What is 'velocity' in classical electrodynamics?
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I have recently noticed that unless the objects in question are conducting, neutral wires, the Lorentz force law along with Maxwell's equations will give you anything and everything as a solution for the force on some charge.
My question can simply be summarised. Given that forces due to magnetism require velocity in their definition, is it possible to deduce the 'correct' frame of reference by which to measure velocity without using special relativity to find what is true for every frame?
The question is best illustrated by examples:
Take a neutral wire with free electrons and fixed positive charges that exists along the y axis. When we set a voltage from source to sink, a still proton in the 'Lab' frame (at rest w.r.t to the positive charges' frame) adjacent to this wire will not be affected according to the Lorentz force law.
Instead, let's set positive charge in motion so that it is at rest w.r.t to the moving electrons' frame, it will begin to repel the wire by magnetic force. The magnetic repulsion is the same in any frame and we can see this using special relativity (Lorentz contraction). However, looking closely at the situation, the two setups were exactly symmetrical except for the fact that the positive charges are fixed in place. I am forced to conclude that this is the cause of the magnetic attraction. Since the electrons are free to move, Lorentz contraction does not affect their charge density in any frame. The exact mechanism by which this happens still baffles me. Nonetheless, we know that the charge repels by experiment. Therefore, the correct velocity is that measured in the fixed positive charges' frame or 'Lab frame'.
This second scenario may help highlight my confusion. Put free positive charges as well as free negative charges, all of which are able to move along the y axis. Well, if we follow the reasoning I used in the first scenario, the free charges will always move until there's no net electric field. We could then pick the frame in which the positive charge is still, no matter what currents exist in the lab frame. By the principle of relativity, this means there must be no force on the charge (no matter what). So the correct frame is the positive charge's, or 'observer frame'.
My question can simply be summarised. Given that forces due to magnetism require velocity in their definition, is it possible to deduce the 'correct' frame of reference by which to measure velocity without using special relativity to find what is true for every frame?
The question is best illustrated by examples:
Take a neutral wire with free electrons and fixed positive charges that exists along the y axis. When we set a voltage from source to sink, a still proton in the 'Lab' frame (at rest w.r.t to the positive charges' frame) adjacent to this wire will not be affected according to the Lorentz force law.
Instead, let's set positive charge in motion so that it is at rest w.r.t to the moving electrons' frame, it will begin to repel the wire by magnetic force. The magnetic repulsion is the same in any frame and we can see this using special relativity (Lorentz contraction). However, looking closely at the situation, the two setups were exactly symmetrical except for the fact that the positive charges are fixed in place. I am forced to conclude that this is the cause of the magnetic attraction. Since the electrons are free to move, Lorentz contraction does not affect their charge density in any frame. The exact mechanism by which this happens still baffles me. Nonetheless, we know that the charge repels by experiment. Therefore, the correct velocity is that measured in the fixed positive charges' frame or 'Lab frame'.
This second scenario may help highlight my confusion. Put free positive charges as well as free negative charges, all of which are able to move along the y axis. Well, if we follow the reasoning I used in the first scenario, the free charges will always move until there's no net electric field. We could then pick the frame in which the positive charge is still, no matter what currents exist in the lab frame. By the principle of relativity, this means there must be no force on the charge (no matter what). So the correct frame is the positive charge's, or 'observer frame'.
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I have recently noticed that unless the objects in question are conducting, neutral wires, the Lorentz force law along with Maxwell's equations will give you anything and everything as a solution for the force on some charge.
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