Old cathode-ray TVs accelerated electrons through about 25 kV, reaching 30% of light speed. Above a few kilovolts the classical formula starts to overestimate the speed noticeably; at 1 MV it would predict a speed faster than light.
How to use the Electron Speed Calculator
- Accelerating potential V
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the accelerating voltage.
What each result means
- Classical speed = sqrt(2×1.602176634e-19×[Accelerating potential V]÷9.1093837015e-31)
- Lorentz factor γ = 1 + 1.602176634e-19×[Accelerating potential V]÷(9.1093837015e-31×299792458^2)
- Relativistic speed = 299792458×sqrt(1 – 1÷[Lorentz factor γ]^2)
- Classical overestimate = ([Classical speed]÷[Relativistic speed] – 1)×100
- Kinetic energy = [Accelerating potential V]÷1000
Example values
How the main result changes when Accelerating potential V changes and everything else stays at the example values above:
| Accelerating potential V (kV) | Relativistic speed (c) |
|---|---|
| 12.5 | 0.2172 |
| 18.75 | 0.2637 |
| 25 | 0.3018 |
| 31.25 | 0.3346 |
| 37.5 | 0.3634 |
| 50 | 0.4127 |
Formula
Classical: v = √(2eV/m_e) · Relativistic: γ = 1 + eV/(m_ec²), v = c·√(1 − 1/γ²)
Worked example
Inputs — Accelerating potential V: 25 kV.
Results — Classical speed = 0.312806 c; Lorentz factor γ = 1.048924; Relativistic speed = 0.301842 c; Classical overestimate = 3.6325 %; Kinetic energy = 25 keV.
Key terms
- Electronvolt
- Energy gained by an electron across 1 volt.
Frequently asked questions
When do relativistic effects matter for electrons?
Above about 5–10 kV the error exceeds 1%.
Can electrons reach the speed of light?
No – they approach it as the voltage grows.
What is 1 eV?
The energy an electron gains crossing 1 V: 1.602 × 10⁻¹⁹ J.
How fast are electrons in an electron microscope?
At 200 kV, about 0.70c.