# Magnetic field as result of length contraction

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• Leureka
Leureka
TL;DR Summary
Magnetic fields around currents in a wire as a result of Lorentz contraction is a popular explanation. Is it right though?
Hi all,
Recently I've tried to wrap my head around a common explanation of magnetic fields that you see online, especially among science educators like veritasium or minute physics.

The setup is as follows: there is a wire, composed of the same number of negative charges (electrons) and positive charges (the lattice ions), and there is a test positive charge next to the wire. A current flows through the wire. When the test charge is at rest, there is no force acting on it. As soon as it starts moving, it experiences force which depends on the direction it's moving wrt the current.

The explanation then goes on like this: let's say the test charge is moving together with the electrons, at the same speed. Then, in the rest frame of the electrons, the test charge is at rest, while the positive ions are moving. Due to the effects of length contraction, the positive charge density is not balanced anymore by the electrons, and there is a net electric field acting on the test charge.

The problems I have with this explanation, which is SO COMMON, are these:

1. If length contraction really were the cause of the force on the test charge, then there should be a force even in the laboratory frame when the test charge is at rest. The electrons would be moving and would be length contracted, creating an excess charge density of opposite sign than before. This is explained away simply by imposing the neutrality of the wire in the lab rest frame, but I'd wager you just can't have it both ways, unless I'm misunderstanding how length contraction applies.

2. In the moving frame of the electrons, the positive ions effectively constitute a current. This in turn should generate a magnetic field. If the charge is at rest with respect to the electrons there would be no resulting force, but if we imagine the test charge is actually still wrt to the positive charges (effectively at rest in the lab frame) then this magnetic field would have an effect, because the test charge would be actually in motion in the electron frame. Again, this would suggest, as in 1, that a current in a wire affects charges at rest.

3. E^2 - B^2 is a Lorentz invariant together with E*B, which means it is impossible for a pure magnetostatic field to look like a pure electrostatic field in another frame. A pure B field would have a negative invariant, which can't be provided by a pure E field. So the whole premise of this explanation seems bogus.

I'm aware that this explanation originated in a book on electromagnetism by Purcell, and it has been perpetrated by the likes of Feynman, and now very impactful educators like Veritasium. Even Wikipedia hosts this exact same explanation under "Length contraction".

What's more, there are a number of response videos to Veritasium's, and they all come to a different conclusion. There doesn't appear to be a consensus on whether this application of relativity to EM makes sense, is wrong, partially wrong, or anything in between. I hope with this thread we could stop the confusion.

Leureka said:
3. E^2 - B^2 is a Lorentz invariant together with E*B, which means it is impossible for a pure magnetostatic field to look like a pure electrostatic field in another frame. A pure B field would have a negative invariant, which can't be provided by a pure E field. So the whole premise of this explanation seems bogus.
That observation shows that either the whole explanation is bogus or you have misunderstood the explanation in some fundamental way.

Leureka said:
TL;DR Summary: Magnetic fields around currents in a wire as a result of Lorentz contraction is a popular explanation. Is it right though?

1. If length contraction really were the cause of the force on the test charge, then there should be a force even in the laboratory frame when the test charge is at rest. The electrons would be moving and would be length contracted, creating an excess charge density of opposite sign than before. This is explained away simply by imposing the neutrality of the wire in the lab rest frame, but I'd wager you just can't have it both ways, unless I'm misunderstanding how length contraction applies.
You are misunderstanding how length contraction applies.

Length contraction is a relationship between two frames. Once you know the distance in one frame then you can transform it to a distance in another frame. But you have to first start with the distance in some specific frame.

Here, we are given that the wire is uncharged in the lab frame. So we start with the knowledge that the distance between electrons and protons is the same in the lab frame. Starting with that given information in the lab frame we can then transform to the moving frame and find that the wire is charged in the moving frame.

What you are mistakenly assuming is that the spacing between the electrons in the electron's rest frame (the electron's proper spacing) should equal the spacing between the protons in the proton's rest frame (the proton's proper spacing). That is not a correct assumption. The proper spacing is different precisely to ensure that the wire is uncharged in the lab frame.

From that fact then length contraction applies.

Leureka said:
What's more, there are a number of response videos to Veritasium's, and they all come to a different conclusion. There doesn't appear to be a consensus on whether this application of relativity to EM makes sense, is wrong, partially wrong, or anything in between. I hope with this thread we could stop the confusion.
The explanation is not wrong. It is also not particularly helpful for most students.

PeroK
Dale said:
Here, we are given that the wire is uncharged in the lab frame. So we start with the knowledge that the distance between electrons and protons is the same in the lab frame. Starting with that given information in the lab frame we can then transform to the moving frame and find that the wire is charged in the moving frame.

