# Electron Speed Calculator

You can accelerate an electron and **calculate its speed as a function of the potential** in which you put it. However, there is a catch; you can't reach a speed higher than the speed of light. Keep reading to learn:

- How to calculate the speed of an electron in an electric field;
- What is the speed of an electron: formula for the classic and relativistic case;
- Some examples of applications of the formulas.

## What is the speed of an electron in an electric field?

Electrons are charged particles: we measure any charge in terms of the electron charge:

The charge of an electron allows us to deflect it with an electric or magnetic field — like the one generated by a solenoid: you can learn more at our solenoid magnetic field calculator.

## What is the formula for the speed of an electron?

To calculate the speed of an electron in an electric field, we use the following formula:

Where:

- $v$ — The
**velocity of the electron in the electric field**; - $e$ — The
**charge of the electron**; - $V_{\text{a}}$ — The
**accelerating potential**; and - $m_{\text{e}}$ — The
**mass of an electron**: $m_{\text{e}} = 9.109 \times 10^{-31}\ \text{kg}$.

We can try the formula for the electron speed: impose a potential of $5\ \text{kV}$:

That's quite fast! Let's try to go faster: impose a potential of $300\ \text{kV}$ . You will find a velocity of $324,853\ \text{km}{s}$: this is well above the **speed of light in the vacuum**. Einstein is not happy now: only light can travel as fast as light, and nothing known by scientists can pass this limit. It's clear that we need to change the electron speed formula.

#### How do we derive the electron speed formula

Starting from the equation for the work done on the electron by the electric field:

We can equate this quantity with the **kinetic energy** of the electron:

From this equation, isolate the velocity to find the formula for the velocity of an electron.

## How do I calculate the speed of an electron in the relativistic case?

If we consider the universe's ultimate speed limit, what is the speed of an electron? The formula undergoes a massive refit:

Where $c$ is the speed of light in the vacuum, $c=299,792\ \text{km}/\text{s}$. As you can see, the term in the square root is always smaller — or asymptotically equal — to $1$, which restricts the speed of the electron in the relativistic case to values smaller than $c$.

You can learn more about this formula at our relativistic kinetic energy calculator.

🙋 The story doesn't end here for an electron: traveling at a speed closer than the speed of light, it would experience many **relativistic effects**: meet them with our length contraction calculator and time dilation calculator.

## The formula for the speed of an electron in action: vintage televisions

The television that once was in every household, the cathode ray tube TV, used **accelerated electrons** to excite phosphors on the screen's glass to create images in a "sequential" fashion.

The typical potential in the **electron gun** was $35\ \text{kV}$: with this voltage, the speed obtained with the formula in the two cases (classic and relativistic) is:

- Classic: $v = 110,958\ \text{km}/\text{s}$; and
- Relativistic: $v = 105,609\ \text{km}/\text{s}$.

The difference (that you can find in our electron speed calculator tool) is $5,350\ \text{km}/\text{s}$. Not as high, but already not negligible.