Attractive Charges

If you didn't read my previous post, this one probably won't make much sense. In my previous post, I provided some pretty graphs of what happens when you try to trap one electron with another. But, what if you try to trap an electron with a proton (i.e. rather than repulsive forces between the particles, the forces are attractive)? Mathematically, the problem is essentially the same; only two negative signs disappear. The behavior of the trapped electron, however, changes significantly. Not only are some of the graphs of its motion way more aesthetically pleasing, but, counter-intuitively, despite the fact that the two charges are attracted to each other, it is much harder to trap the electron due to a sling shot effect that is similar to a gravitational sling shot. (If you want to, instead of thinking of an electron and a proton, think of two similarly sized planets that are gravitationally attracted...it's more or less the same problem.)

By harder to trap the electron (or some negative charge), I mean that I have come across only five speeds so far while running simulations of the problem at which the proton (or whatever positive charge), which orbits about the origin at a constant speed unaffected by the electron, can indefinitely trap the electron. What usually happens in the proton/electron version of this problem is that the electron, although it may be trapped in some region of space for a finite period of time, eventually gets too close to the proton, is accelerated hugely (remember the force between the two: F=kqq/r^2), and then is slingshotted off into the far reaches of the universe.

Here are a couple of graphs of the problem (click on images to see them better):

Blue is the motion of the electron which starts at rest at y=0, x=.2. Red is the motion of the proton which starts at the top of the circle (x=0, y=1) and then moves in a circle at a constant speed.

This is a 1 meter circle. The proton (red) moves at 75 meters per second. The first graph is after 1.5 seconds have passed. The second graph is after only .2 seconds. Notice how the velocity and distance from origin graphs are weird periodic functions. Also, notice how the velocity and acceleration of the electron tend towards infinity when the electron gets very close to the proton. These are obviously very different from the graphs where both particles are electrons...


Here is another beautiful flower. The proton moves slower at 70 m/s. The electron starts at the origin this time.

And here is what usually happens: the electron escapes. In the first one, the proton moves at 225 m/s, the second 785 m/s.


OK, so the graphs, as I mentioned, are cool. But, why do we care about such a problem? Well, we probably don't. But... if a damping term (a constant multiplied by the velocity... just like a damped spring equation) is added into the equations governing the electron's acceleration, the resulting equations can be used to model optical trapping. I am not going to try to explain optical trapping in this post (I currently don't understand half of it), but apparently it can be a very useful tool in biology.

So, I'm getting pretty good at using Mathematica and I will continue to work on this problem as well as the "Homicidal Chauffeur Problem" (I've finally made progress on that front!). Expect the next post to be about the tigers chasing rabbits (Homicidal Chauffeur).