Electron Speed Calculator

Find relativistic electron speed, Lorentz factor, and de Broglie wavelength from kinetic energy or speed fraction.

⚛️ Electron Speed Calculator
Energy Unit
Kinetic Energy50 keV
keV
0.001 keV10,000 keV
Speed (fraction of c)50.000 % of c
% of c
0.001%99.999%
Relativistic Speed
—
Speed (beta = v/c)
—
Lorentz Factor (γ)
—
de Broglie Wavelength
—
Classical Speed (compare)
—
Speed in m/s
—
Lorentz Factor (γ)
—
Kinetic Energy
—
Total Relativistic Energy
—
de Broglie Wavelength
—

⚛️ What is the Electron Speed Calculator?

Electron speed in a physical context is governed by both classical mechanics at low energies and special relativity at higher energies. Unlike a macroscopic object whose speed is straightforwardly related to its kinetic energy through v = sqrt(2K/m), an electron moving at a significant fraction of the speed of light requires the full relativistic treatment: gamma = 1 + K/(m_e c^2) followed by v = c times sqrt(1 - 1/gamma^2). The transition from classical to relativistic behaviour begins to matter around 5 to 10 keV and becomes critical above 100 keV, where classical formulas overestimate the speed by 20 percent or more.

This calculator covers the three quantities most commonly needed in quantum mechanics, particle physics, and electron optics. The first is the relativistic electron speed in m/s and as a fraction of c (the dimensionless parameter beta). The second is the Lorentz factor gamma, which quantifies how strongly relativistic the electron is and enters directly into time dilation, length contraction, relativistic momentum, and total energy calculations. The third is the de Broglie wavelength, which determines the resolving power of electron microscopes and the diffraction conditions for electron crystallography.

Real-world applications span a wide range of energies. Thermal electrons in copper drift at roughly 10^5 m/s (less than 0.1% of c) and are completely non-relativistic. Electrons in a cathode ray tube are accelerated through 15 to 30 kV, giving them kinetic energies of 15 to 30 keV and speeds around 25% of c. Medical X-ray tubes use 50 to 150 kV. Transmission electron microscopes accelerate electrons to 60 to 300 keV, reaching 40 to 70% of c with sub-picometer de Broglie wavelengths. Synchrotron light sources and linear accelerators can push electrons to hundreds of MeV or even GeV, where gamma exceeds 1000 and the electrons travel at 99.99999% of c.

A key subtlety is that accelerating voltage in volts and kinetic energy in electron-volts are numerically identical: an electron accelerated through 50,000 V gains exactly 50,000 eV of kinetic energy, which equals 50 keV. This calculator accepts kinetic energy in eV, keV, MeV, or GeV. The From Speed mode lets you go in the opposite direction: enter any speed as a percentage of c and get the kinetic energy, total energy, and wavelength that correspond to that speed.

📐 Formulas

γ  =  1 + K ÷ (mec²)   |   v  =  c √(1 − 1÷γ²)
γ = Lorentz factor (dimensionless); equals 1 at rest, increases without limit
K = kinetic energy of the electron (eV, keV, MeV, or GeV)
mec² = electron rest energy = 511 keV = 0.511 MeV
v = electron speed (m/s); β = v/c (dimensionless fraction)
Classical (only valid below ~5 keV): v = √(2K/me) = √(2eV/me)
de Broglie wavelength: λ = hc ÷ √(K² + 2Kmec²)
Total energy: E = γmec² = K + 511 keV
Example: K = 50 keV: γ = 1 + 50/511 = 1.0979, v = c√(1-1/1.0979²) = 0.4131c = 1.238×10&sup8; m/s

📖 How to Use This Calculator

Steps

1
Choose a calculation mode - select From Energy to find speed from kinetic energy, or From Speed to find energy and gamma from a known speed as a fraction of c.
2
Enter the electron energy or speed - in From Energy mode, enter the kinetic energy and select the unit (eV, keV, MeV, or GeV). In From Speed mode, enter the speed as a percentage of c (0.001 to 99.99).
3
Read relativistic results - the calculator shows relativistic speed in m/s, beta (v/c as a percentage), Lorentz factor gamma, and de Broglie wavelength. From Energy mode also shows the classical speed for comparison so you can see how much the classical approximation overestimates the speed.

