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In textbooks, such as[1,2], magnetism is taught to be a consequence of relativistic length contraction. The magnetic force is derived in the special case of an infinite straight wire, by finding the total line charge density $\lambda_{tot}=\lambda_++\lambda_-$, and evaluation it as if it was a static charge. The Wikipedia page on the subject[3] states that "The chosen reference frame determines if an electromagnetic phenomenon is viewed as an effect of electrostatics or magnetism". I have found that this is not correct. The textbook analysis will only work in the special case of parallel motion next to an infinite wire.

This can easily be seen by considering perpendicular motion of the test charged with respect to the infinite wire. If we let the test charge move along the x-axis, and the charged wire run along the y-axis, the Lorentz transformations will be independent of y. All charges in the wire will have the same x' and t' coordinate, and there will be no length contraction along the y-axis. The line charge density will not change. $\gamma_-=\gamma'_-$ and $\gamma_+=\gamma'_+$. Hence the wire is electrostatically neutral in the rest frame of the test charge, and no force occurs.

The problem can be resolved by using the dynamic (retarded) electric field in both frames. (eq. 1) See my in-depth analysis in this link: https://drive.google.com/file/d/1HITikNdOX-IbxHmQVZVKQLATOrNXheXp/view?usp=sharing

$$E_D=\frac{q(1-v^2/c^2 )}{4 \pi_0r^2 (1-v^2/c^2 sin^2 ( \theta ))^{(3/2)}} \mathbf{\hat{r}} \tag{1}$$

I come to the conclusion that magnetism must be understood as an electrodynamic phenomenon in the rest frame of the test particle, and as a combined effect of field retardation and relativistic length contraction.

Question: From my analysis, it seems obvious that the magnetism is an electrodynamic effect, and yet I have found no mention of it in either textbooks or on the internet. There seems to be a common agreement that it is an electrostatic effect.

Am I missing something here, or is there really a wide spread misconception about this?

References
1 R. Feynman, The Feynman lectures on physics volume II, chapter 13.6
https://www.feynmanlectures.caltech.edu/II_13.html
2 David J. Griffiths, Introduction to electrodynamics, third edition, chap. 12.3.1
[3] Wikipedia: Classical electromagnetism and special relativity
https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Relationship_between_electricity_and_magnetism

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  • $\begingroup$ There is widespread theory of special relativity. It starts off by saying this asymmetry does not exist in reality, only the theory got it complicated. Einstein doesn't ask the question, he did not laugh either. You are secretly assuming by saying 'my frame' that it is or can be the absolute one, where different fields of physics rest. $\endgroup$
    – user192234
    Commented Dec 13, 2019 at 11:58

2 Answers 2

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Hej Mads,

The electric field and the magnetic field are two projections of a more general object, the electromagnetic tensor. These components take different values in different reference frames, like when the components of a vector change under a rotation.

The question now is can I always find a reference frame where my physics becomes only electrostatics or magnetostatics? Meaning that you will find a reference frame where only one of the fields exists. The answer is NO and you can prove it by calculating an invariant quantity under special relativistic transformations https://en.wikipedia.org/wiki/Classification_of_electromagnetic_fields#Invariants This means that you are right about magnetism not being an electrostatic effect.

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  • $\begingroup$ Thank you Konstantinos. There is a detail i think you are missing. I'm evaluating the force on a specific charge, and not the fields. In the particles rest frame it only experiences an electric force. Not because the magnetic field is absent, but because it's velocity is zero. The question is why Griffith uses the a static expression to find the field of a moving particle, and whether that a correct approach. $\endgroup$
    – Mads Vestergaard Schmidt
    Commented Dec 14, 2019 at 16:38
  • $\begingroup$ Some physicists believe field retardation is included in the Lorentz transformation. A mistaken view in my opinion. All frames must be treated the same, and moving particles gives out retarded fields. I'm referring to this article. "Oleg D. Jefimenko, Retardation and relativity, American journal of Physics 63, 454 (1995)" $\endgroup$
    – Mads Vestergaard Schmidt
    Commented Dec 14, 2019 at 16:40
  • $\begingroup$ Indeed I left out the details because I was not sure which expression I should compare your own to. Can you tell me the year of publication and publisher of your Griffiths edition so I can check? $\endgroup$
    – Konstantinos Tsoukalas
    Commented Dec 14, 2019 at 20:33
  • $\begingroup$ Yes, third edition, chap. 12.3.1. Both Griffith and Feynman calculates the total linecharge density in the rest frame of the test charge, and then treats it as a static linecharge. They ignore the velocity of respectively positive and negative charges in the wire. This only works out because the integral of the Coulomb field and eq. 1, happens to be equal in case of an infinite straight linecharge. (I've edited the question to make this more clear) $\endgroup$
    – Mads Vestergaard Schmidt
    Commented Dec 15, 2019 at 19:34
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There is no claim in The Feynman Lectures on Physics that the phenomenon of magnetism is a "consequence of relativistic length contraction" (applied to electrostatics, or otherwise). The special case to which you refer in Volume II chapter 13 is presented as an example of the relativity of fields in a particular circumstance. No general relationship such as you purport between magnetism, relativistic length contraction, and electrostatics is stated or implied. On the contrary, one finds statements suggesting otherwise, such as "Electric and magnetic forces are part of one physical phenomenon" and "electric and magnetic fields appear in different mixtures if we change our frame of reference." I might add that Volume II chapter 13 is not the book's denouement with regard to the phenomena of magnetism and relativity. The electromagnetic field tensor is introduced in Chapter 26, Lorentz Transformations of the Fields. I agree in sprit with Konstantinos Tsoukalas that the field tensor is fundamental to a proper understanding of how the electric and magnetic fields are related to each other in different frames of reference.

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  • $\begingroup$ Thank you for correcting these points. I agree, it is not directly stated in either The Feynman lectures of Griffiths, that the electrostatic analyses is generally true. On the other hand, it is not pointed out that it is a special case, and that the analysis will not work in other cases. Griffith writes: “Taken together, then, electrostatics and relativity implies the existence of another force” (referring to magnetism). $\endgroup$
    – Mads Vestergaard Schmidt
    Commented Dec 13, 2019 at 20:28
  • $\begingroup$ Feynman writes about the rest frame of the test charge “If there is any force on the particle, it must come from an electric field. It must be that the moving wire has produced an electric field. But it can do that only if it appears charged” $\endgroup$
    – Mads Vestergaard Schmidt
    Commented Dec 13, 2019 at 20:28
  • $\begingroup$ Nothing in what they say is wrong in reference to the case they are treating, but I find the example and the statements misleading. The statement on the Wikipedia page, on the other hand, is directly incorrect, and since no one has corrected this, it assumed it is widely misunderstood. I wanted to correct it, but I have no references to back it up. $\endgroup$
    – Mads Vestergaard Schmidt
    Commented Dec 13, 2019 at 20:29

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