Search for a correct explanation

[DaTario]

Hi All,

Consider please the situation where an electron is moving to the right with velocity Ve parallel to a horixontal wire in which is found electric current I. For the sake of definitness, let positive charges inside the wire go to right and negative go to left, with velocities of equal modulus for an observer at rest in relation to the wire's reference frame. Let's call this observer A (at rest w.r.t. the wire).

This observer is seeing an electron running parallel to the wire which has this rather simmetrical current.

Classical explanation of situation observed by A:
(1) the wire is neutral
(2) the wire, with current inside, produces a magnetic field which is stationary in space according to A.
(3) the electron has a velocity with respect to this reference frame in which the field is stationary.
(4) Consequently, the electron will experience a magnetic force oriented from the wire to the electron, so that it suffers a repulsion from the wire. (F = q (v X B))

Relativistic explanation of situation observed by A:
(1) the wire is neutral, for the stream of positive charges and the stream of negative charges inside the wire have the same velocities and therefore, are contracted by the same factor yielding a wire with a set of two inhomogeneous local densities of opposite charge and the same magnitude in each place.
(2) the wire, with current inside, produces a magnetic field which is stationary in space according to A.
(3) the electron has a velocity with respect to this reference frame in which the field is stationary. But the electron's electric field is no longer decribed by a spherically simmetrical field. However the electron's field will present symmetry w.r.t the line which crosses perpendicularly the wire and pass through the electron.
(4) the interaction between the electron and the system of charges inside the wire now will be given by solving the problem: Find the resultant electric force between two inhomogeneous charge distributions (but equal in each place and practically superimposed) and a particle which has a sort of spherical field contracted along the direction perpendicular to the wire (perpendicular to its velocity, in fact).

OBS: is seems that this resultant force must be zero. My question is how to conduct this two explanations correctly in order to preserve the internal consistency of both descriptions.

Thank you

DaTario

 

[JesseM]

edit: deleting my post, I got confused between the observer A at rest relative to the wire and the electron in motion relative to it. But I'll leave this part:

By the way, you may find the diagrams and explanations here to be helpful since they deal with similar kinds of scenarios.

 

[Dale]

The relativistic explanation is the same as the classical explanation. You only get to relativistic explanations when you change reference frames, here you are dealing only with A's frame.

 

[DaTario]

The relativistic explanation is the same as the classical explanation. You only get to relativistic explanations when you change reference frames, here you are dealing only with A's frame.

I disagree with you in one point.
Classically, the electric field and the magnetic field are independent things. In relativistic analysis magnetic field arises from relativistic corrections in electric field. At leats according to my present understanding of this subject.

Thank you

DaTario

 

[Dale]

There are indeed some scenarios where you can describe a magnetic field as a relativistic correction in an electric field. In order to do that you need to have a current composed of only one kind of charge carrier moving at a well-defined velocity, like a steady beam of protons through empty space. That way you can pick a reference frame where there is no current and describe a purely electrostatic situation. Such a scenario will always involve a net charge in every frame, so what was a purely electric field in one frame will become a mix of electric and magnetic fields in other frames.

However, situations like the one you described are a little more complicated. Because the current-carrying wire is neutral you have two sets of charge carriers moving at different velocities, so there is no reference frame where the situation becomes purely electrostatic. There is a current and therefore there is a magnetic field in every reference frame.

Now, Maxwell's equations are linear and follow the principle of superposition, so you could take the following approach. You could describe the positive charges in their rest frame (electric only) and boost it to the frame of interest (electric and magnetic), then separately consider the negative charges in their rest frame and boost it to the frame of interest, and then add the two results to get the net field as relativistic corrections to two separate fields. But that would require three reference frames, so I would still say that the relativistic explanation is the same as the classical explanation in a single frame and it is only when you talk about a second (or third) frame that you begin to get relativity involved.

 

[DaTario]

I agree with your analysis.
So, it must be correct to say that in all the cases ona can always take separetely each family of charges moving with the same velocity and apply "magnetic is a coulobian (electric)relativitic correction" strategy through the use of a particular boost operation and, at the end, sum up all those contributions to the resultant force and explain the situation just using electricity and relativity.

Do you agree with that?

Thanks

DaTario

 

[Dale]

I agree with your analysis.
So, it must be correct to say that in all the cases ona can always take separetely each family of charges moving with the same velocity and apply "magnetic is a coulobian (electric)relativitic correction" strategy through the use of a particular boost operation and, at the end, sum up all those contributions to the resultant force and explain the situation just using electricity and relativity.

Do you agree with that?

Yes, I think so, although in some scenarios (e.g a curved segment of current-carrying wire) you would need to add together contributions from an infinite number of reference frames.

 

[DaTario]

Yes, I think so, although in some scenarios (e.g a curved segment of current-carrying wire) you would need to add together contributions from an infinite number of reference frames.

Thank you. It seems now to be much more clear to me.

Sincerely

DaTario