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Re: [Phys-l] Electromagnetic waves



Thanks to those who answered. What follows is my qualitative explanation, and two quantitative questions.

Starting with the LC circuit is very useful; the circuit can be said to be a source (antenna). Charging and discharging C is associated with the changing displacement current. In other words, the current is lowing back and worth (clockwise and counterclockwise), in the LC loop. This creates a changing magnetic field (Oersted effect), partially outside the loop. But a changing magnetic field creates an electric field (Faraday effect), also partially outside the loop. That is how an EM disturbance is propagating; it can be detected at some distance from the source (Hertz effect).

By why is the speed of propagation in a vacuum equal to 3*10^8 m/s ? And why is it smaller in a dielectric material (for example, ~2*10^8 m/ s in glass)? I say that I have no answers to these questions. The speeds have been measured and I accept the results as experimental facts. We can calculate the frequency, when L and C are given, but not the speed of the wave propagation. In other words, not all experimental facts are explainable, at our elementary level. Would such admission be appropriate?

Ludwik
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On Feb 28, 2010, at 3:14 PM, John Denker wrote:

On 02/28/2010 11:22 AM, Bob Sciamanda wrote:

Point out the simpler example of the damped oscillation equation, which
applies to both the mass on a damped spring and to the RLC circuit.

Yes!

The
oscillating mass/spring functions through the mechanism of a restoring
force

OK.

this mechanical concept is foreign to the electrical RLC circuit.

I wouldn't have said "foreign".

I suppose you could argue that the equation for the electronic
oscillator is "completely different" from the equation for the
mechanical oscillator ... but I would prefer to argue that it
is exactly the same equation.

I would explain that in upper-division theoretical physics,
students will learn that the choice of "coordinate" is really
quite arbitrary. And the RLC circuit is a beautiful example
of this ... suitable for exploration at the introductory level.

Actually you might as well start with the LC circuit,
without the R, to make things even simpler initially.

In any case, just looking at the equation, you don't need
to be a rocket scientist to see that if we choose the
charge on the capacitor to play the role of coordinate,
then the flux in the inductor plays the role of momentum.

Specifically, Q and φ are dynamically conjugate (in the
electronic oscillator) in *exactly* the same way that
x and p are dynamically conjugate (in the mechanical
oscillator). There is a even a Heisenberg uncertainty
relation for [Q,φ] just as there is for [x,p].

Feynman was fond of saying "the same equations have the
same solutions" and this is a beautiful example of that.

==========

To finish answering Ludwik's original question: Once they
understand the lumped-circuit model for the RC circuit, it
is only a small leap to the distributed RCRCRCRC ladder
circuit, which is an excellent model for a waveguide, which
supports electromagnetic waves, which were the subject of
the original question.

In the limit where the EM wave frequency is large compared
to the cutoff frequency of the waveguide, the latter can
be ignored, and you recover the equation for waves in free
space.

This is so nifty that I would make getting to this point
a goal of the course, no matter what I had to reorganize in
order to support this goal. There may be some personal
bias at work in this paragraph ... but I am not the only
person to think this is nifty.

While we are on the subject, there are other systems that
produce the same equations (and the same solutions!). Also
different equations with different solutions. The list goes
on and on, but here are a few salient items:
-- light waves
-- sound waves
-- waves on a string under tension
-- waves on a piano wire that has some tension plus some stiffness
... you can attach a small mass somewhere on the
string and treat it using the methods of perturbation
theory.
-- waves on a flexible beam, where it's all stiffness
and no tension
-- waves on the surface of water
(very interesting dispersion relation)
-- waves in a waveguide
-- elementary particles

A remark about the last item: The so-called waveguide equation
is also known as the massive scalar Klein-Gordon equation,
according to the particle physics folks. I wouldn't mention
this to students until after they had learned all about it,
but it's true. The cutoff frequency of the waveguide is the
_mass gap_ in the dispersion relation for the particle.

This is fun because it allows you to point out that a photon
in free space is massless, but a photon in a waveguide acts
like it has mass. This allows you to make a small down-payment
on answering questions about "what is mass" and "where does
mass come from". Also it allows you to explain the factor of
2 that shows up (or doesn't!) in the energy-versus-momentum
expression:
E = p v / 2 for slow-moving massive particles
E = p c for photons, *and* for anything moving so
fast that its rest energy is a small fraction
of the total energy.

And so on ..............

========================

One last pedagogical remark: Some students in the introductory
course are always asking, why do we spend so much time on the
harmonic oscillator? Who cares?

Don't forget to mention that they will be seeing this over and
over again. Mechanics, electronics, music, quantum field theory,
and everything in between.
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l

Ludwik Kowalski
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Ludwik's new book--"Tyranny to Freedom: Diary of a Former Stalinist"-- is now available at www.amazon.com

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