Are force fields deformed by length contraction?

[lennon]

Imagine you have an electron traveling at high speeds... would you expect it´s EM field to be contracted following the Lorentz transformations??
If the answer is no, please explain why fields and their shape don´t deform when space-time does. How they retain their shape in a space that is not spacetime.

 

[A.T.]

Imagine you have an electron traveling at high speeds... would you expect it´s EM field to be contracted following the Lorentz transformations?

Yes.

 

[Matterwave]

Imagine you have an electron traveling at high speeds... would you expect it´s EM field to be contracted following the Lorentz transformations??
If the answer is no, please explain why fields and their shape don´t deform when space-time does. How they retain their shape in a space that is not spacetime.

To add a little to A.T.'s answer, in the 4-D world of special (and general) relativity, the electric field is actually 3 components of a larger object, a rank 2 anti-symmetric tensor, called the Faraday Tensor (the other 3 components of which are the magnetic field components). This tensor changes accordingly as a tensor under a Lorentz transformation. Thus, actually a moving electron will have both a magnetic field as well as a transformed electric field.

 

[jtbell]

For the equations, see here:

http://farside.ph.utexas.edu/teaching/em/lectures/node123.html

 

[lennon]

Well, I had this question because I was watching this short video:


at 1:00 he starts explaining how relativity affects an object that moves with the current. According to him, in the frame of reference of an object moving with the current, the protons on a conductor would be/seem moving and hence experience length contraction. I can understand this.
What I don´t get is the effect this generates.
Acording to the video since protons are contracting they are closer to each other, which generate a disbalance with the charges of the electrons, which would favor the protons positive charged field. Thing is that as I see it, although the contraction would make indeed the protons closer to each other, their fields would also contract in proportion so it would seem that you kind of lost the advantage it had over the electrons.

 

[A.T.]

their fields would also contract in proportion so it would seem that you kind of lost the advantage it had over the electrons.

Why would that be? The contracted fields are not weaker overall, just differently distributed directionally.

 

[lennon]

Mmm I see what you mean.

 

[lennon]

It kind of makes me wonder: is there a known explanation for the counter clockwise direction stated in the right hand rule? (sorry I know that´s a different question)

 

[Matterwave]

It kind of makes me wonder: is there a known explanation for the counter clockwise direction stated in the right hand rule? (sorry I know that´s a different question)

The right hand rule is a convention. Most people are right handed, so we go with a right hand rule rather than a left hand rule.

 

[lennon]

What I meant was "why is there a counter clock wise direction, and not a clockwise one?" Why should electrons favor a direction over the other.

 

[A.T.]

What I meant was "why is there a counter clock wise direction, and not a clockwise one?"

Look from the other side, and it's clockwise.

Why should electrons favor a direction over the other.

Because they would be positrons otherwise.

 

[WannabeNewton]

Because they would be positrons otherwise.

What?

EDIT: Sorry, I understand now what you meant. Do carry on.

 

[Matterwave]

What I meant was "why is there a counter clock wise direction, and not a clockwise one?" Why should electrons favor a direction over the other.

This question is analogous to "why is the gravitational force attractive?" or "why do like charges attract while unlike charges repel?" That's just the way things are. If things were the other way around, then we would have a theory describing things the other way around.

A current produces a magnetic field which produces a force on moving electrons through the Lorentz force. In which direction the force acts is simply a property of nature, just like the property that two electrons will repel each other. How we describe this property, e.g. in which direction we define the magnetic field, whether we call the electron positively charged or negatively charged, how we define a current, whether we specify a right hand rule or a left hand rule for cross products, these are all conventions that we invent. We can invent any convention we like as long as they are consistent with the observations and give the right direction for in which direction the moving electron will actually accelerate.

 

[lennon]

Well, the answers where pretty useful. Again, I wasn´t asking about the conventions at all but rather the property of nature and I don´t agree with the epistemological subject that was brought later (but that would be a different topic altogether).
Thanks a lot for clarifying these points.

 

[Dale]

Again, I wasn´t asking about the conventions at all but rather the property of nature

The right handedness is not a property of nature. It is only a convention, just like the negative charge of an electron.

The property of nature is that protons have the opposite charge from electrons, but it is a human convention which is positive and which is negative. No experimental observations would change if we had made the opposite choice.

Similarly, the right or left handedness of the cross product in Maxwell's equations is also a convention. No experimental outcome would change if we chose a left handed cross product. To convince yourself of this calculate the force on an electron due to a set current using a left handed cross product.

