Re: How does an electron gun work

Excellent question!
Your thoughts on the subject are correct, but the trick lies in tricking 
the electrons (or any other charged particle, e.g. a mass spectrometer).  
What you do is build the anode with a hole in it.  This is almost certainly 
what they taught you, and you were completely correct in thinking that the 
electrons or ions would just be reattracted to the anode.  However, what 
they did *not* tell you was that immediately behind the anode was another 
plate with a hole in it, the holes being aligned.  This second plate is 
usually called the suppressor, and the key thing to making this work is 
that the suppressor is either grounded OR at a voltage opposite in polarity 
to the anode.  [Question for you:  if you're working with electrons, what 
polarity would you make the anode and the suppressor?]  So now the 
electrons coming off of the anode are accelerated and those that make it 
through the hole in the anode immediately pass through the hole in the 
suppressor as well.  This suppressor (which can be a plate with a hole in 
it or a grid) shields the electrons that passed through the hole from the 
anode and a rough beam is formed so we've sort of "tricked" the electrons. 
Now, generally you want a fine (collimated) electron beam so you put a 
second anode/suppressor section in.  As before, not all the electrons will 
hit this second hole BUT those that do tend to be well collimated.  You are 
correct in thinking that you can do this stunt any number of times and the 
end result will be a high velocity, finely collimated beam of electrons 
which is generally steered magnetically.  The only limitations on this 
technique are physical size and voltage breakdown between the various 
plates.  But this is the method used.
A variant of this technique is sometimes useful when you want to get as 
many electrons or ions as possible and you don't care about forming a beam. 
 One example would be a vacuum tube.   What you do is make the anode a grid 
so that only a few electrons/ions strike the grid wires;  the rest pass 
right through.  Then you use a suppressor grid and make the final anode 
relatively large.  FYI, this is what high-power radio stations do to get 
the power (several kilowatts/tube) output required.  So as you continue 
your education please don't disdain vacuum tubes;  they can be very useful.