Can you trap electron in a ball charged with more electrons? I have this thought experiment. Imagine you charge an electric ball with electrons - like the one in Van de Graaff generator. There's a hole in the ball.
Now if you shoot an electron in the hole, with enough force to overcome the repulsive magnetic field of the outer electrons, what does it do?
Will it just stick there like on the image?

I assume it can always exit the ball by quantum-tuneling out - that is by randomly "occurring" at the outer side and then leaving repulsed by the electric field of other electrons.
Maybe it can also somehow radiate itself away as elmag waves?
What would really happen?
 A: Since this is a thought experiment, the material is perfectly even, we're in a perfect vacuum, and the hole is negligibly small relative to the size of the sphere, so we can ignore it for charge distribution. 
But is the sphere conductive or insulating?
As you might know, for objects inside a spherical shell of constant density, the gravitational attraction cancels out. If your sphere is an insulator, then its excess electrons can't move around, and their ideally even distribution would mean that the same math applies. Once the free electrons enter the sphere, they will continue moving at constant velocity until they hit the other side of the sphere.
Whereas if your sphere is a conductor, then its electrons will move around in response to the presence of other charges, away from negative ones or towards positive ones, until their mutual repulsion balances out.
I'm not prepared to do the math, but I'm pretty sure the net effect is electric force pushing outwards from the center of the sphere. The force is zero at the exact center, but it would be like trying to balance on the top of a frictionless hemisphere. The slightest perturbation sends you inexorably in motion.
Either way, it won't work as a trap for floating charged particles. They will almost certainly hit the walls.
A: If you shoot an electron into the inside of a hollow metal sphere, it will eventually enter the metal and reside as a surface charge on the outside of the metal. You can charge a hollow sphere almost without limit by mechanically introducing electrons into the interior. This is the principle of the Faradays cup, which is also used in van de Graaff voltage generators. The limit is reached when the electric surface field reaches the breakdown field of air, or in vacuum when the Fowler-Nordheim field emission (tunnel) current becomes significant.
Thus you can definitely "trap" electrons (charge) in a conducting sphere, even when there is already significant charge present! 
