Hong Life CalculatorsUnderstanding the electron speed calculator — concept and principle
This calculator finds how fast an electron moves after being accelerated from rest through a voltage, using relativity, and compares it with the classical estimate. It also gives the Lorentz factor and the de Broglie wavelength, which matter for cathode-ray tubes, X-ray tubes and electron microscopes.
How it is calculated
Kinetic energy K = eV, so K in eV equals the voltage V.
γ = 1 + K ÷ (mₑc²), with mₑc² = 0.51099895069 MeV
v = c √(1 − 1/γ²)
Classical estimate: v = √(2eV ÷ mₑ)
Momentum: pc = √(K² + 2K mₑc²); wavelength λ = h ÷ p
The principle
The field does work eV on the electron, and that work becomes kinetic energy. Classically K = ½mv², but special relativity says K = (γ − 1)mc², where γ grows without limit as v approaches c. That is why adding energy keeps raising γ while the speed creeps ever closer to, but never reaches, the speed of light. The classical formula agrees at low voltage and goes wrong at high voltage: above about 256 kV it even predicts speeds above c. Quantum mechanics assigns every particle a wavelength h/p, and fast electrons have wavelengths far shorter than visible light, which is what lets electron microscopes resolve atoms.
Worked example
Default input: 10,000 V.
- K = 10 keV.
- γ = 1 + 10,000 ÷ 510,999 = 1.01957.
- v ÷ c = √(1 − 1/γ²) = 0.19499, so v = 19.499% of c = 58,455 km/s.
- The classical estimate would give 19.784%, about 1.5% too high.
- pc = √(10,000² + 2 × 10,000 × 510,999) ≈ 101,587 eV, so λ = hc ÷ pc ≈ 12.2 pm.
| Accelerating voltage (V) | Electron speed (% of light speed) |
|---|---|
| 2,500 | 9.856 |
| 5,000 | 13.887 |
| 7,500 | 16.947 |
| 10,000 | 19.499 |
| 12,500 | 21.722 |
| 15,000 | 23.711 |
| 20,000 | 27.187 |
How to read the result
Compare the relativistic and classical percentages. Up to a few kilovolts they agree closely; at tens of kilovolts the gap is noticeable, and at hundreds of kilovolts only the relativistic value makes sense. Transmission electron microscopes run at 100–300 kV, where the electrons travel at more than half the speed of light.
Common mistakes
- Using ½mv² at high voltage and getting a speed near or above c.
- Forgetting that the result assumes a start from rest.
- Confusing keV of kinetic energy with total energy, which also includes the 511 keV rest energy.
Frequently asked questions
How fast is an electron at 1 kV?
About 6.2% of the speed of light, or roughly 18,700 km/s.
Can an electron reach the speed of light?
No. It would need infinite energy, because γ goes to infinity as v approaches c.
Why is the wavelength useful?
It sets the resolution limit of electron microscopes and the angles in electron diffraction.
Sources: OpenStax, University Physics Volume 3 (2016), section 5.9 "Relativistic Energy" · Particle Data Group, pdgLive (electron mass, CODATA 2022)