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Electrical Mobility Calculator

Table of contents

Diffusion constantDrift velocityEinstein-Smoluchowski relation

The electrical mobility calculator explores the Einstein-Smoluchowski relation (also known as the Einstein relation). This relation connects the random motion of electrons in a piece of wire (without a voltage difference applied) to a current flow through a wire (once a voltage difference is applied).

Continue reading to learn about the Einstein-Smoluchowski relation, the diffusion constant, and the drift velocity.

Diffusion constant

Electrons in a wire are in constant thermal motion. If we imagine putting all the electrons in a small region of a wire, the thermal motion quickly spreads them throughout the whole wire. The diffusion constant DD tells us how quickly this happens.

The unit of the diffusion constant is area/time. You can think about the diffusion constant in the following way. Say that, at some moment, electrons occupy a particular area. The diffusion constant is the velocity of growth over time of this area.

Drift velocity

If we apply a voltage difference to a wire, the electrons will start to flow. That's what we call the electric current. There are two effects in play. On one hand, the electrons are accelerated in the electric field; on the other hand, they collide with each other. The result is that the electrons move with a certain velocity, called the drift velocity uu. Try the drift velocity calculator to see how to compute it. The drift velocity depends on the voltage difference ΔV\Delta V. A universal quantity is the electrical mobility μ\mu defined as the ratio of the two:

μ=uΔV\mu = \frac{u}{\Delta V}

Einstein-Smoluchowski relation

The Einstein-Smoluchowski relation connects the diffusion constant with electrical mobility as follows:

D=μkBTq,D = \frac{\mu\, k_{\rm B}\, T}{q},


  • D [m2/s]D\ \rm [m^2/s] – Diffusion constant;
  • μ [m2/(V ⁣ ⁣s)]\mu\ \rm [m^2/(V\! \cdot\! s)] – Electrical mobility;
  • kB=1.3806503×1023 J/Kk_{\rm B} = 1.3806503\times 10^{-23}\ \rm J/K – Boltzmann constant;
  • T [K]T\ \rm [K] – Temperature; and
  • q [C]q\ \rm [C] – Charge of the carriers.

This is the equation that powers this electrical mobility calculator.

In a normal electric wire, the carriers are electrons, so the charge qq is equal to the charge of the electron. The electron mobility in cooper at room temperature is about μ=3000 mm2/(V ⁣ ⁣s)\small \mu = 3000\ \rm mm^2/(V\! \cdot\! s). The resulting diffusion constant is D=77.08 m2/s\small D = 77.08\ \rm m^2/s.

As a second example, consider the sodium ions (Na⁺) in water. The electrical mobility is now μ=0.0519 mm2/(V ⁣ ⁣s)\small \mu = 0.0519\ \rm mm^2/(V\! \cdot\! s), which gives a much smaller diffusion constant of D=0.001333 mm2/s\small D = 0.001333\ \rm mm^2/s.

💡 You might also be interested in our number density calculator to calculate the number density of charge carriers.

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