💡 Example Calculations

Example 1 - Electron in a cathode ray tube (50 keV)

Kinetic energy: 50 keV (50,000 eV accelerating voltage)

1
Lorentz factor: γ = 1 + 50/511 = 1.0979.
2
Relativistic speed: v = c × √(1 - 1/1.0979²) = 0.4131c = 1.238 × 10&sup8; m/s.
3
Classical speed: v = √(2 × 8.011×10-15 / 9.109×10-31) = 1.327 × 10&sup8; m/s. Classical overestimates by 7.2%.
4
de Broglie wavelength: p·c = √(50000² + 2×50000×511000) = 231,517 eV. λ = 1239.84/231.517 pm = 5.36 pm.
Result = 1.238 × 10&sup8; m/s (41.31% of c), γ = 1.0979, λ = 5.36 pm
Try this example →

Example 2 - Photoelectron at 100 eV

Kinetic energy: 100 eV (low-energy photoelectron)

1
Lorentz factor: γ = 1 + 0.1/511 = 1.000196 (essentially classical).
2
Speed: v = c × √(1 - 1/1.000196²) = 0.01979c = 5.932 × 10&sup6; m/s.
3
de Broglie wavelength: p·c = √(100² + 2×100×511000) = 10,110 eV. λ = 1239840/10110 pm = 122.6 pm = 0.1226 nm.
Result = 5.932 × 10&sup6; m/s (1.979% of c), γ = 1.000196, λ = 0.1226 nm
Try this example →

Example 3 - High-energy electron at 1 MeV

Kinetic energy: 1 MeV (strongly relativistic electron)

1
Lorentz factor: γ = 1 + 1000/511 = 2.957.
2
Speed: v = c × √(1 - 1/2.957²) = c × √(0.8856) = 0.9411c = 2.820 × 10&sup8; m/s.
3
de Broglie wavelength: p·c = √(1000² + 2×1000×511) keV = 1422 keV. λ = 1239.84/1422000 nm = 0.872 pm.
Result = 2.820 × 10&sup8; m/s (94.11% of c), γ = 2.957, λ = 0.872 pm
Try this example →

Example 4 - Find energy for an electron at 90% of light speed

Speed: 90% of c (β = 0.9)

1
Lorentz factor: γ = 1/√(1 - 0.81) = 1/√0.19 = 2.294.
2
Kinetic energy: K = (γ - 1) × 511 keV = 1.294 × 511 = 661.2 keV = 0.661 MeV.
3
Total energy: E = γ × 511 = 2.294 × 511 = 1172 keV = 1.172 MeV.
Result = 2.698 × 10&sup8; m/s, γ = 2.294, K = 661.2 keV, E_total = 1.172 MeV
Try this example →

