Question about relativistic effects at synchrotons

[SergioPL]

This 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*k^2/(k+1) )^(1/2) k = 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 so I will know if my model is working or not.

 

[dwintz02]

I may be missing something, but what's wrong with using force balance as follows:

evB = (gamma*m)(v*v/r) where gamma = (1-v^2/c^2)^-.5

B=(gamma*m)*v/(re)

but gamma*m*v is just the relativistic momentum

B=p/(re)

Take this logic with some hesitancy though--synchotrons are pretty complex machines by themselves without making things hairy with relativity. I'm just not 100% sure of my answer but wanted to try and help.

 

[SergioPL]

Thank you dwintz02 for your help

Your answer is logical but I think you have forgotten a fact. Proper time grows the same way than inertial mass, with gamma(v). So the force seek by several bodies moving in diferent inertial frames will be the same, the time this force happens not.

The problem is based on the next: Let's supose 3 bodies, A, B and C so A and B are in the same inertial frame and C is moving at v x from A or B.

If B "boosts" dv (dv << 0) in direction x, so that now B is moving from A with speed dv x then B will be moving from C with speed (-v +dv/gamma^2)x and C will see B moving with speed -(-v +dv/gamma^2)x so Vbc = - Vcb. (B seeks C moving with the oposite speed than C seeks B).

But if B "boosts" dv (dv << 0) in direction y, so that now B is moving from A with speed dv y then, according to relativity B will see C moving with speed (v x - dv y ) whereas C seeks B moving with speed (- v x + dv/Gamma(v) y).

You easly see that Vbc no longer = - Vcb but the acceleration looks bigger for the one that accelarates. This phenomena is related with Thomas precession but with my model it disappears. One of the conclusions of my model is the result I gave in the previous question, so I want to see if it works OK and I succed :D!