358 Week 10: Maxwell’s Equations and Light
for example (you should verify this, obviously, bydoingit). So yes,By(z, t) is also a wave
that propagates at the speed of lightc. The two components were presented together
because they arecoupledby Ampere’s and Faraday’s Laws. The variation ofExin space
and time produces the variation ofByin space and time, so that either one propagates
like a wave, but the waves are not independent. Similarly,EyandBxare coupled as
they vary along thezaxis in time, and obviously they satisfy the same wave equation
and propagate at the same speed as well.
The rest of the course is basically devoted to understanding light asan electromagnetic
wave. Although we will restrict ourselves to “one dimensional” wave forms, we will talk
a bit about how light varies with distance as it spreads out in three dimensions from a
central source. We will think at least a bit about sources, relying heavily on the oscillating
electric dipole as a model source. As a source, the dipole has one ideal feature: It is a
harmonicsource. Consequently, although light in general does nothaveto be harmonic,
we will find it very convenient to focus on understanding it as a harmonic wave^103.10.2: Light as a Harmonic Wave
Before we study light as a harmonic wave, let’s very quickly recapitulate things we know- orshouldknow – about waves based on our study of waves on a string and sound waves
in the first part of the course. Recall that we showed that a very general solution to the
wave equation for waves on a string was:
y(x, t) =f(x±vt) (860)wheref(u) is an arbitrary one-dimensional function. Basicallyanyfunctional form that
propagates to the right or left along thex-axis was a solution to the wave equation.
Since the electric and magnetic fields both satisfy one-dimensional wave equations for
propagation along thez-axis, we can expect this to be true for them as well. Any electric
field that we can create that has some shape at timet= 0 can be made to propagate
in the±zdirection by pairing it with the appropriate magnetic field. However,mostof
those arbitrary shapes are going to be very difficult to arrange, and arranging them to
occur with their correctly paired partner field even more difficult. Wewill thusignore
thisgeneralsolution and concentrate on a much more specific one, one tied to a particular
easy-to-imagine source.
Suppose the source of the wave we observe is indeed an oscillating electric dipole located
at the (distant) origin and aligned with thex-axis. Then we know that at any given
instant in time, if the dipole points up in the +xdirection, its field curls around and
points down in the−xdirection as it passes through thez-axis. At least, this was our
staticresult. Now, however, we see that this result can’t quite be correct. If the electric
field propagates at speedcand the dipole isoscillating, the field itself has to oscillate
too, and furthermore the “up” regions have to move away from the source atc, as do the
“down” regions. In other words, we’d expect the field to have the form of aharmonic
wave:
Ex(z, t) =E 0 xsin(kz±ωt) (861)(^103) Even when we treat light as a non-harmonic wave, we usually begin by transforming e.g. the initial conditions or
boundary conditions into the harmonic/frequency/wavenumber domain, solve the problem for harmonic waves, and
then use theFourier transformto transform back and obtain the general non-harmonic result. Of course this once
again requires more math to pursue. Physics majors, do you get the idea that you will need more math, sooner or
later? Math majors, do you see why you need to take more physics? Everybody else, aren’t you glad youdon’tneed
to in order to pretty much understand light waves perfectly well?