# Electron Speed Calculator

Created by Davide Borchia
Last updated: Sep 14, 2022

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:

$e = 1.60217663 \times 10^{-19}\ \text{C}$

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:

$v = \sqrt{\frac{2\cdot e\cdot V_{\text{a}}}{m_{\text{e}}}}$

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 $5\ \text{kV}$:

\begin{align*} v& = \sqrt{\frac{2\cdot e\cdot V_{\text{a}}}{m_{\text{e}}}}\\ & =\left(\frac{1}{ 9.109 \times 10^{-31}\ \text{kg}} \right.\\ &\left.\cdot2\cdot 1.60217663\! \times \!10^{-19}\ \text{C}\right.\\ &\left.\cdot5,\!000\ \text{V}\right)\\ &=41,\!938\ \frac{\text{km}}{\text{h}} \end{align*}

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:

$W = e\cdot V_{\text{a}}$

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

$W = K = \frac{1}{2}\cdot m_{\text{e}}\cdot v^2$

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:

$v_{\text{rel}}=c\cdot \sqrt{1 - \frac{1}{1 +\left(\frac{e\cdot V_{\text{a}}}{m_0\cdot c^2}\right)^2}}$

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 $c$.

🙋 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.

Davide Borchia
Accelerating potential
V
Classical velocity
km/s
Relativistic velocity
km/s
Velocity difference
km/s
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