❓ Frequently Asked Questions

What is the formula for electron speed from kinetic energy?+
The relativistic formula is: first compute the Lorentz factor gamma = 1 + K/(m_e*c^2) = 1 + K(eV)/511000, then v = c * sqrt(1 - 1/gamma^2). Here m_e*c^2 = 511 keV is the electron rest energy. For K = 50 keV: gamma = 1.0979, v = c * sqrt(1 - 0.8294) = 0.4131c = 1.24 x 10^8 m/s. The classical formula v = sqrt(2K/m_e) is accurate only below about 5 keV.
What is the electron rest energy and how does it affect calculations?+
The electron rest energy m_e*c^2 = 511 keV = 0.511 MeV. This is the reference scale for relativistic effects. When kinetic energy K equals the rest energy (511 keV), gamma = 2 and v = 0.866c (86.6% of light speed). When K is much smaller than 511 keV, relativistic effects are negligible and the classical formula applies. When K is comparable to or larger than 511 keV, the full relativistic treatment is required.
What is the Lorentz factor and what does gamma mean?+
The Lorentz factor gamma = 1/sqrt(1 - v^2/c^2) = 1 + K/(m_e*c^2) quantifies relativistic effects. At rest, gamma = 1. At v = 0.5c, gamma = 1.155. At v = 0.866c, gamma = 2. At v = 0.9950c, gamma = 10. Physically, gamma is the ratio of total energy to rest energy (E = gamma*m_e*c^2), and it appears in time dilation (t' = gamma*t), length contraction (L' = L/gamma), and relativistic momentum (p = gamma*m_e*v).
What is the de Broglie wavelength of an electron at different energies?+
The relativistic de Broglie wavelength lambda = hc/sqrt(K^2 + 2*K*m_e*c^2), where K is kinetic energy. Representative values: 1 eV electron lambda = 1.23 nm (infrared), 100 eV lambda = 0.123 nm (X-ray range), 1 keV lambda = 38.8 pm, 50 keV lambda = 5.36 pm, 200 keV lambda = 2.51 pm (electron microscope), 1 MeV lambda = 0.872 pm, 10 MeV lambda = 0.124 pm. The small wavelength of keV to MeV electrons enables atomic-resolution imaging.
When must I use the relativistic formula instead of the classical one?+
Use the relativistic formula whenever the kinetic energy K exceeds about 5 keV (1% of the rest energy). At K = 5 keV, the classical formula overestimates speed by about 1%. At K = 50 keV, the overestimate is 7%. At K = 511 keV (equal to rest energy), the classical formula gives 1.41c, which exceeds the speed of light and is physically impossible. For any calculation involving X-ray tubes (40+ kV), electron microscopes (60-300 kV), or particle beams (any MeV energy), always use the relativistic formula.
What is beta in the context of electron speed?+
Beta (beta) = v/c is the dimensionless speed of the electron as a fraction of the speed of light. At rest, beta = 0. For a 50 keV electron, beta = 0.4131 (41.31% of c). At the electron rest energy (511 keV), beta = 0.8660 (86.60%). Beta approaches 1 asymptotically as kinetic energy increases, but never reaches 1. The combination beta*gamma appears in relativistic momentum: p = m_e*beta*gamma*c.
How does an electron microscope use electron speed and wavelength?+
Electron microscopes accelerate electrons to 60 to 300 keV. At 200 keV, beta = 0.695 (69.5% of c), gamma = 1.391, and the de Broglie wavelength is 2.51 pm. This is about 40 times shorter than the wavelength of X-rays used in X-ray crystallography (typically 100 pm). The shorter wavelength means the theoretical resolution limit of the electron microscope is at the sub-angstrom scale, enabling direct imaging of individual atoms and even chemical bonds in favourable cases.
What voltage accelerates an electron to 50% of the speed of light?+
At beta = 0.5 (50% of c), gamma = 1/sqrt(1 - 0.25) = 1.1547. The kinetic energy K = (1.1547 - 1) x 511 keV = 79.1 keV. Since kinetic energy in eV equals the accelerating voltage in volts, a voltage of 79.1 kV (79,100 volts) is needed. This is higher than a typical cathode ray tube voltage (15-30 kV) but within the range of older X-ray imaging systems and some electron diffraction instruments.
What is total relativistic energy versus kinetic energy?+
Total relativistic energy E = gamma*m_e*c^2 = kinetic energy K + rest energy m_e*c^2 = K + 511 keV. For a stationary electron, E = 511 keV (pure rest energy). For a 50 keV electron, E = 561 keV. For a 1 MeV electron, E = 1511 keV. The formula E^2 = (pc)^2 + (m_e*c^2)^2 is the relativistic energy-momentum relation that replaces E = p^2/(2m) from classical mechanics.
How fast are electrons in atoms and metals?+
In a hydrogen atom ground state, the electron orbits at approximately 2.19 x 10^6 m/s (0.73% of c), corresponding to about 13.6 eV kinetic energy. This is non-relativistic. Conduction electrons in metals drift slowly under an applied field (millimetres per second) but have a high Fermi velocity of about 1 to 2 x 10^6 m/s due to quantum pressure, corresponding to 3 to 10 eV. Even the fastest conduction electrons in metals are essentially non-relativistic.
Can the classical and relativistic formulas ever give the same result?+
Yes, in the limit of very low kinetic energy (K much less than 511 keV), both formulas converge to the same result. Mathematically, expanding the relativistic formula to first order in K/(m_e*c^2) gives v = sqrt(2K/m_e) plus a small correction term, which equals the classical formula. Below 1 keV, the two formulas agree to within 0.1%. The classical formula is simply the low-energy limit of the more general relativistic formula.
What is relativistic momentum and how does it differ from classical momentum?+
Classical momentum is p = m_e*v. Relativistic momentum is p = gamma*m_e*v. The two agree at low speeds (gamma near 1) and diverge increasingly at high speeds. For a 50 keV electron, p_rel = 1.0979 x p_classical (about 10% larger). For a 511 keV electron (gamma = 2), p_rel = twice the classical value. Relativistic momentum always appears in the de Broglie wavelength: lambda = h/p = h/(gamma*m_e*v). Using classical momentum overestimates the wavelength by a factor of gamma.
Tags: Electron Speed Calculator Relativistic Electron Velocity Lorentz Factor Calculator De Broglie Wavelength Electron Kinetic Energy Calculator Formula Free Online

