Electromagnetism and Relativity

Consider two parallel, identical beams of electrons (same velocity v, same charge/length l ). Consider them infinitely long, to eliminate any problem with end effects. There will be a magnetic field at one beam due to the other equal to
m _oi/2p r, where r is the distance apart. Current i is
l L/t = l v. The magnetic force on each beam is an attractive force equal to iLB, or plug in and get
(l v)(L)(m _o l v/2p r), or
force/length = (m _o/2p r)l ²v². There is also a repulsive electric force. The electric field due to one beam at the location of the other is found (by Gauss's law) to be
E = l /2p e _or. Hence the repulsive force/length, qE/L, is
l ²/2p e _or. The net force is the difference between the electric force and the magnetic force.

Now consider an observer traveling along with the beam velocity. To that observer there is no moving charge, so no current and no magnetic field. Thus we have an apparent paradox that the force is different for different observers. We resolve the paradox by assuming that the length L is different for different observers. Let us call L_o the length of the beam as seen by the observer that has the same v as the beam (called "proper length" because the measurer is at rest with respect to the thing measured). Now equate force/length:

l _o²/2p e _or = l ²/2p e _or - (m _o/2p r)l ²v²

Recall that
c = (e _o m _o)^-1/2, so
m _o = 1/e _oc² and use
l _o = q/L_o, l = q/L. Plug this stuff in, do the algebra and show
L = (1 - v²/c²)^1/2 L_o = L_o/g . Moving bodies are shortened in the direction of motion.

Note that no assumption about c being constant to all observers has been employed. Thus we could base relativity on the reasoning here and derive the fact that c is the same to all. What are the fundamental assumptions in the above derivation?

Now let us consider the relativity of time. A train will pass by and it is an easy matter to time it: From when the front passes a point to the time when the rear goes by the time is D t_o according to an observer at the train station, and D t according to an observer on the train who is relying (somehow) on the clock on the platform. So v is L/D t_o where L is the length of the train to the observer on the platform. According to the observer on the train, v is L_o/D t. Equate the two and show that
D t = g D t_o. What assumptions have been made here?

Now to show that c is constant to all, consider two coordinate systems xyz taken to be at rest, and x'yz moving with velocity v in the +x direction. At t=t'=0, the origins coincide, and a light pulse starts out at the common origin. From the point of view of an xyz system observer, after time t the light would be at x, y, z where
c²t² = x² +y² + z² and x', y, z where
(c't')² = x'² + y² + z²
with x' = x - vt and t' = t/g . Working with the primes,
c'²t²/g ² =x²-2xvt +v²t² + y² + z² =c²t² - 2xvt + v²t².
So c'² = g ²(c² - 2xv/t + v²).
But x/t = v, and
g ² = (1-v²/c²)^-1= c²/(c²-v²), so this reduces to c' = c.

Why this goofy result? If you could travel at the speed of light beside a light wave, conventional thinking would say that if you could detect the light wave, you would find steady-state fields in it. But if that were possible, we ought to be able to create it in the lab, because the physics of a moving system should be the same as the physics "at rest." The stationary E&M waves would violate Maxwell's equations, making it theoretically impossible; but don't trust theory- there is no experimental evidence that this is possible. Find the experimental evidence and get a Nobel Prize. (You can have standing waves in a waveguide, but the fields oscillate in direction. That is different.)

Oops, I just realized that since I used c = (m _o e _o)^-1/2, which must be constant to all because it has no velocity dependence, so my statement above about deriving the constancy of c is bogus. (Did your spider-sense tingle when you read the paragraph stating that "no assumption about c being constant….") So I guess c constant to all is the appropriate fundamental principle. When I started this I was thinking that we could formulate it differently, but now I tend to think that an alternate starting point would be too awkward. In physics we favor simplicity.

So why didn't I delete this thing? Doing this exercise taught me something, so maybe someone else will also benefit from it. Any comments, hate mail, infect me with virus, etc: fredrick.gram at tri-c.edu (remove spaces and replace at with @. This is my defense against spammer software that gets email addresses that are listed on the web).

Back to E&M and all that.

Or go to one of the other main physics pages:
Mechanics
Vibrations and waves
Quantum
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