Deriving Magnetic Force w/ Approximated Gamma-Factor at UVA

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• greypilgrim
In summary: So, if you want to know the force between two particles in a system, you have to use a procedure that ignores time dilation, like the one Feynman gives in the lecture.
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?

"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.

Last edited:
julian
jartsa said:
"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.

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

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.

What is the concept of "Deriving Magnetic Force w/ Approximated Gamma-Factor at UVA"?

The concept of "Deriving Magnetic Force w/ Approximated Gamma-Factor at UVA" involves using the approximated gamma-factor to calculate the magnetic force on a charged particle moving at relativistic speeds in a magnetic field at the University of Virginia (UVA). This concept is commonly used in particle physics and electromagnetism research.

Why is it important to use an approximated gamma-factor in this calculation?

The use of an approximated gamma-factor is important in this calculation because it allows for a more accurate and simplified calculation of the magnetic force on a moving charged particle. The full gamma-factor equation can be complex and time-consuming to calculate, especially for particles moving at relativistic speeds.

How is the approximated gamma-factor calculated in this context?

The approximated gamma-factor is calculated using the Lorentz factor, which is a function of the particle's velocity. The Lorentz factor is then squared and divided by 1 plus the Lorentz factor squared to obtain the approximated gamma-factor.

What are some potential limitations of using an approximated gamma-factor in this calculation?

One potential limitation is that the approximated gamma-factor is only accurate for particles moving at relativistic speeds. For particles moving at non-relativistic speeds, the full gamma-factor equation should be used for a more accurate calculation. Additionally, the approximated gamma-factor may not account for other factors that could affect the magnetic force, such as the particle's direction of motion or the strength of the magnetic field.

Are there any real-life applications of this concept?

Yes, this concept has many real-life applications in fields such as particle accelerators, magnetic confinement fusion, and medical imaging. It is also used in the development of new technologies, such as particle detectors and magnetic levitation devices.

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