# Homework Help: Relativistic Effects in the magnetic field of a synchroton

1. Dec 14, 2007

### SergioPL

[SOLVED] Relativistic Effects in the magnetic field of a synchroton

The Question is about the magnetic field which is needed to keep electrons in uniform circular movement (UCM) when their speed is not negligible compared with the speed of light.
If we ignore relativistic effects, magnetic field will take the following value:
B = (v*m)/(r*q) Where v is linear electron speed, r is the synchroton radius and m q are the electron mass and electric charge respectively.
If we include relativistic effects, I think special relativity cannot explain this experiment because the difference of speed is not the same seen from the electron’s instantaneous inertial frame of reference (IFR) than from the laboratory’s IFR. That is because of Thomas Precession.
Nonetheless, I have been working with a model that is able to determine magnetic field needed to keep electrons in UCM when they reach relativistic speeds. The result I get is the next:
B = (v*m)/(r*q) * ( 2*gamma^2/(gamma+1) )^(1/2) gamma = 1/(1-v^2)^(1/2)
If you know, theoretically or experimentally, the magnetic field need to keep the UCM you will make me a great favour telling me it.

Note: If you think that the force will increase gamma, like the inertial mass, that’s not the solution because the time interval also increases gamma and although we have to do a greater effort to accelerate the inertial mass, this effort spreads in time and both terms compensate one another.
The problem is focused on the variation of speeds and it is related with the Thomas precession.
Let’s suppose 2 particles A and B moving at Vab = V x, Vba = -V x. the particle A’ is in the same inertial frame that A until A accelerates to dv x (seek from A’) with dv<<V. Then the speed between A and B is V-dv/gamma and it’s still true that Vab = -Vba.
But if A accelerates in Y axis taking a value dv y (seek from A’) then the speed Vab is:
Vab = V x – dv y but Vba = -V x + (dv/gamma) y. You can see that Vab is no longer – Vba.
This effect is well known in relativity, two boost can be expressed as a boost and a rotation.
The problem of this result is that the particle which accelerates sees a change of speed in the other particle greater than the change of speed the non accelerating particle sees in the accelerating particle. This doesn’t happen if acceleration is parallel to speed, so it’s said that special relativity doesn’t work with rotating systems.
I have developed a model that can avoid the rotation, including additional transformations when you (A) “boost” a vector which connects particles from another inertial mark (B) to a third inertial mark (C). One of the most important hits in this model is when you calculate the speed between B and C you obtain, in most of the cases, a different result than the velocity addition formula. Other of my model’s results is the expression of the force, and thus the magnetic field, needed to keep UCM as shown above.
I have put 3 matlab scripts that contain this speed transformation algorithm in http://sergiopl81.googlepages.com/home. I have also placed a document explaining it but it’s in Spanish. Unfortunately I haven’t translated it yet.
So I would like to check if my model is working OK, if you can help me I would be very grateful.

Last edited: Dec 14, 2007