# I SR and Magnetism

#### Biker

I remember reaching to the conclusion that changing the reference frame (Galilean transformation) doesn't give the same result in Electromagnetism interactions in highschool, I searched for an answer and didn't look any further than knowing that SR solved this.

Now I am back in an engineering course about this which still talks about these interactions (without SR), So I have some conceptual questions just to understand how this works without getting into the math (Sorry if it is nescessary, Only because we have it next year not this one):

1) Does the magnetic field exist because of SR effects so that it makes the same thing happen in all inertial reference frames? Or does the magnetic, electric field and SR work together to keep everything correct? What I mean here that can you work with SR and electric field only to explain what the magnetic field does in reference frames? and the notion of magnetic field was developed only because we didn't know SR?

2) When you work with high velocity particles with charges, Does for example coloumb's law get affected by this? What about other laws like biot savart law and ampere law (Will take them this semester) or do they inherently compatible with SR?

I just want to get the big picture so I can understand how these laws developed and sorry if this seems non-sense. Thank you in advance.

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#### Dale

Mentor
can you work with SR and electric field only to explain what the magnetic field does in reference frames?
In certain specific circumstances, yes. For example, the magnetic field of a single charge can be understood as the Lorentz transform of the electric field.

However, this does not work in all cases. For example, you cannot get the magnetic field of an uncharged current-carrying wire that way.

#### Biker

You may be interested in this paper https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf from 1905 by Albert Einstein.
I will certainly read that. Thank you.

In certain specific circumstances, yes. For example, the magnetic field of a single charge can be understood as the Lorentz transform of the electric field.

However, this does not work in all cases. For example, you cannot get the magnetic field of an uncharged current-carrying wire that way.
What about Feynman's explanation of that? https://readingfeynman.org/2015/09/06/magnetism-and-relativity/
What I also dont get about this argument is that if we have a current carrying wire stationary to me and the electrons are moving, We get a magnetic field but why doesn't the same length contract in this frame cause higher density of negative charge?! Or why in the stationary reference frame of the charge the density of positive and negative charges dont change by the same factor since the wire is shorted by that? So there should be no effect.

Are there situation were the magnetic field cannot be explained as a result of SR effects? I mean there is nothing more fundamental about the electric field but the notion that only moving charges get affected makes it less to me at least.

So I guess that these laws (Coloumb's law, biot savart law and ampere law) works when you use them on your reference frame but when you try to describe something in another reference frame you get a contradiction that SR solves because our notions of distance and time were wrong and they are only correct for your frame only.

I just want to understand the connection between the two and what happens to the laws when you go in the relativistic speeds. If someone can simply explain that, It would be great.

#### Dale

Mentor
What about Feynman's explanation of that?
That isn’t Feynman’s explanation. That is a blog entitled “Reading Feynman”. The explanation given there is from Purcell.

Note, in that explanation the magnetic field is not given as the Lorentz transform of an electric field. In fact, it is the other way around. They start with a magnetic field and then transform it to show that it leads to an electric field in other frames.

if we have a current carrying wire stationary to me and the electrons are moving, We get a magnetic field but why doesn't the same length contract in this frame cause higher density of negative charge?!
The electrons are not rigidly connected. The voltage source pushes them together such that the wire is essentially uncharged in the lab frame.

Are there situation were the magnetic field cannot be explained as a result of SR effects?
Any situation where $E^2-B^2<0$ in units where c=1

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#### jartsa

2) When you work with high velocity particles with charges, Does for example coloumb's law get affected by this? What about other laws like biot savart law and ampere law (Will take them this semester) or do they inherently compatible with SR?

Coulomb's law: Charged objects exchange momentum at rate: Q1Q2/d2

Charged objects that make a trip in space exchange less momentum than identical charged objects standing on the earth. That's because of the time dilation of the object pair that makes the trip.

Ampere's force law: Parallel wires with electric currents exchange momentum at rate: length* I1I2/d

Parallel wires with electric currents that make a trip in space exchange less momentum than identical wires that are standing on the earth. That's because of the time dilation.

