⚛️ Electron Speed Calculator
Enter an accelerating voltage to calculate the speed an electron reaches, using the relativistically correct formula (electrons become relativistic quickly).
Why electron speed needs relativity
Electrons are so light that even modest accelerating voltages push them to a significant fraction of the speed of light, where classical (Newtonian) mechanics starts giving noticeably wrong answers. This calculator uses the relativistically correct approach.
The formula
An electron accelerated through voltage V gains kinetic energy KE = eV. Adding this to the electron’s rest energy (m0c2 ≈ 511,000 eV) gives total energy, from which the Lorentz factor is γ = Total Energy ÷ Rest Energy. Speed then follows from γ = 1/√(1-β2), rearranged to solve for β = v/c.
At voltages below a few hundred volts the relativistic and classical results are nearly identical, but by 10,000-100,000 volts, the difference becomes significant, which is why devices like electron microscopes and X-ray tubes must account for relativistic effects. This calculator is for general physics education.