# Can't special theory of relativity explain magnetic attraction of conducting wires?

I saw a video of Veritasium and Nick lucid explaining how magnetism in terms of special relativity. Nick Lucid also posted this followup video, which I was not able to understand.

I have these two questions:

1. Why does the charge next to the conducting wire have to be moving. When the charge is stationary, as moving electrons are compressed, the positive charge must experience an attraction. Why is that not happening?

2. What is this???

I know it's probably not a credible source, but how do you answer it?

I am summarizing the above link as many have asked me to do. If you see the video link I have given above in the first part, they show that electrons are compressed and this has resulted in the wire being not neutral, but for this to happen more electrons have to come inside the wire, so if we considered a circular circuit, more electrons cannot come, thereby, in a way negating the claim that SR can explain magnetism, I dont agree with this(intuitively), but I am falling short to reason out why, and was asking help here

• You will probably get better answers to this question if you (a) focus onto ONE of your two questions, and (b), summarize the content of your links, so someone can understand what your question is without clicking through. I think your second question is particularly interesting, so I'd encourage you to edit your question to focus on that, and include the images from the link. – Jahan Claes Aug 28 '18 at 3:29
• Thank you so much. But I'm very poor at articulating things, and I have been hit left and right on this platform, and as a result the QA just went in rounds and also the links offer such a good explanation which I don't wanted to mess with, so I just kinda outlined the question and outsourced the core stuff – VARUN.N RAO Aug 28 '18 at 3:33
• $\operatorname{Question}{\left(2\right)}$ seems pretty ambiguous and references alternativephysics.org. Since non-mainstream physics is off-topic here, that second part might not work on SE.Physics, though it's a bit hard to tell as the question itself is a bit unclear. I'd suggest editing the post to clarify. – Nat Aug 28 '18 at 5:07
• Volume 2 of the Berkeley Physics Course that covers Electricity and Magnetism by author Edward M. Purcell covers the development of the magnetic force between two wires as the special relativity handling of moving electric charges in the wires. I still remember doing the entire derivation of the magnetic force law between two current carrying conductors using only kinematics, special relativity, and the field of the electric charge in motion. It was about a 20 page derivation, by hand, with diagrams and full mathematical detail. – K7PEH Aug 28 '18 at 5:13
• Also, Daniel Schroeder, professor of Physics at Weber State University, has also addressed this same derivation. I think I am choosing the right paper that describes it. Link: physics.weber.edu/schroeder/mrr/MRRnotes.pdf – K7PEH Aug 28 '18 at 5:17

## 3 Answers

This is going to be a long post since there are multiple issues being revealed by this question.

### 1) alternativephysics.org is a bad site

I've responded to another part of that site in another question. The writer misunderstands relativity and comes to bad conclusions from that misunderstanding to "prove" that relativity contradicts itself. Then, the writer proceeds insult anyone who actually understands relativity instead of finding out what they think. I'm not going to write anymore about them.

### 2) The length-contraction argument is a pedagogical tool, not a derivation

Sometimes, instead of a rigorous, math-heavy derivation, an intuitive and simplified argument is given to students to convince them of the real relationship between multiple concepts. For example, the following shows the equivalence of Kepler's third law of planetary motion and Newton's inverse square law of gravity: \begin{align} r^3 = kT^2 &\iff r^3 = k(2\pi r/v)^2 \\ &\iff r = 4\pi^2k/v^2 \\ &\iff v^2/r = 4\pi^2k/r^2 \\ &\iff mv^2/r = 4\pi^2 km/r^2 \\ &\iff F = 4\pi^2 km/r^2 \end{align} where $r$ is the radius of the orbit, $T$ is the period of the orbit, $k$ is some constant, $v$ is the constant speed of the orbit, $m$ is the mass of the orbitting body, and $F$ is the gravitational force. However, this only works for circular orbits. The full derivation of this equivalence for elliptical orbits takes an entire chapter of a graduate-level text book.

Likewise, the length-contraction argument only works when the test particle (the one that feels the magnetic force) moves at the same velocity as the flowing electrons in the wire. That restriction makes the argument much simpler, which is good for convincing students that electric and magnetic fields are the same thing from different points of view (a.k.a., reference frames). But, that same restriction also restricts its generality. In fact, using the same argument, you could conclude that a positive test charge at rest with respect to the nuclei in the wire should be attracted to the wire with a current since the electron spacing should be length-contracted, making the electron density appear greater than the nuclei density. This is obviously not the case.

