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The kinetic energy acquired by the electrons is equal to the electric work done by the field in ripping them away from near the surface of the filament.
So we can say:
Kinetic energy = electrical work done
Where m is the electron mass,
v is its velocity as it leaves the gun,
e is the charge of the electron, and
V is the Voltage on the anode.
Electrons are so light that very quickly they reach speeds close to the speed of light. Trying to accelerate electrons much further starts to bring in relativistic effects: the velocity doesn't increase much more but the electron mass starts to increase instead! Sounds quite complicated, but that's relativity for you!
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