Programming simulations of the "Homicidal Chauffeur Problem" (http://basissrp2010.blogspot.com/2010/02/homicidal-chauffeur.html) has been a bit more difficult than expected (my programs spit gibberish at me 90% of the time though a couple are working). So, to boost my self-esteem, I decided to work on a slightly different problem.

The problem:
It is possible to trap a charge with a second charge if you simply move the second charge very rapidly around the first. So, if you have electron "A" initially at rest at the origin and then start moving electron "B" in a circle around the origin, how fast do you have to move "B" in a circle to indefinitely trap "A"? Also, what is the resulting path of electron "A"?

Before I show you the wondrous math, here are pretty and mysterious graphs:







Answer (skip to end if you dislike math):

Start of math-------------------------------------
The path of "B" since it is moving in a circle is ("t" represents time and b is a subscript):

xb=Sin(t)
yb=Cos(t)

The two electrons will be repulsed by Coulomb's Law (r is the distance between the two electrons, k is a constant, and q is the charge of an electron):

F=ma=kqq/r^2

Divide by m:

a=kqq/(mr^2)

Now, we want to break up the acceleration into its' X and Y components.
If "w" is the angle made between the two electrons (the angle the X axis makes with the line between the electrons) then

ax=[kqq/(mr^2)] Cos(w)
ay=[kqq/(mr^2)] Sin(w)

Using some simple geometry we can rewrite "r", Cos(w), and Sin(w) in terms of x,y, and t .

After that we are left with two ugly second order differential equations that can be plugged into Mathematica's (a super math programming tool) numerical differential equation solver. After that, you end up with some really awesome graphs like the one above. The red circle is clearly the path of electron "B" and the squiggly path represents electron "A".

End of math---------------------------------------------

By varying the speed of electron "B" (the electron moving in a circle), we can change the path of "A" in interesting ways. If "B" doesn't move fast enough, "A" will escape. The faster "B" moves, the more restricted the movement of "A" will be. If you really, really want to relate this problem to the chauffeur problem, you can think of "B" as the tiger and think of "A" as the rabbit who runs away from "B" according to an inverse square rule.

So despite the fact that this electron problem is at best extremely tangentially related to "The Homicidal Chauffeur Problem", for the time being, I will be looking at some variations of this problem (for example, what happens when we add more electrons?).

I hope this post isn't too cryptic. If it is, feel free to ask questions.

Here are more incredible graphs...

The orbiting electron "B" is moving relatively slow and "A" escapes:









Here "B" is moving faster and "A" doesn't escape:

Faster yet and "A" 's movement is restricted even more: