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Today in physics class we were taught about the equivalence of the electric force and magnetic force with respect to a current carrying wire. If an observer is stationary with respect to the system (a current carrying wire and a moving charge along the direction of the wire) then there will be a magnetic force of attraction between the moving charge and the wire, and we can assume that in this state the positive or negative charge is in the wire are balanced so there is no electric force of attraction.

Now switch the frame of reference so that the moving charge is stationary within the frame. Here there is no “current” In the wire because the relative motion of the electrons is zero with respect to the now stationary charge. From this we can apply a length expansion to the now stationary electrons and a relativistic contraction of the positive charges which are now apparently moving in the opposite direction, and from this difference in charge density the stationary charge now perceives an equivalent electric force.

I have no problem understanding the equations and achieving the correct solution. But my confusion is why relativity is necessary at all. In the first case we have stationary positive charges and a flow of negative charges to the right (relative to the test charge) which produces a magnetic field. In the second case we have stationary negative charges and a flow of positive charges to the left (relative to the test charge) which produces an identical magnetic field. The basis of amperes law is symmetric so that a flow of positive charge carriers in the opposite direction will produce an identical magnetic field with respect to our charged particle. No relativity or equivalence of magnetic and electric fields are required to demonstrate this.

The teacher explains this by saying that only electrons could be current carriers that satisfy amperes law, but this is totally unconvincing to me. Wouldn’t a beam of electrons in space produce a magnetic field? The same would be true of a beam of protons.

I know this has to be correct as formulated, but I hope someone can give me some insight to understand the theory in the terms that I have explained.

Thanks.

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  • $\begingroup$ Does the moving test charge have the same speed as the current-carrying electrons? You are implying this when you state "In the second case we have stationary negative charges and a flow of positive charges to the left". What if it is isn't? $\endgroup$ May 30 at 13:43
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    $\begingroup$ The fact that your professor thinks your "proof" of the non essentialness of relativity, is only wrong because "amperes law is violated" is worrying lol. $\endgroup$ May 30 at 13:57
  • $\begingroup$ We don't need relativity to understand this example, but this example useful to understand relativity. We use this example to show that electric and magnetic fields may be the same field in different frames of reference. $\endgroup$
    – Pere
    May 30 at 23:41

2 Answers 2

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You're professor is wrong.

Positive charge moving can clearly cause a magnetic field, to say otherwise actually violates amperes law. Please tell your professor this.

In the stationary frame, electrons are moving right and protons are stationary.

In this frame there is a charge with a velocity that matches the electrons velocity. This charge in this frame would experience a magnetic force.

Now let's switch frames to the frame of reference of the moving charge. In this frame. The electrons are stationary, and the positive charges are moving to the left. There IS a magnetic field in this frame. However, does the now STATIONARY charge experience this magnetic force? No because its not moving.

Hence in one frame it experiences the magnetic force. But in the other frame,it experiences no force. A contradiction.

This is why the lorentz transformation must be applied,and not the gallilean transformation. We see that for the situation in the rest frame to exist. In the moving frame, there is a different charge density (as a result of length contraction)

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  • $\begingroup$ Sorry I posted my answer before I read yours. You are exactly correct I just needed more reflection to realize it. We are covering this in high school so the high school teachers are Probably a bit less qualified when it comes to special relativity. $\endgroup$ May 30 at 14:14
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    $\begingroup$ Ngl, I don't think he should even be teaching physics if he thinks a moving a moving proton doesn't have a magnetic field. IF, that's what he actually meant. $\endgroup$ May 30 at 14:16
  • $\begingroup$ Well he’s very good in a lot of other ways, he explains a lot of things that are not in the curriculum. Thanks for answering my question so quickly by the way. $\endgroup$ May 30 at 14:26
  • $\begingroup$ I the frame in which the electric charges are at rest, doesn’t the test charge experience a magnetic force due to the mòving positive charge? $\endgroup$ May 30 at 14:30
  • $\begingroup$ Hi Felicia, there is no force on the test charge because it is not moving F=VxB. I’m glad someone else has the same misunderstanding as me :-) $\endgroup$ May 30 at 14:44
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OK after further reflection I think I understand now. The magnetic field is unchanged from the first scenario to the second, the difference is that the test charge is not moving in the second scenario since it is stationary within the frame. The teacher had said that the magnetic field is replaced by an electric field, but that is not true, the magnetic field remains but an electric field is necessary in order to account for the force on the test charge. Hopefully my question and answer help somebody who is learning this.

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