Electron Speed Calculator

Background.

Push an electron through 100 volts and it leaves at nearly 6,000 kilometers a second — about 2% of the speed of light — from a voltage you could build on a bench. Electrons are so light that modest energies make them very fast, and that is why the speed has to be worked out with special relativity rather than the schoolbook ½mv².

The deciding number is the electron's rest energy, mₑc², about 511,000 eV. Compared with that, 100 eV is tiny, and the classical formula is only 0.015% off. The error grows quickly: about 0.1% near 700 eV, 1% near 7,000 eV, and 3.6% at 25,000 eV, roughly the voltage that drove a colour television tube. At 511,000 eV the classical formula claims 1.41 times the speed of light, while the true answer is 86.6% of it. Beyond that, extra energy barely moves the speed: a 1 GeV electron travels at 99.999987% of c, and its energy goes into momentum instead.

The calculator therefore uses the exact relations at every energy, from 10⁻¹⁰ eV up to 10¹⁴ eV, so there is no need to decide whether an energy is small enough to treat classically. For an electron that starts at rest and is accelerated through U volts, the kinetic energy in electron volts is simply U, which makes a gun or tube voltage the usual input.

The result describes one free electron carrying exactly the energy you enter. How it got that energy, what it loses along the way, any binding to an atom, scattering, radiation and the direction it travels are all outside the calculation, as the scope note beside the result says.

What is electron speed calculator?

An electron's speed follows from its kinetic energy through the Lorentz factor γ, the ratio of its total energy to its rest energy. Kinetic energy K adds to the rest energy mₑc², so γ = 1 + K ÷ mₑc², and speed follows from γ = 1 ÷ √(1 − v² ÷ c²), rearranged to v = c√(1 − 1 ÷ γ²). The electron volt is the energy one electron gains crossing a potential difference of one volt, exactly 1.602176634 × 10⁻¹⁹ joules. With mₑ = 9.1093837139 × 10⁻³¹ kg from the 2022 CODATA values, the rest energy is 510,998.95 eV.

How to use this calculator.

1. 01Enter the kinetic energy in electron volts. For an electron accelerated from rest through a voltage, that is the voltage itself: 100 V gives 100 eV, 25 kV gives 25,000 eV.
2. 02Read the speed in meters per second. Dividing by 299,792,458 gives the fraction of light speed, often the more telling figure.
3. 03Above about a tenth of light speed, the relativistic result differs noticeably from ½mv², so use it rather than a classical estimate for related calculations.
4. 04For momentum or de Broglie wavelength, use p = γmₑv with the same γ, not mₑv; a separate de Broglie calculator handles the wavelength.
5. 05The rest energy is fixed for the electron. For a proton or ion, substitute that particle's rest energy in the same formulas instead.

The formula.

Special relativity gives a moving particle total energy E = γmₑc², of which mₑc² is present even at rest, so the kinetic energy is K = (γ − 1)mₑc². Solving for γ gives γ = 1 + K ÷ mₑc². The calculator converts K from electron volts to joules with the exact elementary charge, divides by mₑc² computed from the CODATA 2022 electron mass and the exact speed of light, and then applies v = c√(1 − 1 ÷ γ²), the inverse of the Lorentz factor's definition. For small K, expanding the square root gives back ½mₑv² = K, so the classical formula is the low-energy limit of this one rather than a different law. For large K, 1 ÷ γ² shrinks toward zero and v approaches c without reaching it. The result carries 12 significant figures; above roughly 3 × 10¹¹ eV the difference from c falls below that precision, and the display reads exactly 299,792,458 m/s even though the electron is still slower than light.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST, 2022 CODATA Recommended Values of the Fundamental Physical Constants.
2. [2]OpenStax, University Physics Volume 3, section 5.9, Relativistic Energy.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.