Charged objects passing each other at high speed:
Both objects think that there is a sharp impulse (large force lasting a short time) of Coulomb force because of the other object's Lorentz contracted electric field.
Both objects think that the reason the other object thinks that there is a sharp impulse is the time dilation of the other object.

... And then let's say here is a paragraph about two magnets passing each other at high speed: Magnet sees the other magnet's magnetic field to be Lorentz contracted ... blah blah. Same story as in the previous paragraph.

I added the above paragraph in order to point out that it may be a good idea to think that there is a magnetic field around a magnet, although what is around a magnet is really an electric field that depends on frame in such way that another magnet detects forces or torques there where the 'magnetic field' exists.

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#### Dale

Mentor
I added the above paragraph in order to point out that it may be a good idea to think that there is a magnetic field around a magnet, although what is around a magnet is really an electric field that depends on frame in such way that another magnet detects forces or torques there where the 'magnetic field' exists.
If $E^2-B^2<0$ in any frame then there is no frame where the field is purely electric. There is one frame where it is purely magnetic, and in all other frames it is a mixed electromagnetic field.

#### Khashishi

2) When you work with high velocity particles with charges, Does for example coloumb's law get affected by this? What about other laws like biot savart law and ampere law (Will take them this semester) or do they inherently compatible with SR?
Those laws aren't correct for relativistic charges. The correct versions are the Jefimenko's equations (see also the Lienard-Wiechert potential). But Maxwell's equations are correct still with relativity.

#### jartsa

If $E^2-B^2<0$ in any frame then there is no frame where the field is purely electric. There is one frame where it is purely magnetic, and in all other frames it is a mixed electromagnetic field.

So you do not accept the idea of magnets not having any magnetic field around them? Well it sound funny even to me.

But we can explain how a magnet generates an electric field around itself: Electrons move from one side of the magnet to other side of the magnet.

I remember some previous discussions about current loops observed from different frames. A wire loop with current is an electric dipole in all frames except one. So a wire loop without any field around it generates a dipole electric field around it when it accelerates. And a magnet without any field around it generates a dipole electric field around it when it accelerates.

People did not know about that relativistic effect of electrons accumulating on one side of a wire loop at that time when the magnetic field was invented. If they had known, then maybe they would not have invented the magnetic field.

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#### Khashishi

People did not know about that relativistic effect of electrons accumulating on one side of a wire loop at that time when the magnetic field was invented. If they had known, then maybe they would not have invented the magnetic field.
What are you talking about? Electrons don't accumulate on one side of a wire loop. And we know now that the electric and magnetic fields are just components of the electromagnetic field tensor.

#### Dale

Mentor
Electrons don't accumulate on one side of a wire loop
They do, in a reference frame where the loop is moving. If you are familiar with four vectors then the charge density and the current density are the timelike and spacelike parts of a four vector. So if you boost a pure current $(0,\mathbf{I})$ then you get a current and a charge $(\rho’,\mathbf{I’})$

#### A.T.

Electrons don't accumulate on one side of a wire loop.
"Accumulate" is maybe the wrong word. They don't accumulate over time in any inertial frame. Their densities change when you switch inertial frames. Here is a good explanation by DrGreg:

They do [accumulate on one side of a wire loop], in a reference frame where the loop is moving.
Maybe in a reference frame where the loop is accelerating. When the loop moves at constant speed, the densities are different in the two wire segments, but constant over time. So they are not continuously "accumulating" any more. Again, I think this is a wording problem.

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#### Dale

Mentor
When the loop moves at constant speed, the densities are different in the two wire segments, but constant over time.
Yes, I agree. $(0,\mathbf{I})$ boosts to $(\rho’,\mathbf{I’})$ not $(\rho’ t’,\mathbf{I’})$

#### Khashishi

OK. I had previously answered without looking at the necessary context in the other thread, and didn't realize you were talking about a moving loop of current.

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