1

Reasons for the existence of a magnetic force can be found in the rest frame of a moving electron next to a current carrying wire as explained in Chapter 5.9 "Interaction between a moving charge and other moving charges" of Purcell: Electricity And Magnetism.

However, I am looking for an explanation in the lab frame, since an explanation should not depend on observations in a specific reference frame. An explanation should be possible in any reference frame, hence also in the lab frame, where two current carrying wires attract each other. I would like to understand how moving charges can create a magnetic force there.

Frank Breitling
  • 148

  • since an explanation should not depend on observations in a specific reference frame. All physics explanations ultimately come to generalizations of empiric evidence, and that is always obtained in some specific reference frame. Maybe you're not looking for an explanation, but for a mathematical derivation, which does not use any concrete reference frame? For that you have to adopt some assumptions, such as special relativity with the Lorentz transformation. Then magnetic force is just that part of relativistic electromagnetic force which depends on velocity.

     – Ján Lalinský Jan 03 '24 at 23:04
  • Lorentz-force. Between electrons. Not between ions and electrons. Even though in the electron frame there was this 'extra' force between ions and electrons. I will not tell that it's a time-dilation of momentum exchange, because nobody ever gets it. – stuffu Jan 03 '24 at 23:12
  • @JánLalinský I'm just looking for an explanation of the magnetic force in the lab frame without involving reasons from the electron's rest frame, because the lab frame should be self-consistent. – Frank Breitling Jan 03 '24 at 23:53
  • @stuffu I have not heard of "time-dilation of momentum exchange" in this context. Do you have a reference? – Frank Breitling Jan 03 '24 at 23:57
  • 1
    Related : Why don't stationary charge feel force from a current carrying wire?. – Frobenius Jan 04 '24 at 03:42
  • ‘ Reasons for the existence of a magnetic force can be found’ These can only be found in experiment. – my2cts Jan 04 '24 at 12:54
  • @my2cts Yes, found in this thought experiment. – Frank Breitling Jan 04 '24 at 13:49
  • @FrankBreitling Thought experiments are not to me confused with real experiments. – my2cts Jan 04 '24 at 14:09

1 Answers1

0

Let us say we construct a clock that consists of two very large freely moving positively charged plates, with large inertia.

This clock reads zero when the plates are stacked together. The plates accumulate momentum at constant rate. We read the clock by measuring this accumulated momentum.

Okay now we send this clock to a trip to Alpha Centauri and back at constant speed 0.87 c.

Let's say Alpha Centauri is in the up-direction, while plates repel each other in the left-right-direction.

The trip takes 9.74 years earth time. But the clock that took the trip proceeded only 4.85 years.

Compared to a charged-plates-clock that stayed on earth the charged-plates-clock that made the trip accumulated half the time. And half the momentum.

The force on the plates in the lab (earth) frame was the change of momentum of the plates divided by 9.74 years. (I am talking about the clock that made the trip)

The force on the plates in the rest frame of the center of mass of the charged plates was the change of momentum of the plates divided by 4.85 years. (I am talking about the clock that made the trip)

The reduction of the force in the lab frame is the magnetic force.

stuffu
  • 1,978
  • 11
  • 11
  • Why do the clocks accumulate momentum and how? – Frank Breitling Jan 04 '24 at 13:51
  • @FrankBreitling Plates repel each other by a constant Coulomb force. Force is constant thanks to the large size and the large inertia. Oh yes I need to add that plates move parallelly to Alpha Centauri while exerting a force to each other transversely to the motion to said star, like moving electrons in parallel wires. – stuffu Jan 04 '24 at 15:02
  • I see. But could this also explain a repellent force in case of opposite currents? – Frank Breitling Jan 04 '24 at 18:58
  • @FrankBreitling I don't know ... But to calculate the force between two parallel electron beams in lab frame we can go to the rest frame of electron beam and notice that the length-stretching of the beams causes a reduction of momentum exchange in that frame. And in the lab frame that momentum exchange is stretched over long time, so the force changes as 1/gamma^2 in the lab frame. I mean Coulomb force changes like that when beams are accelerated by electric field. Such acceleration is such that it causes no length-contraction in lab frame and length-stretching in electron frame. – stuffu Jan 05 '24 at 16:06
  • Ok, but how can this explain a repellent force in case of opposite currents? – Frank Breitling Jan 05 '24 at 22:32
  • @FrankBreitling I don't know ... I noticed that I can't understand the length-contraction effects in the electron frame in the lab frame. That's maybe the problem. – stuffu Jan 06 '24 at 14:17