Force on charge particle with constant velocity

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Discussion Overview

The discussion revolves around the forces acting between two equal charges moving with constant velocity. Participants explore the nature of electromagnetic interactions, particularly focusing on the electric and magnetic forces involved, and the implications of the angle between the charges and their velocity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the assumption that the force between the charges would be zero, prompting a discussion on the electromagnetic field associated with moving charges.
  • Another participant suggests that the electric forces cancel each other while the magnetic fields exerted by each charge are in opposite directions, raising a question about the net effect.
  • A different viewpoint emphasizes that the inquiry should focus on the force on one particle due to the other, rather than the net force on the system.
  • One participant proposes a specific case where the angle A is \(\frac{\pi}{2}\), detailing a method to analyze the forces by transforming from a stationary frame to a moving frame, invoking Coulomb's law and Lorentz transformations.
  • The same participant notes that the transformation leads to a reduction in force, which they interpret as a magnetic attractive force counteracting the electric repulsive force, while acknowledging that the situation becomes more complex for angles other than right angles.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the forces involved and the appropriate approach to analyze the problem. There is no consensus on the resolution of the forces acting between the charges, and multiple competing models and interpretations are presented.

Contextual Notes

The discussion includes assumptions about the behavior of electric and magnetic fields in different reference frames and the implications of relativistic effects, which remain unresolved. The complexity of the situation for angles other than right angles is also noted but not fully explored.

elsafo
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I have a problem. Anyone can help me?
Two equal charges q move with equal velocity v. What is the force acting between two charges?
The distance between charges is R and the angle between R and v is A.
 
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Why would it be zero? What is the electromagnetic field associated with a moving point charge?
 
The electric force cancel each other and the magnetic field exerted by each charges are opposite but opposite direction. How?
 
I believe that what is sought is not the net force on the two-patricle system but the force on one of the particles induced by the other.
 
So how to solve it?
 
Start by trying to answer the second question of my first post.
 
elsafo said:
So how to solve it?
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I'll deal just with the case of A = \frac{\pi}{2}, that is the velocity is at right angles to the line joining the charges. The easiest way to do this is to start in the frame of reference in which the charges are stationary. The force between them in this frame is simply the ordinary Coulomb's law force. Now transform this force to the lab frame, in which the charges are moving. The force transformation is very easy as

transverse force = change in transverse momentum / time taken to change.

The change in transverse momentum is the same in both frames, as it is a Lorentz invariant. The time is dilated by the usual gamma factor in the lab frame, compared with that in the frame in which the charges are at rest. So the force between the charges in the lab frame is, in SI units
\frac{\sqrt{1 - \frac{v^2}{c^2}} Q^2}{4 \pi \epsilon_0 d^2}.
This reduction in the force can be interpreted as a magnetic (Ampère) attractive force coming into play in the opposite direction to the electric (Coulomb) repulsive force. But it's not as simple as that… As Zoki's equations show, the electric field is also changed. You'll note that Zoki's equations boil down to give the result I've derived above, in the special case of velocity at right angles to the line joining the charges. When the angle is not a right angle things get more complicated, but, again, you can either go for a relativistic force transformation approach (treating components parallel to, and transverse to, the velocity separately), or you can use Zoki's equations (having first derived them?)
 
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