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How fast does an electron move in a typical atom?

In a hydrogen atom, the electron in the ground state orbits with a speed of approximately 2.19 x 10^6 m/s, which is about 0.73% of the speed of light. This corresponds to kinetic energy of roughly 13.6 eV. At this low speed, relativistic corrections are negligible and the classical formula gives accurate results. Electrons in heavier atoms with high nuclear charge move significantly faster.

What is the formula for electron speed from accelerating voltage?

For non-relativistic electrons (below about 10 keV), v = sqrt(2eV/m_e), where e = 1.602 x 10^-19 C, m_e = 9.109 x 10^-31 kg, and V is the voltage in volts. For higher voltages, use the relativistic formula: gamma = 1 + eV/(m_e*c^2), then v = c * sqrt(1 - 1/gamma^2). The accelerating voltage V in volts equals the kinetic energy in electron-volts (eV), so 50 kV gives 50 keV of kinetic energy.

What is the electron rest energy and why does it matter?

The electron rest energy (m_e c^2) is 511 keV or 0.511 MeV. It is the reference point for relativistic effects. When an electron's kinetic energy equals its rest energy (511 keV), the Lorentz factor gamma equals 2 and the electron travels at 86.6% of the speed of light. Kinetic energies below about 50 keV (less than 10% of rest energy) allow the classical approximation; above 100 keV, the relativistic formula is essential.

What is the Lorentz factor gamma and how do I interpret it?

The Lorentz factor gamma = 1/sqrt(1 - v^2/c^2) quantifies how strongly relativistic an electron is. At low speeds, gamma is very close to 1 (classical regime). At v = 0.5c, gamma = 1.155. At v = 0.866c, gamma = 2. At v = 0.995c, gamma = 10. A gamma of 2 means the electron's total energy (kinetic plus rest) is twice its rest energy. Gamma appears in time dilation, length contraction, and momentum calculations.

What is the de Broglie wavelength of an electron?

The relativistic de Broglie wavelength is lambda = h/p = hc/sqrt(K^2 + 2Km_ec^2), where K is kinetic energy and h = 6.626 x 10^-34 J*s. For a 100 eV electron, lambda = 0.123 nm (close to X-ray wavelengths). For a 50 keV electron, lambda = 5.36 pm. For 200 keV (electron microscope), lambda = 2.51 pm. The small wavelength of high-energy electrons enables atomic-resolution imaging in electron microscopy.

When can I use the classical formula and when must I use the relativistic one?

The classical formula v = sqrt(2K/m_e) is accurate within 1% for kinetic energies below about 5 keV (beta less than 0.14). It is within 10% up to about 50 keV. For electron energies used in medical X-ray tubes (50 to 150 keV), the classical formula overestimates speed by 7 to 20 percent. For any accelerating voltage above 50 kV, always use the relativistic formula. For electron microscopes (60 to 300 keV) and cathode ray tubes, relativistic corrections are significant.

What is beta in particle physics?

