# Motion under repulsive electrostatic force

1. Sep 14, 2006

### lark

Is there a simple curve that 2 particles follow when there's a repulsive electrostatic force - like there is for gravitational forces?

I don't know how to solve the differential equation that you get for the
motion.

Laura

2. Sep 14, 2006

### Staff: Mentor

When two particles have the same polarity (either both are + charged or - charged), they repel (rather than attract if charges are opposite) each other. The force is proportional to the inverse of the square of the distance (similar to the gravitational force law), and is described by Coulomb's law.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html

Using F = ma = m$\ddot{x}$, where x is relative to some reference point, e.g. the distance from one charge to the midpoint between the charges, which is then half of the distance between the charges. So there must be an equation of motion for each charge, for which the force is common, and which varies according to the separation of the charges.

If the charges have different masses, then they will have different accelerations, but the force one is the same as the other, even if the charges are of different magnitude.

3. Sep 19, 2006

### Meir Achuz

In the cm system, each particle follows a hyperbola.
The DE's are the same as for attractive force (like gravity), but with a different sign for qq'. The full derivation is in advanced mechanics books (like Goldstein), in connection with the classical derivation of the "Rutherford Scattering" cross section.

4. Sep 19, 2006

### lark

Yes, if one particle is fixed in place, it's at a focus of the other
particle's hyperbola! At the other focus than the one it would be at if the force were attractive.

But I don't know what the time parameterization looks like, though I separated the variables to get an integral for time as a function of distance.

Laura

5. Sep 19, 2006

### Meir Achuz

If neither particle is fixed, you can use center of mass variables.
I suggest you look into Goldstein for more detail.