Electron Speed Calculator
Enter an accelerating voltage (or pick kinetic energy mode) to find how fast an electron travels. The calculator gives you the relativistic speed, the classical prediction, the Lorentz factor, relativistic momentum, and the de Broglie wavelength. Because electrons reach a significant fraction of the speed of light even at modest voltages, the gap between the classical and relativistic answers is large and immediately visible.
Formula
Worked example
An electron accelerated through 100 kV gains 100 keV of kinetic energy. The rest energy of the electron is 511 keV, so the Lorentz factor is gamma = 1 + 100/511 = 1.196. Beta = sqrt(1 - 1/1.196^2) = 0.548, giving a relativistic speed of 54.8% of c. The classical formula gives sqrt(2 * 1.602e-14 / 9.109e-31) = 1.876e8 m/s = 62.6% of c, an overestimate of ~14%.
Why electrons go relativistic at modest voltages
An electron has a rest mass of 9.109 x 10^-31 kg, corresponding to a rest energy of just 511 keV. Because that rest energy is so small, even a 10 kV accelerating voltage boosts an electron to almost 20% of the speed of light, the regime where Newtonian mechanics begins to diverge from reality. By 511 kV, the Lorentz factor reaches 2 and the electron behaves as though it were twice as massive. By 5 MV it is travelling at 99.6% of c. In contrast, a proton has a rest energy of about 938 MeV, so it needs nearly 2000 times more energy to reach the same Lorentz factor. Electrons are therefore the lightest, fastest-moving particles routinely encountered in practice.
Classical vs relativistic formulas
The classical formula, v = sqrt(2eV / m_e), treats the electron as a Newtonian billiard ball. It works well below a few kilovolts (beta below ~0.06) but predicts speeds exceeding the speed of light above about 261 kV, which is physically impossible. The relativistic formula uses the Lorentz factor: gamma = 1 + eV / (m_e c^2), then beta = sqrt(1 - 1/gamma^2), then v = beta * c. This correctly caps the speed below c no matter how large the voltage. The Lorentz factor also appears in the relativistic momentum p = gamma * m_e * v and governs time dilation and length contraction experienced by the electron.
The Lorentz factor and what it tells you
The Lorentz factor gamma quantifies how far the electron has departed from the Newtonian world. At rest, gamma = 1. At 50% of c, gamma = 1.155. At 99% of c, gamma = 7.09. Each doubling of gamma requires a much larger energy increment: reaching gamma = 2 takes 511 keV, reaching gamma = 10 takes about 4.6 MeV, and reaching gamma = 100 takes about 51 MeV. Because gamma appears in every relativistic formula (momentum, energy, time dilation, length contraction), knowing gamma immediately tells you the relativistic correction factor for every other quantity.
de Broglie wavelength and electron microscopy
Louis de Broglie proposed in 1924 that every moving particle has an associated wavelength lambda = h / p, where h is Planck's constant and p is the relativistic momentum. For electrons at 100 keV, the de Broglie wavelength is about 3.7 pm, orders of magnitude smaller than visible light (~500 nm) and smaller than a typical atom (~100-300 pm). This wave-like nature enables transmission electron microscopes (TEM) to image individual atoms. Higher accelerating voltages shrink the wavelength further and improve the theoretical resolution limit, which is why research-grade TEMs operate at 80 to 300 kV and some reach 1 MV.
Electron speed at common accelerating voltages
| Voltage | Energy (keV) | Classical (% c) | Relativistic (% c) | Lorentz factor (gamma) |
|---|---|---|---|---|
| 1 kV | 1 | 6.3 | 6.3 | 1.002 |
| 10 kV | 10 | 19.8 | 19.5 | 1.020 |
| 35 kV | 35 | 37.0 | 35.4 | 1.069 |
| 100 kV | 100 | 62.6 | 54.8 | 1.196 |
| 200 kV | 200 | 88.5 | 69.5 | 1.391 |
| 511 kV | 511 | 141.6 | 86.6 | 2.000 |
| 1 MV | 1000 | 198.0 | 94.1 | 2.957 |
| 5 MV | 5000 | 443.0 | 99.6 | 10.78 |
The relativistic column is always correct. Classical values exceed the speed of light at high voltages, showing where Newtonian mechanics breaks down.
Frequently asked questions
Why does the classical formula sometimes give a speed greater than the speed of light?
The classical formula v = sqrt(2eV/m_e) has no built-in speed limit. It predicts a speed proportional to the square root of the voltage, so above about 261 kV it returns a value larger than c. Real electrons cannot travel faster than light; the classical formula simply breaks down in this regime. The relativistic formula, derived from special relativity, correctly limits the speed to a value that approaches but never reaches c, regardless of how much energy is added.
At what voltage do relativistic effects become important?
A common rule of thumb is that relativistic corrections exceed 1% when the electron speed is above about 14% of c, which corresponds to a kinetic energy of roughly 5 keV or an accelerating voltage around 5 kV. By 10 kV the classical formula is already about 1.5% high. By 100 kV the error grows to around 14%. Any instrument using electrons above a few kilovolts, such as a TEM, SEM, or X-ray tube, must account for relativistic effects to maintain accuracy.
What is the de Broglie wavelength of an electron and why does it matter?
The de Broglie wavelength is the quantum mechanical wavelength associated with the electron's momentum: lambda = h / p. For a 100 keV electron it is about 3.7 pm. This is far smaller than the wavelength of visible light (~400-700 nm), which is why electron beams can resolve atomic-scale features that are invisible to optical microscopes. Higher energy means shorter wavelength and finer potential resolution, which drives the design of high-voltage electron microscopes.
What is the Lorentz factor and how does it affect the electron?
The Lorentz factor gamma = 1 / sqrt(1 - v^2/c^2) is a measure of how far an object has departed from Newtonian behaviour. At rest, gamma = 1 and the electron behaves normally. At 86.6% of c, gamma = 2: the electron has twice its rest-mass inertia, its internal clock runs at half the rate, and it is length-contracted to half its rest length in the direction of motion. Relativistic momentum is gamma * m_e * v, so at gamma = 2 the electron has twice the momentum a Newtonian calculation would give for the same speed.
How is the electron speed related to its kinetic energy?
In the relativistic picture, kinetic energy KE = (gamma - 1) * m_e * c^2. Rearranging gives gamma = 1 + KE / (m_e c^2), and then v = c * sqrt(1 - 1/gamma^2). An electron with KE equal to its rest energy (511 keV) has gamma = 2 and travels at 86.6% of c. Each doubling of KE beyond that yields diminishing returns in speed because the velocity asymptotically approaches c while the energy can increase without limit.