Drift Velocity Calculator
Calculate electron drift velocity in conductors using v_d = I/(nAq). Includes material presets, AWG wire gauges, current density, and traversal time analysis.
Electron Speed Calculator
Calculate electron speed from kinetic energy or accelerating voltage. Includes relativistic corrections, de Broglie wavelength, classical vs relativistic comparison.
V
Electron Speed⧉
90,487,099 m/s
Relativistic calculation used
Fraction of c (β)⧉
0.301825
v/c — fraction of the speed of light
Lorentz Factor (γ)⧉
1.048918
γ = 1/√(1−β²) — relativistic mass increase factor
de Broglie Wavelength⧉
7.6639 pm
λ = h/p — wave nature of the electron
Kinetic Energy⧉
0.0000 J
25,000.00 eV
Classical Error⧉
3.63%
How much the classical calculation deviates from relativistic
Classical vs Relativistic Comparison
⚠️ Classical error is 3.6% — relativistic correction is important at this energy.
| Quantity | Classical | Relativistic |
|---|---|---|
| Speed (m/s) | 93,773,669 | 90,487,099 |
| β (v/c) | 0.312787 | 0.301825 |
| Speed exceeds c? | No ✓ | No ✓ (always < c) |
Energy vs Speed Table
| Energy (eV) | Classical v (m/s) | Relativistic v (m/s) | β | γ | Error % |
|---|---|---|---|---|---|
| 1 eV | 593,077 | 593,076 | 0.001978 | 1.0000 | 0.00% |
| 10 eV | 1,875,473 | 1,875,446 | 0.006256 | 1.0000 | 0.00% |
| 100 eV | 5,930,768 | 5,929,897 | 0.019780 | 1.0002 | 0.01% |
| 1.0 keV | 18,754,734 | 18,727,262 | 0.062466 | 1.0020 | 0.15% |
| 10.0 keV | 59,307,676 | 58,453,323 | 0.194974 | 1.0196 | 1.46% |
| 100.0 keV | 187,547,337 | 164,349,200 | 0.548196 | 1.1957 | 14.12% |
| 511.0 keV | 423,956,553 | 259,629,379 | 0.866009 | 1.9999 | 63.29% |
| 1.00 MeV | 593,076,755 | 282,132,696 | 0.941070 | 2.9567 | 110.21% |
| 10.00 MeV | 1,875,473,373 | 299,445,425 | 0.998817 | 20.5672 | 526.32% |
| 100.00 MeV | 5,930,767,550 | 299,796,125 | 0.999987 | 196.6719 | 1,878.27% |
Electron rest energy: 511.06 keV (511 keV). Relativistic corrections become significant above ~5 keV (1% of rest energy).
Planning notes, formulas, and examples
About the Electron Speed Calculator
When an electron is accelerated through a potential difference V, it gains kinetic energy equal to eV (electron charge times voltage). The resulting speed depends on whether classical or relativistic mechanics applies. For accelerating voltages below about 5 kV, classical mechanics (v = √(2eV/m)) is sufficiently accurate. Above that, the electron's speed becomes a significant fraction of the speed of light and relativistic corrections are essential.
At the electron's rest mass energy of 511 keV, the classical formula predicts a speed exceeding c (the speed of light), which is physically impossible. The relativistic formula correctly ensures the speed asymptotically approaches but never reaches c, no matter how much energy is added.
This calculator handles both classical and relativistic calculations, automatically detecting when relativistic corrections are needed. It also computes the de Broglie wavelength, which is crucial for electron microscopy — a 100 keV electron has a wavelength of about 3.7 picometers, far shorter than visible light, enabling atomic-resolution imaging.
When This Page Helps
Relativistic calculations involve the Lorentz factor γ and careful handling of rest mass energy versus kinetic energy. Mistakes are easy when working with tiny numbers like electron mass (9.109 × 10⁻³¹ kg) and elementary charge (1.602 × 10⁻¹⁹ C). This calculator also provides the de Broglie wavelength and a comprehensive energy-speed comparison table showing exactly where classical mechanics breaks down.
How to Use the Inputs
- Choose input mode: accelerating voltage (V) or kinetic energy (eV).
- Enter the value — for voltage mode, the energy in eV equals the voltage numerically.
- Select relativistic correction mode: auto (recommended), always, or classical only.
- Read the electron speed, fraction of c, Lorentz factor, and de Broglie wavelength.
