Electron Speed Calculator

Electron Speed Calculator — fast, accurate results online. Enter your values and get instant answers.

V

Results

Calculated
Speed (relativistic)
—
Metres per second, exact relativistic result
Fraction of light speed
—
v / c, relativistic
Speed (classical)
—
Non-relativistic estimate, m/s
Lorentz factor (γ)
—
Dimensionless

What the electron speed calculator does and when to use it

This calculator finds how fast an electron is moving after it has been accelerated through a given voltage, the situation inside a cathode-ray tube, an electron microscope, an X-ray tube or an electron gun. You enter the accelerating voltage in volts and it returns the exact relativistic speed, the speed as a fraction of the speed of light, the classical (non-relativistic) estimate, and the Lorentz factor gamma. Showing the two speeds side by side makes it obvious when the simple textbook formula stops being trustworthy.

Use it for physics homework, for sizing up electron-beam equipment, or to see why particle accelerators cannot push an electron past the speed of light no matter how much energy is added. At a few hundred volts the classical formula is accurate to well under one percent. From tens of kilovolts upward the classical answer overshoots noticeably, and above roughly 260 kV it would even exceed the speed of light, which is impossible; that is where the relativistic result is essential.

Formula and variables

An electron with charge e falling through a potential difference V gains kinetic energy KE = eV. The calculator then uses special relativity.

  • e = 1.602176634 × 10−19 C, the elementary charge.
  • me = 9.1093837015 × 10−31 kg, the electron rest mass, and c = 299,792,458 m/s.
  • Lorentz factor: γ = 1 + eV / (mec2). The electron rest energy mec2 is about 510,999 eV.
  • Speed fraction: β = v / c = √(1 − 1/γ2), so v = βc.
  • Classical estimate: v = √(2eV / me), which ignores relativity.

Worked example: 100 kV (100,000 V)

  • Rest energy is 510,998.95 eV, so γ = 1 + 100,000 / 510,998.95 = 1.1957.
  • β = √(1 − 1/1.19572) = √(1 − 0.6994) = 0.5482, so v = 0.5482 × 299,792,458 ≈ 1.6435 × 108 m/s.
  • Classical: v = √(2 × 1.602 × 10−19 × 105 / 9.109 × 10−31) ≈ 1.8755 × 108 m/s, or 0.6256 c.

The classical formula overestimates the true speed by about 14 percent at 100 kV. At 1,000 V the two answers differ by only about 0.15 percent (6.247 percent of c versus 6.256 percent).

Common mistakes and how to interpret the result

  • Entering kilovolts as volts. A 20 kV microscope needs 20000 in the field, not 20.
  • Trusting the classical value at high voltage. If the classical result is a large fraction of c, use the relativistic result; the calculator flags any classical value that reaches c.
  • Confusing energy with speed. Speed rises steeply at first and then flattens toward c; doubling the voltage at high energy barely changes the speed.
  • Forgetting the idealizations. The result assumes a single electron starting at rest in a vacuum with no collisions, and it treats V as the full potential drop.

Frequently Asked Questions

Why can an electron never reach the speed of light?
As speed approaches c, the energy needed to accelerate further grows without limit because the Lorentz factor diverges. Extra voltage adds energy and momentum but the speed only creeps closer to c.
At what voltage does the classical formula fail badly?
There is no sharp threshold. The classical estimate is already off by about 0.15 percent at 1 kV and 14 percent at 100 kV. It gives a speed above c near 256 kV, where it is plainly wrong.
What does the Lorentz factor tell me?
Gamma is the ratio of total energy to rest energy. A gamma of 1.1957 means the electron's total energy is about 19.6 percent greater than its rest energy at 100 kV.
Can I use this for protons or other particles?
No. The mass is fixed at the electron mass. For other particles the rest energy in the gamma formula would change.

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