Attempt to find the magnetic field of a moving charge

[Hiero]

First let me say THANK YOU to anyone who reads my work and tries to help, I know it can be a pain to decipher the work of others (although I've tried to make it as clear as possible). If any part is unclear I will try to explain it.

I cannot for the life of me see any flaws in the argument, so help is much appreciated.

Also note that [v-u] stands for the relativistic difference of speeds.

EDIT: on the second to last expression (on the bottom line) I left out a 1/c^2 factor on accident but the result is unchanged

 

[Dale]

Why didn’t you just use the Lienard Weichert fields?

https://en.m.wikipedia.org/wiki/Liénard–Wiechert_potential

 

[jartsa]

When a moving charge passes a still charge, there is some "extra Coulomb force", because the electric field of the moving charge is "length contracted". I mean, in a frame where the charge moves its electric field is "length contracted".

So include that to your calculation ... or make the relative speed of the charges zero.

 

[Hiero]

@Dale Well I don't just want the answer, I want to know why this simple setup yields a wrong answer.

@jartsa I already accounted for that effect. In the picture I said, "but we know F_E = ϒ_vF₁," I didn't explain why, but it's because of the effect you're referring to. The electric force in the third frame (F_E) is ϒ_v times larger than the electric force in the first frame (F₁).
Any other ideas as to why this went wrong?

 

[Hiero]

@jartsa I was mistakenly thinking I had to break it up into three frames with two speeds but you are right we can just work with q and Q moving together and one frame moving v relative to them.

Lets still call the coloumb force F₁, which is now the proper force on q, so in the other frame the force must come out to be smaller: F_E+F_B = F₁/ϒ where we also have (by the effect you mentioned) F_E = ϒF₁

Putting those together and solving for B gives the exact same wrong answer as in my OP.

Surely someone can point out the error now that the derivation is this simplified?

 

[jartsa]

@jartsa I was mistakenly thinking I had to break it up into three frames with two speeds but you are right we can just work with q and Q moving together and one frame moving v relative to them.

Lets still call the coloumb force F₁, which is now the proper force on q, so in the other frame the force must come out to be smaller: F_E+F_B = F₁/ϒ where we also have (by the effect you mentioned) F_E = ϒF₁

Well in this case "length contraction" of the electric fields is considered magnetism. Whatever force the "length contraction" may cause, that is counted as a magnetic force.

Any difference in the net force in different frames is considered a magnetic force:

F' - F = F_magnetic

And the difference in the net force according to special relativity is: F-F/ϒ

 

[Hiero]

Well in this case "length contraction" of the electric fields is considered magnetism. Whatever force the "length contraction" may cause, that is counted as a magnetic force.

Any difference in the net force in different frames is considered a magnetic force:

F' - F = F_magnetic

And the difference in the net force according to special relativity is: F-F/ϒ

You have me thoroughly confused now. Is it not true that the transverse component of a proper electric field becomes larger (by gamma) when boosting out of the proper frame? But you're saying this is the source of the magnetic field? That doesn't make sense to me for various reasons and it also doesn't give the right magnetic field.
Actually, looking at the other thread, I don't think I made any mistakes. I think whoever told me I was wrong was wrong. I say this on the basis of the link given by @Khashishi (http://farside.ph.utexas.edu/teaching/em/lectures/node125.html) in the other thread.

In fact I think this link also agrees with my answer, given the simplifications of my setup. https://en.m.wikipedia.org/wiki/Liénard–Wiechert_potential (although at first I thought it disagreed).
Edit:
Wait no, this latter link does disagree. So now I'm confused as to why

 

[jartsa]

You have me thoroughly confused now. Is it not true that the transverse component of a proper electric field becomes larger (by gamma) when boosting out of the proper frame? But you're saying this is the source of the magnetic field? That doesn't make sense to me for various reasons and it also doesn't give the right magnetic field.

Hmmm electric field becomes stronger as you say, which causes Coulomb force to be larger, well I never thought of that. Oh dear.In post #1 when calculating F₁ is this taken into account?