TITLE: Construction of an optimal electron cage
QUESTION [17 upvotes]: I will describe the question first in 2D, but my interest is in $\mathbb{R}^3$.
An electron $x$ will shoot from the origin along an initial vector $v$. You know the speed $|v|$ but not the direction.
Your task is to arrange a minimum number of point-charge electrons 
at fixed locations on an origin-centered unit circle
so that, no matter in which direction $v$ is pointing, $x$ cannot escape the disk.
All the charges are equal, and repel each other via inverse-square Coulomb force.
Below there are four charges on the circle. The left electron escapes,
the right, which is shot along the same direction but with a lesser speed, does not. (Of course aiming $v$ along a diagonal is the best escape strategy in this
configuration.)

   


Now here is my question. Consider the same problem in $\mathbb{R}^3$.
Given $|v|$, arrange fixed electrons on a sphere to cage-in the electron $x$
starting from the origin with velocity $v$ in any direction.
It seems natural to think that an optimal arrangement is a type
of disk-packing on a sphere, 
for example, the solutions to the
Thompson problem
or the
Tammes problem.
(Here I am mining an earlier MO question.)

          


          

(Image from Paul Sutcliffe.)



Q. Is an optimal electron cage for a given speed $|v|$
  a configuration that minimizes electrostatic potential?
  Or maximizes the minimum distance between any pair of electrons,
  i.e., an optimal disk packing on the sphere?


Answered by Robert Israel: No such cage is possible in $\mathbb{R}^3$!

($\mathbb{R}^2$ is rather different and not a reliable guide to $\mathbb{R}^3$.)

REPLY [3 votes]: Here is Robert Israel's animation.