# Are force fields deformed by length contraction?

1. Dec 22, 2014

### lennon

Imagine you have an electron travelling 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.

2. Dec 22, 2014

### A.T.

Yes.

3. Dec 22, 2014

### Matterwave

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.

4. Dec 22, 2014

### Staff: Mentor

5. Dec 22, 2014

### 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 lenght 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, wich generate a disbalance with the charges of the electrons, wich 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.

6. Dec 22, 2014

### A.T.

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

7. Dec 22, 2014

### lennon

Mmm I see what you mean.

8. Dec 22, 2014

### 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)

9. Dec 22, 2014

### Matterwave

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.

10. Dec 22, 2014

### 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.

11. Dec 22, 2014

### A.T.

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

Because they would be positrons otherwise.

12. Dec 22, 2014

### WannabeNewton

What?

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

Last edited: Dec 22, 2014
13. Dec 22, 2014

### Matterwave

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.

14. Dec 23, 2014

### 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.

Last edited: Dec 23, 2014
15. Dec 23, 2014

### Staff: Mentor

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.

16. Dec 24, 2014

### 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 wich 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 wich 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 wich I need no assistence. But please, do correct me if I understood anything wrong about the phenomenon per se.

17. Dec 24, 2014

### Staff: Mentor

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.

18. Dec 25, 2014

### 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 (∇), wich 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,

19. Dec 25, 2014

### Staff: Mentor

The laws I posted above are the mechanical explanation.

20. Dec 25, 2014

### 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?