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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