The following physics problem has been proposed on Momes (via Perfectly Reasonable Deviations):

Two adjacent electrons move parallel to each other, in the same direction in a vacuum. What happens? Consider the situation at varying speeds from 0 to c.

The most intuitive response is to say that they attract since it has been taught to us that two parallel wires carrying electric current in the same direction will attract each-other. The question is somewhat ill-posed until the frames of reference are defined. Do the electrons move at the same speed? What speed does the observer move? If we assume that the electrons are stationary with a fixed distance between then and the reference frame of the observer moves at some **v** that varies from 0 to c, we can write the Lorentz force equation to describe the force that one electron exerts on the other *as observed in the moving reference frame*. The thing to remember here is that the magnetic field is the Lorentz transform of the electric field, or rather that it is the result of a charge moving with respect to the observer. Since the Lorentz force equation, **F**=q**E** + **v** x **B**, has both an electric and magnetic component, it can be shown that the force is repulsive when the observer is stationary with respect to the electrons and becomes attractive after the speed (I know this should be velocity, but we are assuming that the direction is away from the electrons) becomes a certain v’ such that **F** = 0. This is different from parallel cables that attract for any measurable current in the same direction because the metal cables are electrically neutral for any macroscopic volume. That is, for every electron, there is a proton, so there is no net radial electric field and the total force component is due to the magnetic field only.

One final thought to keep in mind is that two electrons stationary in vacuum without external fields is nothing more than a mathematical construct. Like plane waves, this configuration would not be static in time without some external force keeping the electrons a fixed distance from each-other. The reason that I bring this up is that, although they are good exercises, one must not try to infer laws of nature from mathematical constructs.

[Image is of density measurements of an electron cloud from Physics Central.]