausible guess for what the fields look like is shown in figure 12.4.
The base conclusions we can take from this simple, qualitative analysis are that
Believe it or not, with just these three observations in hand, plus some calculus, we can come up with a startling conclusion as to how the fields interact with each other.
If we extrapolate these fields out far enough, they look like plane waves, i.e. the E and B fields stretch out enough that, for a sufficiently small section of space, they look approximately straight rather than curved as shown in figure 12.5. In this case, we can identify a small section of space as a wave front, a plane which is perpendicular to the direction of propagation of the fields. It is this wave front which we use for our quantitative analysis.
First, let's draw a portion of the plane wave. The wave propagates along the x axis which means the E and B fields lie along the y and z axes, respectively. We then pick two plane wavefronts which are of sizes Dx by Dy and Dx by Dz as shown in figure 12.6. We keep the sizes of Dx, Dy, and Dz smaller than the range of distance over which the curvature of E and B are nonnegligible.
To find the quantitative description of the E and B fields as
functions of time and x position, we make use of AmpereMaxwell's
Law and Faraday's Law. First consider the case of the B field's
effect on the E field. We assume that B field moves along the x
axis. It is also changing along the z axis in
figure 12.6 above, but we do not yet
have a description of how it is changing. For the moment, we take
the average value of B_{z} along the x axis as being somewhere
between B_{z, 1} and B_{z, 2}. For simplicity, we assume
that the average is the field all along the Dx distance
between the two plane fronts. Since B_{z} is, in general, not
zero, we have a flux through the area bounded by E_{y, 1}
and E_{y, 2}. That flux is
 (12.1.2.1) 
 (12.1.2.2) 

We can perform the same analysis for the effect of the E
field on B. If we note that the E field is not
constant over the Dx, Dz area bounded by
B_{z, 1} and B_{z, 2} in
figure 12.6, but use the average
value E_{y} over this interval, then we have a nonzero
electric flux (usually) over this interval. Since the electric
flux will also not be constant in time or space across this
interval (the E field is moving along x and probably has time
variation in magnitude as a function of x), the AmpereMaxwell
relation will come into play. Again, in doing the line integral
of B around the Dx, Dz boundary, remember that
the B field is parallel to the z axis and has no components along
x. In this case, we evaluate the line integral by going along
B_{z, 1}, parallel to +x, down along B_{z, 2},
and along x.



 (12.1.2.7) 
 (12.1.2.8) 

 (12.1.2.10) 
Finally, we note that if we consider this plane wave as
propagating for a very short period of time dt, then
we can assume that the electric and magnetic fields are
essentially constant over the distance dx = c dt traveled.
Considering equation 2,
we can integrate the nearly constant E field over an
infinitesimal distance dy. We find that there must be
a relationship between the magnitude of the electric and
magnetic fields as follows.

We expect that electromagnetic (EM) waves should carry
energy just as mechanical waves do. This is, in fact,
true. Any region of empty space that the wave travels
through will have an energy density, u, associated with
the electric and magnetic fields so that
 (12.1.3.12) 

 (12.1.3.14) 


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