Electron Speed Calculator: Classical and Relativistic Velocity
Are you looking for an electron speed calculator to estimate both the classical and relativistic speed of an electron? Be our guest! In this calculator, you can not only estimate the relativistic and non-relativistic velocities of an electron under a given accelerating potential, but you will also learn:
- The events which led to the discovery of electrons;
- The Newtonian velocity equation for an electron in an electric field and its derivation; and
- How to find the relativistic speed of an electron from its energy.
Last but not least, you'll learn answers to some interesting questions, like "Do electrons move near the speed of light?" Let's go!
What is an electron?
Atoms consist of three basic components: electrons, protons, and neutrons. Based on the results of 𝛼-scattering experiments, the physicist Rutherford suggested that negatively charged electrons move in circular orbits around the positively charged region called the nucleus.
An electron has a mass of 9.109 × 10⁻³¹ kg and a charge of 1.602 × 10⁻¹⁹ C.
Acceleration of a particle in an electric field is possible if it carries a charge. Therefore, electromagnetic fields accelerate electrons. If you know the value of the electric field's potential, you can calculate the speed of an electron moving under its influence using the equation for kinetic energy.
💡 Did you know that the motion of electrons in magnetic and electric fields helped determine the sign of their charge? In 1879, the English physicist and chemist William Crookes discovered that magnetic fields bend cathode rays, and the direction of deflection indicated that they were negatively charged particles. Then in 1897, J. J. Thomson observed cathode rays bending towards the positive plate and deviating away from the negative plate when allowed to pass between them. Thomson's discovery established that electricity involved the flow of negatively charged particles. Thomson called these particles electrons.
How do I calculate the classical/non-relativistic velocity of electrons in an electric field?
We calculate the classical or non-relativistic velocity of an electron under the influence of an electric field as:
vₙ = √(2eVₐ / m),
where:
- vₙ — Classical or non-relativistic velocity;
- e — Elementary charge, or the charge of an electron (e = 1.602 × 10⁻¹⁹ C);
- Vₐ — Accelerating potential, or the potential difference that is applied to accelerate the electron; and
- m — The mass of an electron (m = 9.109 × 10⁻³¹ kg).
How do I derive the Newtonian velocity equation for an accelerated electron?
In an electric field of potential Vₐ, an electron experiences a force of e × E, where E is the intensity of the electric field. Therefore, the work done on the electron is:
W = force × distance = (e × E) × r.
Here, the product E × r is the electric potential Vₐ. Therefore, the work done on the electron is:
W = e × Vₐ.
By the work-energy theorem, the net work done by the forces on an object equals the change in its kinetic energy. Therefore, the kinetic energy of an electron accelerated through a potential difference Vₐ is:
m vₙ² / 2 = e Vₐ,
or
vₙ = √(2 e Vₐ / m).
How do I find the relativistic speed of an electron from its energy?
Under the influence of electric fields, electrons accelerate to speeds at which relativistic effects become evident — the velocity, momentum, and energy take on values different from those calculated using classical physics. At speeds greater than 1/10th of the speed of light, we must use the relativistic formula to find an electron's velocity in an electric field.
The total energy of the electron equals its rest energy plus the kinetic energy it gained from the accelerating potential:
γ = 1 + eVₐ / (mc²),
where γ is the Lorentz factor and c is the speed of light (c = 299,792,458 m/s). The relativistic speed then follows from:
v = c × √(1 − 1/γ²).
Notice that, unlike the classical formula, this relativistic equation can never give a speed equal to or greater than the speed of light, no matter how large the accelerating potential becomes.
How to use the electron speed calculator?
- Enter the accelerating potential (Vₐ) and choose its unit (V, kV, MV, or GV).
- Select your preferred unit system — Metric (m/s, km/s) or American (mi/s, ft/s, mph).
- Read the results: the calculator instantly shows both the classical (Newtonian) and the relativistic speed, the Lorentz factor γ, and the kinetic energy in eV, keV, MeV, and joules.
- Compare the two velocities. When the speed is below 10% of c, the classical and relativistic results almost match. Above that, you should rely on the relativistic value.
Do electrons move near the speed of light?
Yes — they easily can. An electron accelerated through only about 5 kV already reaches roughly 14% of the speed of light, and in devices such as electron microscopes (100–300 kV) or particle accelerators (MV–GV), electrons travel at well over 99% of c. That is exactly why a relativistic treatment is essential for these energies.
FAQs
Why does the classical formula sometimes give a speed above the speed of light?
The Newtonian equation vₙ = √(2eVₐ/m) has no upper limit, so at high potentials it predicts speeds greater than c. This is unphysical and simply marks the point where classical mechanics breaks down and the relativistic formula must be used.
What is the kinetic energy of an electron accelerated through 1 volt?
Exactly 1 electron-volt (1 eV = 1.602 × 10⁻¹⁹ J). In general, an electron accelerated through Vₐ volts gains a kinetic energy of Vₐ electron-volts.
When do I need the relativistic formula?
Whenever the electron's speed exceeds about 0.1c (roughly above a few keV of accelerating potential). The calculator flags this for you automatically.