Beta (symbol beta) is the ratio of a particle's speed to the speed of light: beta = v/c. For an electron at rest, beta = 0. For a highly relativistic electron approaching the speed of light, beta approaches 1. Beta is dimensionless and appears in many relativistic formulas. A beta of 0.5 means the electron travels at half the speed of light. The combination beta-gamma (beta times gamma) appears in momentum calculations: p = m_e * gamma * beta * c.

Can an electron reach the speed of light?

No. According to special relativity, any particle with mass (including electrons) would require infinite energy to reach the speed of light. As an electron's kinetic energy increases, its speed approaches c asymptotically but never reaches it. At 511 keV (equal to the rest energy), the electron reaches only 86.6% of c. At 10 times the rest energy (5.11 MeV), it reaches 99.5% of c. At 100 times the rest energy (51.1 MeV), it reaches 99.995% of c.

What voltage is needed to accelerate an electron to 50% of the speed of light?

At beta = 0.5 (50% of c), the Lorentz factor gamma = 1/sqrt(1-0.25) = 1.1547. The kinetic energy K = (gamma - 1) * m_e * c^2 = 0.1547 * 511 keV = 79.1 keV. So an accelerating voltage of 79.1 kV is needed. This is in the range of typical cathode ray tubes and older X-ray imaging systems. To reach 90% of c requires 661 kV, and 99% of c requires 3.11 MV.

How does an electron microscope use electron speed?

An electron microscope accelerates electrons to 60 to 300 keV using a high-voltage electron gun. At 200 keV, electrons travel at 69.5% of the speed of light and have a de Broglie wavelength of only 2.51 pm, far smaller than X-ray wavelengths (0.01 to 10 nm). This small wavelength determines the theoretical resolution limit of the instrument. The relativistic mass of the electrons also affects the magnetic lens design needed to focus the beam.

What are thermal electrons and how fast are they?

Thermal electrons in a conductor at room temperature (300 K) have an average kinetic energy of about (3/2) * k_B * T = 0.039 eV, giving a thermal speed of about 1.17 x 10^5 m/s (0.04% of c). This is completely non-relativistic. The Fermi velocity of conduction electrons in metals is much higher, around 1 to 2 x 10^6 m/s (0.3 to 0.7% of c), corresponding to the Fermi energy of 3 to 10 eV. These are still non-relativistic.

How is relativistic momentum different from classical momentum?

Classical momentum is p = m_e * v. Relativistic momentum is p = m_e * gamma * v = m_e * gamma * beta * c. For a 50 keV electron with gamma = 1.098, the relativistic momentum is 9.8% larger than the classical value. For a 511 keV electron (gamma = 2), the relativistic momentum is twice the classical momentum. The de Broglie wavelength always uses relativistic momentum: lambda = h/p = h/(m_e * gamma * v), so using classical momentum overestimates the wavelength for energetic electrons.

What is the total relativistic energy of an electron?

Total relativistic energy E = gamma * m_e * c^2 = kinetic energy + rest energy = K + 511 keV. For a stationary electron, E = 511 keV (all rest energy, no kinetic energy). For a 50 keV electron, E = 50 + 511 = 561 keV and gamma = 561/511 = 1.098. For a 1 MeV electron, E = 1000 + 511 = 1511 keV and gamma = 2.957. The relationship E^2 = (pc)^2 + (m_e*c^2)^2 is the relativistic energy-momentum relation.

📌 Quick Tips

💡Relativistic effects become noticeable above about 10 keV, where the classical formula overestimates electron speed by more than 1%. At 511 keV (one electron rest energy), the electron travels at 86.6% of light speed.
💡The de Broglie wavelength of a 100 eV electron is about 0.123 nm, similar to atomic spacings in crystals. This is why electron diffraction and electron microscopy can resolve atomic structures.
💡At 50 keV (typical for a cathode ray tube), the classical formula overestimates speed by about 7%. Above 100 keV, always use the relativistic formula for accurate results.
💡The Lorentz factor gamma equals 2 when the electron's kinetic energy equals its rest energy of 511 keV. At gamma = 10, the electron carries 9 times its rest energy as kinetic energy.
💡Electron microscopes use 60 to 300 keV electrons. At 200 keV, beta = 0.695 (69.5% of c) and the de Broglie wavelength is 2.51 pm, enabling sub-angstrom resolution imaging.