Electron Thermal Velocity Calculator
Find a plasma's electron thermal velocity, v_te = √(Te·e/mₑ), the characteristic electron speed used throughout plasma physics.
🏃 What is the Electron Thermal Velocity Calculator?
This electron thermal velocity calculator finds the characteristic speed of electrons in a plasma from its temperature. Enter the electron temperature, and it returns the thermal velocity in metres per second, plus the equivalent percentage of the speed of light.
Electron thermal velocity, v_te = √(Te·e/mₑ), is one of the most fundamental building blocks in plasma physics, the natural speed scale that appears directly in collision frequency, mean free path, and wave-particle interaction formulas throughout the field.
Because electrons are so light, even modest temperatures translate into enormous speeds, a plasma at just a few electronvolts already has electrons moving at hundreds of kilometres per second, while a hot fusion plasma pushes electron speeds to a meaningful fraction of the speed of light.
This calculator is useful for plasma physics and fusion engineering students building up the standard toolkit of plasma parameters, and anyone curious just how fast plasma electrons actually move.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Tokamak fusion plasma core
Example 2 - Solar corona
Example 3 - Cool laboratory plasma
❓ Frequently Asked Questions
🔗 Related Calculators
What is electron thermal velocity?
Electron thermal velocity is the characteristic speed of electrons in a plasma due to their random thermal motion, derived from the electron temperature. It sets the natural speed scale for a huge range of plasma physics formulas, including collision frequency, mean free path, and wave-particle interaction physics.
What is the formula for electron thermal velocity?
v_te = √(Te·e/mₑ), where Te is the electron temperature in electronvolts, e is the elementary charge, and mₑ is the electron mass. This is the convention used directly in the standard NRL Plasma Formulary collision frequency and mean free path formulas.
Why are there different conventions for 'thermal velocity'?
Different physical questions call for slightly different statistical averages of the Maxwell-Boltzmann speed distribution: the most probable speed, the mean speed, and the root-mean-square speed, all differ by small numerical factors close to 1. Plasma physics conventionally uses v_te = √(Te·e/mₑ) because it appears cleanly in the standard transport coefficient formulas.
Why do electrons move so much faster than ions at the same temperature?
Thermal velocity scales as 1/√m for a fixed temperature (fixed thermal energy), and since electrons are about 1836 times lighter than protons, electrons move about √1836 ≈ 43 times faster than protons at the same temperature. This large speed disparity is why electron-scale phenomena (plasma frequency, cyclotron resonance heating) are so much faster than ion-scale phenomena.
Where does electron thermal velocity appear in other plasma formulas?
It appears directly in the mean free path formula (mean free path = thermal velocity / collision frequency), and closely related quantities enter the Debye length derivation, plasma wave phase velocities, and Landau damping calculations. It is one of the most frequently reused building blocks in plasma physics.
How fast is a typical fusion plasma electron?
In a 10 keV tokamak core, electrons move at roughly 42 million metres per second, about 14% of the speed of light, fast enough that relativistic corrections become a modest but non-negligible consideration in precise calculations.
Does electron thermal velocity depend on density?
No, thermal velocity depends only on temperature and the electron's mass, not on how many electrons are present. Density instead controls related quantities like collision frequency and plasma frequency.
Is thermal velocity the same as drift velocity?
No, thermal velocity describes the random, disordered motion of electrons due to their temperature, while drift velocity describes any organized, net motion of the electron population (such as current flow or an E×B drift). A plasma can have electrons with a large thermal velocity but zero net drift velocity, or vice versa.
How is this related to the ion thermal velocity?
Ion thermal velocity follows the identical functional form, √(Ti·e/mᵢ), but with the much larger ion mass substituted, giving a correspondingly slower characteristic speed. The related <a href="/science/plasma-physics/ion-thermal-velocity-calculator/">Ion Thermal Velocity Calculator</a> computes it directly.
Can electron thermal velocity approach the speed of light?
At very high temperatures (hundreds of keV or more), this non-relativistic formula begins to overestimate the true speed, since velocity cannot exceed the speed of light; a proper relativistic treatment is needed at those extreme temperatures. For temperatures below a few tens of keV, typical of most laboratory and fusion plasmas, the non-relativistic formula remains an excellent approximation.