Electron Speed Calculator

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!

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.

We calculate the classical or non-relativistic velocity of an electron under the influence of an electric field as:

• v_n = √(2eV_a / m),

where:

• v_n — Classical or non-relativistic velocity;
• e — Elementary charge, or the charge of an electron (e = 1.602 × 10^-19 C);
• V_a — 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).

In an electric field of potential V_a, 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 V_a.

Therefore, the work done on the electron is:

• W = e × V_a.

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 V_a is:

• m v_n² / 2 = e V_a,

or

• v_n = √(2 e V_a / m).

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 1/10^th of the speed of light, we must use the relativistic formula to find an electron's velocity in an electric field:

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 eV_a indicates the electron's kinetic energy in the electric field.

Curious to learn more about relativistic effects? Check out our velocity addition calculator and the time dilation 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.