The electron speed calculator above converts an accelerating voltage into the speed the electron actually reaches. It uses the relativistic relation, because electrons are light enough that the classical formula starts failing at voltages found in ordinary laboratory and industrial equipment — not at exotic accelerator energies, but at the tens of kilovolts inside an electron microscope or an X-ray tube.
Arb Digital publishes free physics calculators that show where a familiar formula stops being valid rather than quietly using it anyway. Here that means computing both results and displaying the gap between them, so the point at which the classical answer becomes unusable is visible instead of assumed.
What This Electron Speed Calculator Does
It takes the potential difference an electron is accelerated through from rest and returns its final speed in metres per second and as a fraction of the speed of light. Alongside that it gives the Lorentz factor, the speed the non-relativistic formula would have predicted, and the percentage by which that classical answer is too high.
It is worth being clear about the boundary with the neighbouring tool. The electron volt calculator converts an accelerating voltage into an energy — that conversion is exact, purely definitional and involves no dynamics at all, because one electron-volt is defined as the energy an elementary charge gains falling through one volt. This page takes that energy and asks how fast the particle is consequently moving, which is a dynamics question, and it is the step where relativity enters. Energy scales linearly with voltage forever; speed does not.
How to Use It
- Enter the accelerating voltage in volts, kilovolts or megavolts. This is the full potential difference traversed, starting from rest.
- Leave the charge state and rest mass at 1 for an electron. Raise the mass to see how a heavier particle behaves at the same voltage.
- Read the speed and the fraction of c. The fraction is usually the more informative of the two.
- Compare the classical figure against the relativistic one. The overestimate percentage is the honest measure of whether the simple formula was usable.
- Check the Lorentz factor. A γ noticeably above 1 means the kinetic energy is a significant fraction of the rest energy, and relativity is not optional.
The Formula: Why the Classical Version Fails
The classical treatment equates the work done to the Newtonian kinetic energy: qV = ½mv², giving v = √(2qV ÷ m). It is simple, it is what most people learn first, and it has no upper bound — raise V far enough and it returns speeds above c, which is a clear signal that it is the wrong equation rather than a slightly inaccurate one.
The relativistic treatment equates the work done to the increase in total energy. The kinetic energy is (γ − 1)mc², where γ = 1 ÷ √(1 − v²/c²) is the Lorentz factor. Setting qV = (γ − 1)mc² gives γ = 1 + qV ÷ (mc²), and rearranging the definition of γ gives the speed:
v = c √(1 − 1/γ²). The HyperPhysics page on relativistic energy sets out the total-energy and kinetic-energy expressions this comes from, and shows how the relativistic kinetic energy reduces to the familiar ½mv² at low speed.
The constants used are the exact defined value 299,792,458 m/s for the speed of light in vacuum published by NIST and the CODATA electron mass of 9.1093837139 × 10−31 kg. Together those give an electron rest energy of about 511 keV, which is the number that decides everything on this page: when the accelerating voltage becomes an appreciable fraction of 511 kV, relativity matters.
Work the default through by hand. At 100 kV, γ = 1 + 100,000 ÷ 510,999 = 1.19570. Then γ² = 1.42969, 1/γ² = 0.69945, and 1 − 0.69945 = 0.30055, whose square root is 0.54822. So v = 0.5482c = 1.6435 × 108 m/s — the value quoted in standard electron-microscopy tables for a 100 kV instrument. The classical formula gives √(2 × 1.6022 × 10−19 × 105 ÷ 9.1094 × 10−31) = 1.8755 × 108 m/s, which is 0.6256c — too high by about 14 per cent.
Where Exactly the Classical Formula Stops Being Usable
There is no sharp threshold, only a growing error, so the honest answer is stated as a tolerance. Running the two formulas against each other gives a clear picture. At a few hundred volts the classical result is high by well under a tenth of a per cent and nobody would care. By around 7 kV the overestimate reaches roughly one per cent. By 100 kV it is about 14 per cent, and by 300 kV it is far worse than that. Above about 511 kV the kinetic energy exceeds the rest energy entirely and the classical picture has no meaning left.
The practical rule that follows: below roughly 5 kV the classical formula is fine for most purposes. Between 5 and 20 kV it is a usable approximation if you can accept a few per cent. Above about 20 kV, use the relativistic result — and note that this covers ordinary equipment, not just accelerators. Electron microscopes, X-ray tubes and older high-voltage display tubes all sit in or above that range.
The reason the boundary is so low for electrons is simply that they are light. The rest energy of a proton is about 938 MeV, nearly two thousand times higher, so a proton accelerated through 100 kV has a γ of 1.0001 and is entirely classical. Raise the mass input on this page to 1,836 to see that directly.
Why Speed Is the Wrong Variable at High Energy
Once an electron is above a few hundred kilovolts, quoting its speed stops being informative. The speed is asymptotically pinned near c and barely changes: at 1 MV an electron is at about 0.94c, at 10 MV about 0.9987c. Two beams with a tenfold difference in energy differ in speed by a few per cent.
