My first post so please forgive me if I am already breaking some rules ;) This may apply to positive or negative particles: We know that equal charges repel each other to the value of Coulomb force F=kqq/r^2. What I would like to know is if a magnetic force between two (non-moving, but rotating) equal charges is also always repelling? (or any other force?) Assume that the particles have E and B fields, and they rotate in one place. It is allowed to orientate the particles in different directions if necessary. So if for example a negative particle facing in the x-direction is close to a negative particle facing in the y-direction, their E and B fields are perpendicular. Because they are rotating, and their fields are crossing, I am expecting a magnetic effect, but will it also be a repulsive force? Or is there some orientation that could make this magnetic force negative? Are there equations that I could use to analyse such a situation? Is there another force, like gravitational force, that could have two equal charges attracting each other? It does not need to overcome the Coulomb force.
No, the force depends on the rotation, charge distribution and so on. However, the electric repulsion will be stronger (and the magnetic force can approach this value for velocity-->c), unless you add opposite charges to the system. For static setups (constant currents and charge distribution): Biot-Savart law For non-static setups: Retarded potentials
Thanks for the reply mfb When I look at the potential energies, and the equation: Uem = Ue + Um Ue generally overshadows Um, and Ue is positive or negative depending on the +/- product of e of the particles. So for two positive particles Ue is positive, and the force between them is repellent. The question is: Is there any way that Um can be negative between two positive particles? Will Um always add to the repelling force, or can it be a negative force(energy)? I'm hoping there's some relation to spacial orientation, to get an ExB effect between the 2 particles that makes Um<0? I am not concerned at getting Uem<0, but only to reduce it by making Um<0 for the 2 positive particles. Is this at all theoretically possible?
You asked "[...] is also always repelling?" and I answered "no". It did not change since then. Yes, you can have attractive components in the calculation. Let both rotate in parallel planes, in the same direction, for example. However, it is questionable how useful a separation in "this is electric" and "this is magnetic" is.
This being the classical physics forum, you may consider e.g. two homogeneously charged spheres. If the spheres are rotating, they will have a magnetic moment and the two magnetic moments may attract, if the moments are anti-parallel. Non-relativistically, the magnetic moments can be arbitrary large, as we can make the spheres spin with arbitrary velocity. Relativistically, the speed of the particles must be lower than c. Then I would suppose that the magnetic attraction can be shown to be smaller than the repulsion of the charges.
Even if the speed is limited, couldn't the current loop be made arbitrarily large by putting enough charge on the spheres?
With more charges, you increase both the magnetic and the electric forces by the same factor. You can indeed get a net attractive force, if you somehow get the electrons to rotate in one direction and the nuclei to rotate in the other direction, combined with a small net charge. Depends on the orientation of the rotation axes relative to the displacement of the spheres.
The magentic moment m of the rotating sphere of radius R and charge q is [itex]m=const. qvR[/itex] where v is the velocity of the sphere and const. is a constant of order unity. The attraction of the two dipoles falls of like 1/r^{3} with the distance r of the dipoles and the strength of the attractive force is proportional to [itex]m^2/c^2[/itex] while the repulsion of the two charges is proportional to q^{2} and falls off like [itex]1/r^2[/itex]. The attraction falls off more rapidly with r than the repulsion, but r cannot be smaller than 2R. Substituting this distance into the forces, we see that the attraction of nearly touching spheres scales like 1/R with the radius but the repulsion falls off like [itex]1/R^2[/itex]. Hence for small distances and sufficiently big spheres, there will be always attraction. On the other hand as v cannot be larger than c, there is a minimal radius below which the force between the two spheres is repulsive at all distances.
Quantum mechanically, It is possible in the case of electrons when they pass through a Bose - Einstein condensate. The electrons form cooper pairs and behave like bosons without following the Pauli exclusion principle and the electron spins do not prevent them from flowing together through the lattice. You could say that they behave like they are attracting but in reality there are no magnetic forces that cause the attraction. It only happens because of the formation of cooper pairs as a result of the electromagnetic interactions with the condensate's lattice and the exchange of bosons called lattice phonons or simply phonons.