See, the problem I have with that is that the wire is uncharged both with and without a current. Which means, going by that assumption, the spacing between electrons and between protons , in the lab frame, is the same, regardless of the motion of the electrons (this MUST be so, otherwise the wire would be charged).

Let's remove the electrons for a second: in the moving frame, the spacing of the protons is length contracted wrt the lab frame, and that's a fact clearly stated in the video. But since there is no way to tell whether it's you or the protons that are moving (motion is relative), you should be able to definitely say that the protons are length contracted if they were moving in the lab frame, i.e. moving charges get length contracted. For some reason, we don't apply this reasoning to electrons: regardless if they move or not, their spacing stays the same.

What you are mistakenly assuming is that the spacing between the electrons in the electron's rest frame (the electron's proper spacing) should equal the spacing between the protons in the proton's rest frame (the proton's proper spacing). That is not a correct assumption.

When exactly did I assume this? I think I clearly stated the opposite: the spacing of protons is larger in the lab frame (proton proper spacing) than in the electron's frame. They need to be different, otherwise there would be no extra charge density.

Also, this does not address points 2 and 3.
PeroK said:
That observation shows that either the whole explanation is bogus or you have misunderstood the explanation in some fundamental way.

This doesn't help answer the question much, does it?

Leureka said:
See, the problem I have with that is that the wire is uncharged both with and without a current. Which means, going by that assumption, the spacing between electrons and between protons , in the lab frame, is the same, regardless of the motion of the electrons (this MUST be so, otherwise the wire would be charged).
Yes. Why is that a problem? That is what we can actually do with real circuits in real labs.

Leureka said:
For some reason, we don't apply this reasoning to electrons: regardless if they move or not, their spacing stays the same.
It isn't a matter of reasoning. With reasoning you start from some premise and reason out the logical consequence. The fact that in the lab frame the electron spacing is the same as the proton spacing is the premise. It is given in the setup of the problem, and it is consistent with what we actually observe in real circuits. From that given we then apply reasoning. You are trying to reason out the premise, and it doesn't work that way. You start with the premise and reason from there.

Leureka said:
Also, this does not address points 2 and 3.
Yes, one thing at a time. There is no point in addressing 2 and 3 until you get 1 straightened out.

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Leureka said:
See, the problem I have with that is ...
Looking through your previous threads, it looks like @Dale is in for a long battle here!

Dale said:
Yes. Why is that a problem? That is what we actually do with real circuits in real labs.It isn't a matter of reasoning. With reasoning you start from some premise and reason out the logical consequence. The fact that the electron spacing is the same is the premise. It is given in the setup of the problem, and it is consistent with what we actually observe in real circuits. From that given we then apply reasoning.

But the logical consequences lead you to treat different charges inconsistently, as I just explained above. If you think that I used faulty logic you're welcome to point where.

PeroK said:
Looking through your previous threads, it looks like @Dale is in for a long battle here!

Wow, and you got that conclusion from a whole 1 post in my history? You contributed 0 to the discussion, and yet felt the need to have some sort of dig at me?

Leureka said:
But the logical consequences lead you to treat different charges inconsistently, as I just explained above.
I believe that by "treat different charges inconsistently" you mean that the proper spacing of protons is is the same whether there is a current or not, while the proper spacing of electrons changes whether there is a current or not. That is physics. Electrons are not protons. They are physically differently and they behave differently.

Leureka said:
If you think that I used faulty logic you're welcome to point where
Your logic is faulty in two places. First, and most importantly, it is faulty where you are trying to derive something that is a given. Second, it is faulty where you assume that electrons and protons should behave the same even though they are different.

Dale said:
First, and most importantly, it is faulty where you are trying to derive something that is a given.
I feel that I should delve a bit more into this.

For reasoning and logic, a given or a premise is a starting point. You do not try to derive it.

What you can do is that you can show that a set of premises is inconsistent. For example, you can show that assuming premises "A" and "B" leads to both proposition "C" and "not C". Then that shows that A and B are not logically consistent with each other.

You can also try to show that some premise is unrealistic (doesn't hold in the real world). This is not a matter of reason but a matter of experience. Any reasoning based on that premise may be correct but will not have any implications for the real world. For example, I could give as a premise that the earth has less mass than the moon. That premise could be rejected by citing measurements of the mass of the earth and the moon. Correct reasoning from the incorrect premise would not be meaningful.

The premise that the wire is both uncharged and current-carrying is neither inconsistent nor is it unrealistic. So since it is a valid premise we simply start from there and do our reasoning based on that assumption.

Dale said:
The premise that the wire is both uncharged and current-carrying is neither inconsistent nor is it unrealistic. So since it is a valid premise we simply start from there and do our reasoning based on that assumption.
There's also the simple explanation that when the electrons were set in motion the proper distance between them increased, so that the wire remained uncharged in the lab frame. That is then the physical scenario under discussion.

