Deriving Magnetic Force w/ Approximated Gamma-Factor at UVA

[greypilgrim]

Hi.

On the webpage of a lecture at the University of Virginia, I found this derivation of the magnetic force using Lorentz contraction and electrostatic force. They approximate the gamma factor to the order of $v^2$ and get the correct result. How is it possible to get to the same result using an approximation as one would get without approximating anything?

Of course, if multiple approximations are made to terms that get subtracted or divided, the errors introduced by the approximations might cancel out. But I can't see this happening here, using the same approximation for the increase in positive charge density and decrease in negative charge density and adding them should even increase the error.

What am I missing here?

 

[jartsa]

"So observers in the two frames will agree on the rate at which the particle accelerates away from the wire"

That sentence from the web page is is not true at all, because anything the particle does is time dilated in the wire frame, that includes accelerations.

And the result of the calculation is wrong. Forces should be different in the two frames, not the same.

Edit: Oh yes the problem was that the result should have been slightly wrong, but it seemed to be exactly right. That problem is solved now.

 

[stevendaryl]

"So observers in the two frames will agree on the rate at which the particle accelerates away from the wire"

That sentence from the web page is is not true at all, because anything the particle does is time dilated in the wire frame, that includes accelerations.

And the result of the calculation is wrong. Forces should be different in the two frames, not the same.

He's doing a non-relativistic approximation, keeping only the terms of order $v^2/c^2$ or lower. The difference in the acceleration in the two frames would be of order $v^4/c^4$. So it's correct to that order.

 

[Battlemage!]

This is similar to when you use the binomial theorem to get the Newtonian kinetic energy equation from the relativistic one by neglecting the higher order terms- and the reason you can ignore those terms is because (v/c)⁴ << (v/c)²

 

[julian]

The exact analysis is given in section 13-6 of the second volume of the Feynman Lectures on Physics.

The difference is because, as jarsta alluded to, transverse forces change as you go from one system to another due to time dilation.