Week 10: Maxwell’s Equations and Light 353
magnetic fields make electric fields. Maxwell showed us thatat the same time, changing
electric fields make magnetic fields! Why is this significant? Because a changing electric
field can make a changing magnetic field that makes a changing electricfield that makes
a changing magnetic field that makes – wait a minute! Is it possible thatwe could have
anelectromagnetic wave?
It is!
To see this is a bit tricky. It is tricky because we are taking an intro course where we have
to avoid “real” differential multivariate calculus and the dread∇~ differential operator.
We have learned only the integral equation forms, which means basically that we have
to convert them into derivatives in order to end up with a wave (differential) equation
for the electric and magnetic field. Let’s get to it.
We start by doing away with one complication – the sources. Note thatultimatelyboth
electric and magnetic fields have to come fromelectric charges– the only in Maxwell’s
Equations that get electric or magnetic fields into the Universe in thefirst place are those
pesky little charges, but again, to understand how they make themcorrectlywe really
need to lose the integrals and work with differential equations and we’re not ready to do
that yet (and never will be, in this course). So here is a very short discursion on sources
of electromagnetic fields:10.1.1: Accelerating Charge
Against my custom I’m not derivinganythingin this short section. Either you believe
me or you don’t, or you read a book or take an advanced course that does it right^97.
Just be sure that you take two or three fairly serious courses in ordinary and partial
differential calculus and maybe complex variables first...
If one takes an electric charge and accelerates it, it radiates awayelectric and magnetic
energy in the form of electromagnetic waves of the sort we’re about to derive. Charge
moving at a constant velocity (which is a frame transformation awayfrom being charge
at rest) does not radiate energy. It may produce an electric and magnetic field, but
that field is guaranteed not to carry any energy away. Only when it accelerates does
the charge radiate (and of course, there is no inertial frame thatcan get rid of that
acceleration, so the radiation occurs in all frames).
That’s it. Not complicated at all (although the derivation of this factis a bit hairy).
Well, when do charges accelerate? Well, they’d accelerate if they (for example) went
around in a circular orbit. That pesky centripetal acceleration qualifies as one that
would radiate energy. They’d also accelerate if they were just oscillating harmonically,
as a harmonic oscillator in one dimension is nearly always accelerating.
These two observations are among the most profound in all of physics. What they add
up to is this:There is no obvious way to make a model for an atom that does notinvolve
orbiting, oscillating charge!No non-obvious way either, at least not classically, especially
not one that agrees with the observation that atomsdoradiate electromagnetic energy,
but only at certain fairly sharp energies and frequencies!
In fact, if you build a simple model for an atom consisting of a proton being orbited by
a light electron, you find that it collapses, with the electron spiralling into the proton
while it radiates away energy, in around 10−^20 seconds. A classical Universe based on
Maxwell’s equations would last just about that long.(^97) Such as my grown-up graduate E&M book online, used in a graduate course in Classical Electrodynamics. Even
most undergrad intermediate E&M courses do a sloppy job of treating radiation from sources, partly because the
math required is relativelydifficult