Dale
PeroK said:
There's also the simple explanation that when the electrons were set in motion the proper distance between them increased, so that the wire remained uncharged in the lab frame. That is then the physical scenario under discussion.
Yes. And it is rather simple to draw the circuit diagram corresponding to this scenario. When students object to this, it usually indicates a failure to understand circuits rather than a problem specific to relativity. I believe that is the case here.

PeroK said:
There's also the simple explanation that when the electrons were set in motion the proper distance between them increased, so that the wire remained uncharged in the lab frame. That is then the physical scenario under discussion.

And how would that happen? And why doesn't this also apply to protons, when they are "set in motion" by switching to the electron rest frame?

Dale said:
I believe that by "treat different charges inconsistently" you mean that the proper spacing of protons is is the same whether there is a current or not, while the proper spacing of electrons changes whether there is a current or not.

Yes, except that you should replace "current", which makes it sound I always refer to electrons, with "motion".

Second, it is faulty where you assume that electrons and protons should behave the same even though they are different.

What a preposterous statement. Length contraction happens to everything with motion. A particle's nature has nothing to do with it, if it is moving it is subject to length contraction and time dilation.

And it is rather simple to draw the circuit diagram corresponding to this scenario

I'm not really sure what circuit diagrams have to do with this query, which is specifically asking about special relativity applied to idealized currents.

Leureka said:
Yes, except that you should replace "current", which makes it sound I always refer to electrons, with "motion"
Interesting. This is actually a wrong distinction. "Motion" distinguishes between two different views of the same physical scenario, i.e. in one reference frame the protons are moving and in one they are not but it is the same physical scenario. "Current" distinguishes between two physically different scenarios, i.e. the current on is a physically different scenario from the current off, regardless of which frame you are analyzing.

Different frames cannot change the proper spacing between electrons. But current on or off can, because that is a physically different scenario. I am not certain but this misunderstanding may be the core.

Leureka said:
What a preposterous statement. Length contraction happens to everything with motion. A particle's nature has nothing to do with it, if it is moving it is subject to length contraction and time dilation.
While this is all true it is actually irrelevant to the specific issue at hand, which is simply establishing the givens in the scenario. The givens are set in a single frame.

Leureka said:
I'm not really sure what circuit diagrams have to do with this query,
Do you understand that it is possible to make a circuit with a wire that is both current carrying and uncharged?

If so, then the relativity can be applied from that starting condition in the rest frame of the circuit. If not, then I can post a circuit diagram that accomplishes it.

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(1) Watching a YouTube video does not mean you understand the argument. I recommend books. Purcell is one of the first textbooks to treat magnetism this way.,

(2) "I'm sure if I actually did the calculation it would come out..." is not going to convince anyone of anything. Think there's a problem? Do the calculation and demonstrate it - show that two numbers that should be equal are not. If you can't do the calculation, it means you don't understand the argument.

sophiecentaur and Dale
Leureka said:
And how would that happen?
If you consider a finite loop of wire, then it must happen. If the loop is 100m long and there are 1000 electrons at rest, the electrons must be 10cm apart. If a steady current is created in the wire, then the electrons must stay 10cm apart - otherwise, you would have to change the number of electrons in the wire.
Leureka said:
And why doesn't this also apply to protons, when they are "set in motion" by switching to the electron rest frame?
There is no frame where the electrons are at rest if there is a current in a loop of wire. This is because the electrons must change direction. If the circuit is rectangular, then there are four different rest frames involved. There is no absolute symmetry for this reason.

If we consider one section of wire, then the length of that section is length contracted. Let's say there are 250 protons in 25m of wire. In the frame of the electrons moving in that section of wire, the wire itself is length contracted and the protons are closer together than 10cm.

The idealised scenario is an "infinite" wire. But, in practical terms, we are looking at a short section of a long length of wire that must eventually change direction and form a circuit.

If you look at the whole circuit, you see the asymmetry between the single (inertial) proton frame and the more complicated (four-way-inertial with a change of direction at the corners) electron frame.

In that sense, the current is absolutely a motion of electrons. The electrons must have been accelerated by a battery, whereas the protons have remained inertial.

Dale and Leureka
vanhees71 said:

Very interesting, thank you. I'm not surprised you get a correct answer using four vectors... But it's simply not what the original video was saying, which is only using a very basic special relativity approach. This paper is actually stating clearly that the wire is uncharged in the electron rest frame, not the wire frame, in direct opposition of the video and other answers here. I'm not surprised I was confused. I guess the lesson is to never oversimplify a problem.

vanhees71

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