- Check the classical vs relativistic comparison table to see the error magnitude.
- Review the energy-speed table spanning from 1 eV to 100 MeV.
Formula used
Classical: v = √(2eV/mₑ) = √(2KE/mₑ)
Relativistic: γ = 1 + KE/(mₑc²)
β = √(1 − 1/γ²)
v = βc
de Broglie: λ = h/p = h/(γmₑv)
Where:
mₑ = 9.109 × 10⁻³¹ kg (electron mass)
e = 1.602 × 10⁻¹⁹ C
c = 2.998 × 10⁸ m/s
h = 6.626 × 10⁻³⁴ J·s
mₑc² = 511 keV (rest energy)Example Calculation
Result: 9.38 × 10⁷ m/s
A 25 kV accelerating voltage gives KE = 25 keV = 4.0 × 10⁻¹⁵ J. Classically: v = 9.39 × 10⁷ m/s (0.313c). Relativistically: γ = 1.049, v = 9.38 × 10⁷ m/s (0.313c). The classical error is about 2.4%.
Tips & Best Practices
- The electron rest mass energy of 511 keV is a key reference: that is where KE equals rest energy and γ = 2.
- For electron microscopy, the de Broglie wavelength matters more than the speed itself.
- Classical mechanics overestimates electron speed — at 100 keV, it predicts 0.625c vs the correct 0.548c.
- Protons are 1836× heavier, so they need 1836× more energy to reach the same speed.
- In accelerator physics, particles are often described by their γ factor rather than their speed.
- The electron beam in a CRT television has about 25 keV energy, reaching 31% of the speed of light.
Classical vs Relativistic Mechanics
At everyday speeds, Newton's mechanics works perfectly. Kinetic energy is ½mv² and speed is simply v = √(2KE/m). But this breaks down as speeds approach c. The classical formula predicts v > c for electrons above 256 keV, which violates special relativity.
Einstein's special relativity replaces the classical kinetic energy with KE = (γ - 1)mc², where γ = 1/√(1 - v²/c²). This correctly limits v < c for any finite energy. The transition from classical to relativistic behavior is gradual — there is no sharp cutoff, just increasing error in the classical approximation.
Electron Microscopy Applications
Scanning and transmission electron microscopes (SEM, TEM) accelerate electrons to high voltages to achieve atomic resolution. The resolution limit is fundamentally set by the de Broglie wavelength. A 200 keV TEM electron has λ ≈ 2.5 pm, theoretically enabling sub-angstrom resolution (though lens aberrations typically limit practical resolution to ~0.5-1 Å).
Higher voltages give shorter wavelengths and better resolution, but also cause more radiation damage to samples. This is why cryo-EM (used for biological samples) typically operates at 200-300 keV as a compromise between resolution and damage.
Particle Accelerators
Modern particle accelerators push electrons to extreme relativistic speeds. The Large Electron-Positron Collider (LEP) at CERN accelerated electrons to γ ≈ 200,000 (speeds of 0.999999999988c). At these energies, each electron had the kinetic energy equivalent to a flying mosquito concentrated in a particle billions of times smaller than a grain of sand.
Sources & Methodology
Last updated:
Frequently Asked Questions
Relativistic corrections become significant when the electron's kinetic energy exceeds about 1% of its rest mass energy (511 keV), which is around 5 keV. Above 50 keV, classical mechanics gives errors greater than 5%.
The de Broglie wavelength determines the resolution limit of electron microscopes. Shorter wavelengths (higher-energy electrons) enable higher resolution. A 200 keV electron has λ ≈ 2.5 pm, enabling atomic-scale imaging.
As an electron approaches c, its relativistic mass increases without bound (γ → ∞), requiring infinite energy to reach c. The speed asymptotically approaches but never equals c.
γ = 1/√(1-v²/c²) quantifies relativistic effects. At γ = 1, the particle is at rest. At γ = 2 (KE = rest energy), time dilation and length contraction are significant. γ also represents the total energy divided by rest energy.
One electron volt is the kinetic energy gained by an electron accelerated through a 1-volt potential difference: 1 eV = 1.602 × 10⁻¹⁹ joules. It is a convenient energy unit in atomic and particle physics.
At room temperature (300 K), the average kinetic energy is about 0.026 eV, giving a classical speed of about 1.17 × 10⁵ m/s — fast, but only 0.039% of the speed of light.
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