I am currently trying to understand the classical consequences of more than one charged particles in a magnetic field. So a thought experiment which came to mind was an electric dipole being placed in a magnetic field and given a slight push. Since there is Coulomb attraction between the two individual monopoles, and the magnetic forces acting on the two of them are in opposite directions due to their charge, does that mean they will eventually meet and annihilate (assuming no radiation, relativistic or quantum effects)? Does that mean a dipole is unstable in a magnetic field? Is that physically reasonable? Thank you!
First point: there is not ANY static distribution of electric charge that is stable under its own electrostatic forces. So either your two charges have to be held apart from one another by a rod or spring or gluey substance or something, or else they will not be stable even in the absence of any external field (of whatever kind).
Second point: opposite charges do not always annihilate. For example, when an electron meets a proton, they do not annihilate; rather one settles into a spherical distribution around the other (neutral hydrogen atom in its ground state). On a classical model (i.e. ignoring quantum physics) they would spiral around one another and eventually stick together and to avoid inconsistencies you would need to invoke a model in which they are not point-like. An electron and positron, on the other hand, would annihilate.
Finally, then, if we place two classical particles of opposite charge near one another in an external magnetic field, with no rod keeping them apart, then how will they move? If released from rest then the forces on them are initially purely Coulomb attraction so they begin to accelerate towards one another. If they are moving in the plane perpendicular to $\bf B$ then they begin to experience a magnetic force which, in the first instance, is in the same direction on both (since they have opposite velocity and opposite charge). If the $\bf B$ field is weak they each follow a curved path and soon collide. If the $\bf B$ field is strong I think (but have not calculated) that they each follow approximately a cycloid (i.e. curly line) imposed on a more gently curving shape as they move towards one another.
If the initial position is such that the particles are separated along the direction of $\bf B$, then if they start from rest they never get any magnetic contribution to the force (because their velocities are aligned with $\bf B$) so they move directly towards one another. If you set them off with a push to the side, then they get opposite magnetic forces and I guess they will do a beautiful little spiral dance as they move towards one another, and they may never collide ... a nice project for a fun computer simulation if anyone has the time?