What is total energy

Acceleration in an electron gun

As a rule of thumb it is said that relativistic effects from speeds of 10% of the speed of light should be considered. This is the case with electrons at acceleration voltages from approx. 2.6 kV.
Relativistically, you can calculate the flight speed of the electrons as follows:
The kinetic energy is the total energy minus the rest energy: \ [E _ {\ text {Kin}} = m _ {\ text {rel}} \ cdot c ^ 2 - m_ {e} \ cdot c ^ 2 \] Equation of kinetic energy with the performed Work \ (W _ {\ rm {el}} = U _ {\ rm {b}} \ cdot e \) of the E-field: \ [U _ {\ text b} \ cdot e = m _ {\ text {rel}} \ cdot c ^ 2 - m_ {e} \ cdot c ^ 2 \ quad (1) \] The masses \ (m _ {\ text {rel}} \) and \ (m_ {e} \) are determined by the Lorentz factor \ (\ gamma \) linked together: \ [m _ {\ text {rel}} = \ gamma \ cdot m_ {e} = \ frac {m_ {e}} {\ sqrt {1- \ frac {v ^ 2} { c ^ 2}}} \ quad (2) \] Inserting (2) into (1), excluding and dividing by \ (m_ {e} \ cdot c ^ 2 \) yields: \ [\ frac {U_ { \ text b} \ cdot e} {m_ {e} \ cdot c ^ 2} = \ frac {1} {\ sqrt {1- \ frac {v ^ 2} {c ^ 2}}} - 1 \] Add of 1 and squaring it leads to: \ [\ left ({1+ \ frac {U _ {\ text b} \ cdot e} {m_ {e} \ cdot c ^ 2}} \ right) ^ 2 = \ frac {1 } {1- \ frac {v ^ 2} {c ^ 2}} \] Form reciprocal values, multiply by -1 and add 1 yields: \ [\ frac {v ^ 2} {c ^ 2} = 1- \ frac {1} {\ left ({1+ \ frac {U _ {\ text b} \ cdot e} {m_ {e} \ cdot c ^ 2}} \ right) ^ 2} \] Multiplication by \ (c ^ 2 \) and pull the Root leads to: \ [\ bbox [5px, border: 2px solid red] {{v _ {\ text {relativistic}} = c \ cdot \ sqrt {1- \ frac {1} {\ left ({1+ \ frac {U _ {\ text b} \ cdot e} {m_ {e} \ cdot c ^ 2}} \ right) ^ 2}}}} \]