For a better derivation, I've found this (non-peer-reviewed) paper that uses the relativity of simultaneity to derive the magnetic force and the lack of force on a non-moving test charge. In this paper, the test charge is not restricted to moving at the same velocity as the current in the wire. The math is more difficult and even makes reference to a textbook simply referred to as Jackson--a tome much feared amongst physics grad students (me included). However, the first two sections should be manageable for most people familiar with special relativity.

In conclusion, the length-contraction argument serves a purpose: to relate electric and magnetic fields in different reference frames to students who are first encountering these ideas. It is not rigorous or general, but it is useful.

• Thanks for the sophisticated answer which I was expecting. I am in no way defending alternativephysics.org, I had suspected that it is not a credible source at all, but that particular article made sense, I couldn't explain it, so ended up asking it here. – VARUN.N RAO Aug 28 '18 at 8:02
• @ChakrapaniNRao You're welcome. Since I've seen that site mentioned before in other questions, I felt I had to make a clear statement about it. – Mark H Aug 28 '18 at 8:31
• You seem to have some misconceptions about what this style of derivation does and what the underlying logic is. I haven't viewed the video, but for a correct derivation at the freshman level, see the classic textbook by Purcell. There is no restriction to the case where the test particle's velocity equals the drift velocity of the charge carriers in the wire. – user4552 Oct 14 '18 at 22:01

Unless I misunderstand your question, then this is just basic physics. Perhaps you should include a link to the original video?

1. Magnetic force (the Lorentz Force) is a deflecting force, meaning it deflects something that is already moving. If that thing is not already moving, then there's no deflection. Thus, the electrons must be moving to be effected by the magnetic field of the other wire. Likewise, in order to generate a magnetic field one must produce a current and the way to do this is to move electrons. So either way, electrons are moving!

2. That is not a credible source AT ALL. Firstly, notice how they cite nothing, and the author does not have their credentials anywhere - that's just sketchy. Beyond that, there are too many errors in that article to even begin.... Not worth anybody's time here. What the people over at alternativephysics.org don't get is something called "empiricism." That is, they believe something is true simply because they feel it is the best explanation, and everyone who disagrees with them is part of a grand conspiracy.... stick to academic websites, for your own wellbeing :)

EDIT:

Assuming that the charge outside the wire is at rest with respect to the protons in the wire, while also assuming that protons in the wire are at rest with respect to the wire itself, implies that the charge outside the wire is at rest with respect to the wire itself, meaning that, via the Lorentz force law, the charge feels no force on it. The Lorentz force law is special relativistic, so I'm not sure what more you want.

• I have given the link to video. 1. You have not understood the question 2. You have not even attempted to answer the question, you have done the same thing what you accused them of, simply assuming you are correct just because you feel like it. – VARUN.N RAO Aug 28 '18 at 4:17
• If you show that you have done research and ask a more specific question, then you'll get a better response. Oh, i did answer your second question... that website is nonsense. – Daddy Kropotkin Aug 28 '18 at 4:54
• The source for 2 isn't all that bad. It has mistakes, but then it is a confusing subject. He has rediscovered the Ehrenfest paradox, and applied it to electrons in a circular wire. He concludes that the apparent paradox means that Lorentz contraction is not the explanation for magnatism. This is a lot more reasonable than some people get. See en.wikipedia.org/wiki/Ehrenfest_paradox – mmesser314 Aug 28 '18 at 4:54
• @ChakrapaniNRao Also, I did answer your question 1., since if the charge is at rest it will not feel a force from the wire. The Feynman Lectures Volume 2 pgs 13-6,7,8,9,10 cover this quite extensively. – Daddy Kropotkin Aug 28 '18 at 5:08
• @N.Steinle, but the question wasn't whether charges at rest feel a force, the question was how does this follow from relativity? – The Photon Aug 28 '18 at 5:19

Why does the charge next to the conducting wire have to be moving. When the charge is stationary, as moving electrons are compressed, the positive charge must experience an attraction. Why is that not happening?

It's a special case where length contraction does not happen.

Whenever we have electron gas that is being accelerated by an electric field, then we have that "special case".

I'll draw a picture that might clarify what exactly happens:

Here are some still standing electrons:

o o o o

And here are those electrons after an electric field has accelerated them to relativistic velocity, the velocity is towards the right side of the screen:

0 0 0 0