# Electron Speed Calculator

Created by Mehjabin Abdurrazaque
Reviewed by Wojciech Sas, PhD and Rijk de Wet
Last updated: Jun 05, 2023

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:

1. The events which led to the discovery of electrons;
2. The Newtonian velocity equation for an electron in an electric field and its derivation; and
3. 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. Read on to learn how to calculate the speed of an electron accelerated by an electric field.

💡 Do 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);
• VaAccelerating 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, the electrons accelerate to speeds at which relativistic effects become evident such as the velocity, momentum, and energy having values different from those calculated using classical physics. You can read more about it in the article accompanying the relativistic kinetic energy calculator.

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:

$v_\text{rel} = c\sqrt{1-\frac{1}{(1+\frac{eV_\text{a}}{m_0c^2})^2}} .$

Here:

• $v_\text{rel}$ is the electron's relativistic speed;
• $c$ is the velocity of light;
• $e$ is the electron's charge;
• $V_\text{a}$ is the accelerating potential; and
• $m_0$ is the electron's rest mass (9.109 × 10-31 kg).

The product eVa indicates the electron's kinetic energy in the electric field.

## How to use the electron speed calculator?

Good news — it's simple! To find the Newtonian (classical) and Einsteinian (relativistic) speeds of an electron under a given accelerating potential, just enter the value of accelerating potential in the respective input box.

For example, to find the speed of an electron at a potential difference of 11 kV:

1. Change the unit of Accelerating potential to kV - just click on the unit beside the input box for accelerating potential and select kV.
2. Enter 11 in the input box for accelerating potential.

Omni's electron speed calculator gives you:

• Classical velocity: 62,205 km/s;
• Relativistic velocity: 61,221 km/s; and
• Difference between two velocities: -984 km/s.

## FAQ

### At what speed do relativistic effects occur?

Relativistic effects occur at any speed. However, they are extremely small for speeds below 1/10th of the speed of light (30,000 km/s) and are negligible. When an object approaches speeds of 1/10th of the speed of light, relativistic effects become relevant.

### Do electrons move near the speed of light?

Electrons naturally move with a speed less than 1% of the speed of light, but we can accelerate them to speeds of more than 99% of the speed of light by providing billions of electron volts of energy! Interestingly, we couldn't accelerate electrons to 100% of the speed of light as the faster the electrons move, the heavier they become. At those speeds, they require a tremendous amount of energy to increase their speed, even by 0.000000001%.

### Can a magnetic field accelerate electrons?

Magnetic fields cannot accelerate an electron by changing its speed. They can only deflect an electron.

### How do I calculate the kinetic energy of an electron accelerated through a potential difference of 100 V?

To obtain the kinetic energy of an electron accelerated through a potential difference of 100 V, multiply the elementary charge e (9.109 × 10-31 kg) with the given potential difference Va:

• K E = e Va = 100 eV
Mehjabin Abdurrazaque
Accelerating potential
V
Classical velocity
km/s
Relativistic velocity
km/s
Velocity difference
km/s
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