# Electron Speed Calculator

URL: https://speedycalculator.com/calculators/physics/electron-speed/
Last updated: 2026-10-30

> The electron speed calculator finds the speed of an electron accelerated from rest through a given potential difference, both classically and relativistically, and the error of the classical formula.

## Inputs

- Accelerating potential V (kV, V, MV)

## Results

- Classical speed
- Lorentz factor γ
- Relativistic speed
- Classical overestimate
- Kinetic energy

## Formula

```
Classical: v = √(2eV/m_e) · Relativistic: γ = 1 + eV/(m_ec²),  v = c·√(1 − 1/γ²)
```

## Explanation

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)12.50.217218.750.2637250.301831.250.334637.50.3634500.4127

## 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.

## FAQ

**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.


## Sources

- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)
- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)