 

[lennon]

As long as I know, if I were to place a bunch of magnets around a conductor at 90 degrees with a current passing through, I´d see the positive pole of each magnet pointing in the direction described by right hand rule. If I were to use my left hand with my thumb pointing to the current, my fingers wiould curl in the opposit direction and they would describe the direction in which the negative poles of the magnets would point.
If you tell me that this is a convention because positive and negative have been set by us I´d still would argue that the way in which we call them is irrelevant to me, I just want to know why one of the poles goes ALWAYS in one direction with respect to the current flow and not to the other. Since positive charge here is related to the protons and negative to the electrons I think there should probably be a mechanic reason to explain that preference, a mechanic reason that I don´t understand (that´s why I asked). I´ll accept this as a rule if the reason it´s still not known. Wether if we can ever know this, or if it´s something that we´ll always need to accept as an axiom is an epistemological question for which I need no assistence. But please, do correct me if I understood anything wrong about the phenomenon per se.

 

[Dale]

I just want to know why one of the poles goes ALWAYS in one direction with respect to the current flow and not to the other.

This is indeed a physical observation which is not a matter of convention. It depends on two laws (assuming appropriate simplifying conditions):

Ampere's law: $\nabla \times B = \mu_0 J$

Lorentz' force law: $f= J \times B$

Since the force depends on two cross products it doesn't depend on the "handedness" of the cross product. If you keep the current in the wire the same and if you keep the bound currents of the magnets the same then the force will remain the same. If you have a right handed cross product then the B field is in the direction indicated in the diagram, and the force on an upward bound current will be inward. If you have a left handed cross product then the B field is in the opposite direction, but the force on an upward bound current remains inward.

 

[lennon]

I see. Both equations describe the realtionship between the current (J) and the magnetic field (B) -with Ampere´s law taking into consideration the permeability of air (μ0)-. I can also see the curl operator there (∇), which implies the crossproduct you were talking about (the crossproduct that brought the confusion). But now that you can understand further what was the nature of my question, can you confirm wether the reason of this preferencial direction flow of the magnetic field with respect to the current is unknown, or if there is any mechanical explanation for it? thanks,

 

[Dale]

The laws I posted above are the mechanical explanation.

 

[Matterwave]

I think maybe OP should take a look at the video of Feynman explaining "why" questions.

What kind of an answer would satisfy you lennon? For example, when you put two electrons near each other, they repel. What kind of an "explanation" would satisfy you for why they repel?

 

[lennon]

First let me say that I know that this is quite tiring and a little frustrating, and that I appreciate the answers

DaleSpam
My question was why one of the poles goes ALWAYS in one direction with respect to the current flow and not to the other
you say you can explain why this happen with this equations?
or is it how Matterwave states that there is no explanation for this (i.e. it´s just a property of nature that has been observed and that has to be accepted as it is)?
If there is an explanation and you can read that explanation in this equations, Would you translate it for me?

Matterwave,
There are certain things that -to your epistemological view of the world- are fundamental axioms, I understand that (even if I don´t agree), but I don´t believe the phenomenon I´m referring falls in that category of natural axioms. Take this problem into consideration:

Q: Why would a charged moving object running parallel (and close) to a conductor with electric current passing through experiments a positively charged magnetic force?

A: Because in the moving object´s frame of reference, the protons inside the conductor are experiencing length contraction, and as the fields get closer they exert more force than the electron´s.

[sorry for the bad formulation]
--Now, are you really sure, that if I ask you this question in a parallel universe where you never heard or read the relativistic explanation, you wouldn´t told me “it´s just how things are?”. Please notice that what I´m asking is a problem that looks more like this one, than the one you´re suggesting. In any case, I respect your view, and fully understand it. Keep in mind that epistemology is not science; science may rest in it, may depend upon it, but both are different things with different rules and methods. Feynman is definitely not an authority in the field, but even if you were to bring a quote by Kant or Popper or any other epistemologist, I´d still would keep my opinion in this regard.
If today´s scientfic view of this matter of the direction of the orientation with respect to the current goes as far as "the phenomenon you describe -to us- represents an axiom of nature, a caprice of the universe", fine, thanks for the answer that´s all I needed to know, I just want to be sure that there's consensus on this matter, that´s all. Don´t think I haven´t taken note of your comment or that I don´t apreciate your opinion.

 

[Matterwave]

First let me say that I know that this is quite tiring and a little frustrating, and that I appreciate the answers

Matterwave,
There are certain things that -for your epistemological view of the world- are fundamental axioms, I understand that (even if I don´t agree), but I don´t believe the phenomenon I´m referring falls in that category of natural axioms. Take this problem into consideration:

Q: Why would a charged moving object running parallel (and close) to a conductor with electric current passing through experiments a positively charged magnetic force?