What continues to change dramatically is the Lorentz factor, and with it the momentum and total energy. That is why accelerator physics quotes energies rather than speeds, and why γ is the more useful number on this page once it climbs above about 1.5. If you are working with the energy rather than the speed, the relativistic kinetic energy calculator handles that side directly.
What This Page Does Not Model
Three assumptions are baked in. The electron starts from rest, so any initial thermal energy from a hot cathode — a fraction of an electron-volt — is ignored, which is negligible except at the very lowest voltages. The acceleration happens in a vacuum, so there are no collisions; an electron accelerated through a gas loses energy continuously and never reaches the speed computed here. And the field does no work beyond the stated potential difference, so magnetic deflection, space charge and lens fields are all outside the model.
None of that is a limitation of the physics, only of what a single formula can express. The relation between voltage and final speed is exact for a charge crossing a potential difference in vacuum, and the three caveats are about whether your situation matches that description.
Where This Sits Next to the Other Electron Tools
The electron volt calculator is the energy-unit tool: voltage in, energy in electron-volts and joules out, with no dynamics. This page is the dynamics tool: voltage in, speed out, with relativity applied. The two are deliberately separate because the first conversion stays exact at any energy while the second changes character entirely once the particle becomes relativistic.
For a charged particle accelerated under the classical treatment, the charge acceleration calculator covers the Newtonian case. The relativistic kinetic energy calculator and the kinetic energy calculator handle the energy side in both regimes, and the de Broglie wavelength calculator turns the resulting momentum into the wavelength that sets an electron microscope's resolution. For electrons freed by light rather than by a field, see the photoelectric effect calculator. None of this describes electrons in a wire, which crawl — that is the drift velocity calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the classical formula above a few kilovolts — it overestimates by about one per cent at 7 kV and about 14 per cent at 100 kV.
- Confusing the accelerating voltage with the electron's energy in joules — the number in electron-volts equals the number in volts, but joules is a different quantity by a factor of about 1.6 × 10−19.
- Quoting speed instead of energy for relativistic electrons — above a megavolt the speed barely changes while the energy keeps rising.
- Applying this to electrons in a conductor — drift velocity in a wire is a fraction of a millimetre per second, a completely different situation.
- Forgetting that the particle must start from rest — an electron entering the field already moving reaches a higher final speed than this calculation gives.
Related Free Tools From Arb Digital
Pair this with the electron volt calculator for the energy conversion and the relativistic kinetic energy calculator for the energy side of the same problem. The kinetic energy calculator covers the classical case, the charge acceleration calculator the Newtonian charged-particle problem, and the de Broglie wavelength calculator the wave side. See also the photoelectric effect calculator, the electric field calculator and the drift velocity calculator. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
It depends on the accuracy you need, because the error grows smoothly. The classical formula is high by about one per cent at 7 kilovolts and about 14 per cent at 100 kilovolts. Below roughly 5 kilovolts it is fine for most purposes; above about 20 kilovolts you should use the relativistic result.
About 1.64 times ten to the eighth metres per second, which is 0.548 times the speed of light. That is the figure quoted in electron-microscopy tables for a 100 kilovolt instrument. The classical formula would have given 1.88 times ten to the eighth, which is too high by roughly fourteen per cent.
Because it contains no mechanism to prevent it. Newtonian kinetic energy rises as the square of speed with no upper bound, so any energy has a corresponding speed. The relativistic expression instead makes energy diverge as speed approaches c, so no finite energy can reach it. The classical formula is not slightly wrong at high energy; it is the wrong equation.
That page converts a voltage into an energy, which is a definitional conversion requiring no dynamics and staying exact at any energy. This page takes the energy and works out how fast the particle is moving, which is where relativity enters. Energy scales linearly with voltage indefinitely; speed does not.
Because it is the rest energy of the electron, its mass multiplied by the speed of light squared. Relativistic effects become significant when the kinetic energy is an appreciable fraction of the rest energy, so an electron becomes noticeably relativistic at tens of kilovolts. A proton, with a rest energy nearly two thousand times larger, does not.
No, and the difference is enormous. Electrons in a conductor drift at a fraction of a millimetre per second because they collide constantly with the lattice. This page describes a free electron accelerated across a vacuum with nothing in its way, which is why it reaches a substantial fraction of the speed of light.
The modern convention is to treat rest mass as invariant and to say that momentum and energy grow faster than the classical expressions predict. The older language of relativistic mass gives the same numerical answers but causes confusion, so this page works in terms of the Lorentz factor and leaves the mass alone.
The kinetic energy exceeds the rest energy, the Lorentz factor passes 2, and the speed is already above 0.86c. Beyond that point speed becomes a poor way to describe the beam, because it barely changes while energy and momentum keep rising. Accelerator work quotes energy for exactly that reason.
This tool is provided for educational use. It assumes a particle starting from rest, accelerated through the stated potential difference in a vacuum with no collisions, and does not model space charge, thermal emission energy, magnetic fields or radiative losses.
