Motion of an electric charge within an electric field with two electric charges

Question

Assume that air resistance and gravity are negligible and the only significant force acting in the scenario is the electric force.

There are two electric charges, both with an equal and positive charge of +q. They are spaced 2x apart. At some distance away, along the perpendicular bisector of the line connecting the two charges, lies another charge with an equal but negative charge of -q at distance d away from the midpoint of the line connecting the two positive charges. Given that the positive charges are fixed in place while the negative charge is released at its position with a velocity that is less than the escape velocity required to exit the system, describe the motion of the moving negative charge.

The negative charge's initial velocity has no limitation of direction.

Is the motion going to be similar to an object orbiting around the barycenter, just like a gravitational field?

Answer 1

Although the situation is quite simple, and equations of motion can be written without much difficulty, these equations cannot in general be solved in terms of simple functions, except in a few special cases. Numerical solution is necessary in most cases, and 'orbits' will not be stable.

Motion along the perpendicular bisector will be an oscillation but not SHM, because the restoring force is not proportional to the displacement from the equilibrium position. It will also be unstable, because a small deviation will increase the force pulling this charge to the side. A simple circular motion, in the plane perpendicular to the line joining the fixed charges and passing through their midpoint, is possible. But again this will not be stable.

Other motions will be very difficult to predict even in descriptive terms, except in extreme cases. This is not a situation where there is a spherically symmetric force, so motion will not in general be like that of a planet orbiting the Sun.

When the distance between the fixed charges is much smaller than the distance from the moving charge, a sort of orbital motion is possible but it is very likely to be unstable and non-repetitive.

As Declan points out, accelerating charges emit radiation, but unless the speed of the moving electron is a significant fraction of light speed, the effect is negligible. Taking this into account would complicate the problem enormously, when it is already very difficult.