A: Because in the moving object´s frame of reference, the protons inside the conductor are experiencing length contraction, and as the fields get closer they exert more force than the electron´s.

But that just begs the question "why is the forces on an electron due to protons directed towards the protons, while the forces due to electrons is directed away from the electrons?"

Maybe you can go one level deeper, but then what?

 

[lennon]

Well, if you explain this thing up to a level like the quoted one, I´d be glad enough.
There must be something, generating the asymmetry. It seems to be the flowing current. Perhaps the electrons are also making a movement in this fashion, like some helix. There must be some reason, I don´t know, perhaps there´s some relationship between the blank spaces of the suborbitals in relation to the current (it must be something like that, probably some hybridization that favor a helix path of the electronic flow).
I´d be glad if there was an explanation that would reduce it to the attractive nature of opposite charges.

 

[Dale]

My question was why one of the poles goes ALWAYS in one direction with respect to the current flow and not to the other
you say you can explain why this happen with this equations?

Yes.

or is it how Matterwave states that there is no explanation for this (i.e. it´s just a property of nature that has been observed and that has to be accepted as it is)?

I think that Matterwave was essentially answering the related question "why those equations, and not some other equation?" That kind of question can be answered by appealing to a more fundamental theory. In this case, the more fundamental theory is Quantum Electrodynamics (QED). QED can be used to derive those equations, but then you can ask "why the QED equations, and not some other equation?" So you always wind up, in the final analysis, just using the equations because that is what has been observed to work.

If there is an explanation and you can read that explanation in this equations, Would you translate it for me?

Let's simplify things. Let's say that you have a uniform B field pointing upwards, and you have a square-shaped loop of wire oriented like a window or a door. Let's say that the current on the loop is going clockwise, so on the top the current is going to the right, on the right the current is going down, on the bottom the current is going to the left, and on the left the current is going up. So, the second equation says that the force is $f=J \times B$. For the currents on the side, going up and down, those are parallel to the upwards B field, so those have 0 force. For the top, the current to the right crossed with the upwards field gives a force towards you. For the bottom, the current to the left crossed with the upwards field gives a force away from you. The two opposing forces give a net torque, which tends to rotate the loop.

That is the basic idea why the magnet rotates. There are some slight complications in the scenario you asked about since the B field is not uniform and the current loops are bound currents instead of free currents, but the principle is the same and the complications just make the description more cumbersome without adding any insight.

 

[lennon]

So, this is what I understand of your epxlanation

 

[Dale]

Wow. Nice drawing! Looks right on.

 

[lennon]

Question: Our society draws the X positive axis as going to the right. But this IS a human convention, isn´t it?
If the convention were instead that the X positive axis would go to the left side of the origin then the torque force here would go into the opposite direction (and it wouldn´t match the torque that we see in our experience) Isn´t that right? In other words, as much as I see it -and maybe I´m very wrong about this-, your explanation gives a good account of why there is a torque (i.e. to opossite forces in the same plane), but doesn´t explain why this torque goes in one direction and not the other (since the direction of the resultant of the cross product is -as far as I see- a convention).
Still I´m a little confuse about some other things about what you say, I have to examine all that a little closer.

 

[Dale]

Question: Our society draws the X positive axis as going to the right. But this IS a human convention, isn´t it?

Yes. That is a convention, and it is not even a strong convention. It gets violated all of the time.

If the convention were instead that the X positive axis would go to the left side of the origin then the torque force here would go into the opposite direction (and it wouldn´t match the torque that we see in our experience) Isn´t that right?

No, changing the coordinate axes doesn't change the direction of the force. The cross product is defined on vectors, and you can use any set of basis vectors to represent your vectors. You will have different components in a different basis, but the vector itself is still the same.

In other words, as much as I see it -and maybe I´m very wrong about this-, your explanation gives a good account of why there is a torque (i.e. to opossite forces in the same plane), but doesn´t explain why this torque goes in one direction and not the other (since the direction of the resultant of the cross product is -as far as I see- a convention).
Still I´m a little confuse about some other things about what you say, I have to examine all that a little closer.

The explanation explains both the torque and also the direction of the torque.

The coordinates/basis are a convention that doesn't change anything. The other important conventions are the sign of the charge on an electron, and the handedness of the cross product. Those two conventions would change the direction of J and B respectively, but they would not change the direction of the force.

 

[lennon]

Ok, I´ll try to examine all this a little bit further.