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=1.60217663×1019 Ce = 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=2eVamev = \sqrt{\frac{2\cdot e\cdot V_{\text{a}}}{m_{\text{e}}}}


  • vv — The velocity of the electron in the electric field;
  • ee — The charge of the electron;
  • VaV_{\text{a}} — The accelerating potential; and
  • mem_{\text{e}} — The mass of an electron: me=9.109×1031 kgm_{\text{e}} = 9.109 \times 10^{-31}\ \text{kg}.

We can try the formula for the electron speed: impose a potential of 5 kV5\ \text{kV}:

v=2eVame=(19.109×1031 kg21.60217663 ⁣× ⁣1019 C5, ⁣000 V)=41, ⁣938 kmh\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 kV300\ \text{kV} . You will find a velocity of 324,853 kms324,853\ \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=eVaW = e\cdot V_{\text{a}}

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

W=K=12mev2W = 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:

vrel=c111+(eVam0c2)2v_{\text{rel}}=c\cdot \sqrt{1 - \frac{1}{1 +\left(\frac{e\cdot V_{\text{a}}}{m_0\cdot c^2}\right)^2}}

Where cc is the speed of light in the vacuum, c=299,792 km/sc=299,792\ \text{km}/\text{s}. As you can see, the term in the square root is always smaller — or asymptotically equal — to 11, which restricts the speed of the electron in the relativistic case to values smaller than cc.

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 kV35\ \text{kV}: with this voltage, the speed obtained with the formula in the two cases (classic and relativistic) is:

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

The difference (that you can find in our electron speed calculator tool) is 5,350 km/s5,350\ \text{km}/\text{s}. Not as high, but already not negligible.

Davide Borchia
Accelerating potential
Classical velocity
Relativistic velocity
Velocity difference
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