The speed of electrons
In an electron gun, electrons are boiled off the surface of a hot metal plate. They leave the plate with very small speeds, and then the electric field accelerates them towards the anode. See the guidance note Electronic guns.
You can calculate the electrons' speed by thinking of the energy changes in the system.
Each electron has a charge of e coulombs, and the potential difference between the filament and the anode is V volts.
The electrical energy transferred to each coulomb of charge is V joules.
So the energy transferred to electrons is eV joules.
The electrons gain kinetic energy. Unlike electrons in a wire, these electrons have nothing to hit, nothing to lose energy to, as they travel towards the anode. So each electron gains kinetic energy equal to the amount of energy transferred from the electrical supply.
The electron starts from rest (near enough) so the kinetic energy gained is given by 1/2 mv2where m is its mass and v is its speed.
So we can say that: 1/2mv2 = eV
The mass of the electron is m = 9 x 10-31k g
The electronic charge is e = 1.6 x 10-19C
For an electron gun with a voltage between its cathode and anode of V = 100V the electron will have a speed of about v = 6 x 106 m/s. (Relativistic effects have not been taken into account.)
There will be no more acceleration once the electrons have passed through the anode.
A crude model would be a collection of marbles running down a sloping board to crash into a wall at the bottom, except for a few that might hit a gap in the wall and would continue along on the flat ground on the other side of the wall. The slope corresponds to the electric field we apply inside the gun to accelerate the electrons. The flat ground corresponds to the region beyond the anode where the electrons continue at a constant velocity.
A TV picture tube has just such a gun, to fire electrons straight out to the screen in the tube. There the electrons make a bright spot by exciting a glow on the screen, but on their way they can be pulled out of a straight line by magnetic fields.