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Born2bwire
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lugita15 said:Just to clarify, the whole point of my thread is to try to understand whether or not the work done by the magnetic force on an object can ever be nonzero. Usually, people say that the work done by magnetic forces is always zero. But if statements A through E are true, then the work done by magnetic forces CAN be nonzero. So I am trying to find out which, if any, of the statements A through E are true.
Work is not done by the magnetic contribution to the Lorentz force. This is because the Lorentz force is directly acting upon the currents, not the wire. The wire is moved in reaction to the wire's ionic lattice being attracted via Coulombic forces between the displaced currents. That is, the Lorentz force displaces the electrons in the wire, the movement of the electrons causes them to pull the wire with them via electric fields (to first order). So when we are talking about the path of displacement over which the Lorentz force is acting, it is not in say the x direction (assuming that our wires run in the z direction). That is the direction of the movement of the wires but since the force is not acting on the wires it is not relevant. The force is acting on the moving charges that make up the current, now these charges had an initial velocity so they are actually going to be moving in a circular trajectory in response to the magnetic field. But as their velocity vector changes, so does the force from the magnetic field. The Lorentz force from the magnetic fields always changes its direction as the direction of the charge's movement changes too.
This circular trajectory is not noticed because the movement of the electrons is not impeded along the wire, so there is no movement in the wire along its axial direction. In addition, the wire is infinitely long, so we are looking at a superposition of charges that are all moving and reacting identically. On the whole, what we see is a line current moving together in one direction towards to the other wire.
But the other point to note is that if we allow the wires to be attracted over a distance, then we now have a changing set of magnetic fields since the currents are moving in space. This means that we now have a set of electric fields. So we need to now take into account the force that will arise from these electric fields. It actually becomes a complicated problem that is not apparent from the static force problem.
EDIT: I would say that statements A-E are correct. It is just when you assume that the path that we take our integral for the force is not the path that the wire moves. Ignore the wire, think of what would happen if we just had lines of electrons that were moving in place of the wires and currents. What happens to just one electron in response to the magnetic fields from the other wire? Keep in mind that when start the problem off, the electron already has an initial velocity, corresponding to being a current.