Parallel line currents & Lorentz transformation

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SUMMARY

This discussion centers on the interaction between two parallel infinite line currents and the implications of the Lorentz transformation on their behavior. In frame K, these currents attract each other due to their equal linear densities. However, when transitioning to frame K', where the current is perceived as zero, the nature of the forces shifts from magnetic to electric. The participants conclude that despite the symmetry in linear densities, the transformation leads to a repulsive force between the currents in K', contradicting the initial attraction observed in K.

PREREQUISITES
  • Understanding of Lorentz transformation principles
  • Knowledge of electromagnetic theory, specifically the behavior of currents
  • Familiarity with the concept of charge density and its implications in different reference frames
  • Basic grasp of the Lorentz-Fitzgerald length contraction
NEXT STEPS
  • Study the implications of Lorentz transformation on electric and magnetic fields
  • Explore the relationship between charge density and force interactions in different frames
  • Investigate the concept of electromagnetic duality in particle physics
  • Learn about the mathematical formulation of the Lorentz-Fitzgerald contraction
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism and special relativity will benefit from this discussion, particularly those interested in the behavior of currents in different reference frames.

Preno
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Okay, I'm sure I must have overlooked something very trivial, so please help me with this:

Two parallel infinite line currents of equal magnitude attract each other. The current can be thought of as consisting of positive and negative particles with equal linear densities, but each moving with a different velocity (wlog let's assume the positive ones are static in some frame of reference K). Let us now move to the frame of reference K' which moves with the current so that the current in K' is zero. The force which was purely magnetic in K is now replaced with a purely electrical one - the linear densities of the positive and negative particles undergo Lorentz transformation, so that the total density in K' is non-zero.

Here's my problem: if linear densities and the velocities of particles (and hence the currents) have the same magnitude in K, then they must also have the same magnitude in K' (the transformation respects the symmetry between the two lines). But like charges repel each other. So it would seem that in K' (hence also in K), the currents actually repel each other.
 
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Hi Preno! :smile:
Preno said:
The current can be thought of as consisting of positive and negative particles with equal linear densities, but each moving with a different velocity …
Let us now move to the frame of reference K' which moves with the current so that the current in K' is zero. …

Sorry, I'm not following what K' is …

as you say, there are two lots of charge, moving with a different velocity. :confused:

(and don't forget the Lorentz-Fitzgerald length-contraction)
 
tiny-tim said:
Hi Preno! :smile:


Sorry, I'm not following what K' is …
K' is the frame in which the current vanishes.
as you say, there are two lots of charge, moving with a different velocity. :confused:
Well, the line is electrically neutral, so there need to be both positive and negative charges.
(and don't forget the Lorentz-Fitzgerald length-contraction)
No, I'm not forgetting it. That's the whole point. The linear density of those charges changes when transforming to K' but it remains the same on both lines. Hence they should repel each other.
 

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