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Magnetic field of charge moving at constant velocity 
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#19
Jul3111, 04:09 AM

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I don't understand what you mean by different approaches to solve for the full Maxwell equations. If you mean the BiotSavart Law from magneto statics, of course it's not applicable to the magnetic field of accelerated charges since it works only in the special case of magnetostatics, i.e., stationary currents.
In any case, you get the right answer by using the retarded Green's function (LienardWiechert potentials for an arbitrarily moving point charge). In the case of a uniformly moving charge you get the fields also from Lorentz boosting the Coulomb field for the charge at rest. Of course, both solutions must be identical since Maxwell's electromagnetics is a relativistically covariant classical field theory. 


#20
Jul3111, 07:25 AM

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#21
Jul3111, 01:36 PM

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#22
Aug111, 03:19 AM

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Dale, thanks for all your feedback, much appreciated. An issue remains for the low speed approximation. At low speeds there are still electrodynamic and magnetodynamic terms dE/dt and dB/dt in Maxwell and yet the solution is a magnetostatic. So why is the low speed solution magnetostatic? (1) are these dynamic terms simply both small at low speeds, or (2) do the effects of dE/dt and dB/dt cancel (and if they cancel, is there some insight about that)?



#23
Aug111, 03:23 AM

P: 18

RedX, in terms of deeper insight, my suspicion is that the equations can be separated, perhaps Helmholtz decomposition, into parts that behave in different ways, and such a decomposition will explain the result. But I haven't figured out what that decomposition is yet.



#24
Aug111, 06:22 AM

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#25
Aug111, 12:22 PM

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How would you think this specific result was viewed by Maxwell and his peers, if it was known? It surely would have seemed odd that in moving from a static particle to a frame at constant speed, a magnetic field mysteriously appeared yet the electric field was essentially the same. Did this get commented on?
Of course everyone believed in the Ether and absolute frames but I'm surprised that nobody noticed or commented on this odd effect. If they had and had investigated, maybe they would have scooped Einstein by 50 years! 


#26
Aug111, 12:32 PM

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#27
Aug111, 12:54 PM

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That is, [itex]\int E \cdot d\vec{l}=\frac{d \Phi}{dt}=0 [/itex] since phi is zero. 


#28
Aug311, 02:21 PM

P: 262

(If needed, use an online calculator to get the maths out of the way that’s more than fine by me.) Just some of my thoughts: I can see that you will be able to derive the relativistic equation for the electric field of a point charge, which is useful for high speed particles but I’m not sure about deriving its magnetic field and especially not its relativistic magnetic field. This last one always seems a “double up” effect to me since a magnetic field is in the first place understood as a result of relativity wrt moving charges. 


#29
Aug311, 05:19 PM

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[tex]{\bf E}= \frac{q{\bf r}} {\gamma^2[{\bf r}^2({\bf v\times r})^2]^{\frac{3}{2}}}[/tex] and [tex]{\bf B}={\bf v\times E} =\frac{q{\bf v\times r}} {\gamma^2[{\bf r}^2({\bf v\times r})^2]^{\frac{3}{2}}}[/tex] 


#30
Aug411, 02:13 PM

P: 262

And exactly what are you doing here?
Copy & paste, how clever! Why couldn’t I think of that? 


#31
Aug411, 04:57 PM

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#32
Aug511, 08:58 AM

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Jtbell that paper your referring to doesn’t look like “ a Lorentz boosted coulomb field” to me. It looks a lot more complicated including a lot of maths and Greek. Oh well, someday someone will come up with something a bit more consumer friendly.



#33
Aug611, 02:48 AM

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I find the manuscript by Fitzpatrick excellent. It's also awailable as a textbook. I can only recommend to read his chapter on the fully relativistic treatment of electromagnetics. I never understood, why there is no textbook on classical electromagnetism that from the very beginning strictly uses the relativistic framework. Instead all authors copy more or less the classical textbooks of the first half of the 20th century (although there are excellent books among them, first of all Sommerfeld's Lectures on Theoretical Physics and Becker's book). The only exception is LandauLifgarbages in his vol. II, but in vol. VIII he treats the constitutive equations nonrelativistic as usual. But that lamento becomes offtopic now...



#34
Aug611, 06:36 AM

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#35
Aug911, 01:08 PM

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Thank you all for the comments and feedback it was most helpful and has spurred me on to do some of the maths for myself. One of my difficulties is that many textbooks "fudge" the solution by ignoring the fact that a point charge is essentially a delta distribution and they skip essential steps in the calculation. However, you all encouraged me to make the effort, after 40 years of abstinence. Thanks!



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