Electron Velocity Calculator: Formula & Steps

This calculator finds the velocity of an electron when you enter its kinetic energy in electronvolts (eV). You get the speed in meters per second, plus the relativistic factor (gamma) so you can see whether the electron is moving fast enough for relativity to matter. The calculation uses the standard relativistic kinetic energy formula, which is accurate for all energies from near-zero up to extremely high values.

Formula

The relativistic kinetic energy of a particle is:

KE = (γ − 1) m₀c²

where:

• KE = kinetic energy in joules (J)
• γ = Lorentz factor = 1 / √(1 − v²/c²)
• mâ‚€ = rest mass of the electron ≈ 9.1093837015 × 10â�»Â³Â¹ kg
• c = speed of light ≈ 299,792,458 m/s
• v = velocity of the electron in m/s

Rearranged to solve for velocity:

v = c √(1 − 1/γ²) where γ = 1 + KE/(m₀c²)

If you enter energy in electronvolts (eV), convert to joules first: 1 eV = 1.602176634 × 10�¹� J.

Worked Example

Using the default value of 100 eV:

1. Convert energy to joules: 100 eV × 1.602176634 × 10�¹� = 1.602176634 × 10�¹� J
2. Compute rest energy: m₀c² = 9.1093837015 × 10�³¹ × (299792458)² ≈ 8.187105776 × 10�¹� J
3. Find γ = 1 + (1.602176634 × 10�¹�) / (8.187105776 × 10�¹�) ≈ 1.0001957
4. v = 299792458 × √(1 − 1/1.0001957²) ≈ 5,928,000 m/s

The calculator shows 5.928 × 10� m/s, which is about 1.98% of the speed of light. The Lorentz factor is 1.0002, so relativistic effects are minor at this energy.

When to Use This Calculator

Electron velocity calculations appear in many physics and engineering contexts:

• Electron microscopy: estimating electron speed in an electron beam
• Particle accelerators: checking whether electrons are relativistic before designing magnetic focusing
• Cathode ray tubes and vacuum electronics: understanding transit time of electrons
• Photoelectric effect problems: finding the speed of emitted electrons given their kinetic energy
• Plasma physics: estimating electron thermal velocities

For most introductory physics problems, the non-relativistic formula v = √(2KE/m) is used, but that formula becomes inaccurate above about 1% of the speed of light. This calculator uses the relativistic formula, so it is correct at all energies.

How to Read the Results

The first output is the velocity in meters per second. The second output, the Lorentz factor γ, tells you how strong relativistic effects are:

• γ ≈ 1.0: non-relativistic, classical formula is fine
• γ = 1.1: about 10% relativistic correction
• γ = 10: highly relativistic, velocity is 99.5% of light speed

The third output gives the same energy in joules, which is useful if you need to plug into other equations.

Common Mistakes and Misconceptions

Using the wrong mass: The electron mass is 9.109 × 10�³¹ kg, not the proton mass (1.673 × 10�²� kg) or the neutron mass. A common error is using the atomic mass unit (1.66 × 10�²� kg) which is 1800 times heavier.

Forgetting to convert eV to joules: Most physics problems give energy in electronvolts. If you plug eV directly into the formula with mass in kg, you get nonsense. Always multiply by 1.602 × 10�¹�.

Using the classical formula for high energies: If you use v = √(2KE/m) for a 1 MeV electron, you get v = 5.93 × 10� m/s, which is faster than light! The relativistic formula correctly gives 2.82 × 10� m/s (94% of c).

Confusing velocity with drift velocity: In a current-carrying wire, electrons have a random thermal speed of about 10� m/s but a drift velocity of only millimeters per second. This calculator gives the speed corresponding to the kinetic energy, not the drift velocity.

Limits of the Formula

The formula assumes the electron is in free space with no potential energy. In a real material, electrons have a different effective mass and band structure. For example, in graphene, electrons behave as massless particles, and in semiconductors, the effective mass can be a fraction of the free electron mass.

The calculation also assumes the kinetic energy is known. In practice, you might know the accelerating voltage (V) instead. In that case, the kinetic energy in eV equals the voltage if the electron starts from rest: KE = eV. So a 100 V accelerating potential gives 100 eV kinetic energy.

For extremely high energies (above several GeV), you may need to consider synchrotron radiation losses, but for most laboratory situations this is negligible.

Frequently Asked Questions

What is the velocity of an electron with 1 eV kinetic energy?

Using the relativistic formula, 1 eV gives v ≈ 5.93 × 10� m/s, about 0.2% of the speed of light. The non-relativistic formula gives the same to three significant figures because the energy is low.

How fast is an electron accelerated through 100 volts?

An electron starting from rest and accelerated through 100 V gains 100 eV of kinetic energy. Its velocity is about 5.93 × 10� m/s, or about 2% of the speed of light. This is typical for electrons in a cathode ray tube.

When do I need to use relativistic formulas for electrons?

When the kinetic energy exceeds about 1% of the rest energy (511 keV), relativistic effects become noticeable. Above about 10 keV, the classical formula overestimates velocity by more than 1%. For electron microscopes operating at 100-300 keV, relativistic correction is essential.

Why can’t I just use v = √(2KE/m)?

That formula comes from classical mechanics and works well at low speeds (v < 0.1c). But as velocity approaches the speed of light, mass effectively increases and the kinetic energy grows faster than v². Using the classical formula at high energies can give velocities exceeding the speed of light, which is physically impossible.

What is the maximum velocity an electron can have?

According to special relativity, no massive particle can reach exactly the speed of light. As kinetic energy approaches infinity, velocity approaches c but never reaches it. For practical purposes, an electron with 10 MeV energy has v ≈ 0.9988c, and with 1 GeV it has v ≈ 0.999999c.

How do I convert electron velocity to kinetic energy?

Rearrange the formula: KE = (γ − 1)m₀c², where γ = 1/√(1 − v²/c²). If you measure velocity, you can compute γ and then find the energy. This calculator works in the forward direction (energy to velocity), but you can use the same equations in reverse.

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in E | Kinetic energy | eV | 100 | 0
out v | Velocity | m/s | 299792458 * sqrt(1 - (1 / (1 + E * 1.602176634e-19 / (9.1093837015e-31 * 299792458^2))^2)) | 3
out gamma | Lorentz factor | - | 1 + E * 1.602176634e-19 / (9.1093837015e-31 * 299792458^2) | 4
out KE_j | Kinetic energy | J | E * 1.602176634e-19 | 3