Electrical Mobility Calculator

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.

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 $D$ 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.

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 $u$. Try the drift velocity calculator to see how to compute it. The drift velocity depends on the voltage difference $\Delta V$. A universal quantity is the electrical mobility $\mu$ defined as the ratio of the two:

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

where:

• $D\ \rm [m^2/s]$ – Diffusion constant;
• $\mu\ \rm [m^2/(V\! \cdot\! s)]$ – Electrical mobility;
• $k_{\rm B} = 1.3806503\times 10^{-23}\ \rm J/K$ – Boltzmann constant;
• $T\ \rm [K]$ – Temperature; and
• $q\ \rm [C]$ – 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 $q$ is equal to the charge of the electron. The electron mobility in cooper at room temperature is about $\small \mu = 3000\ \rm mm^2/(V\! \cdot\! s)$. The resulting diffusion constant is $\small D = 77.08\ \rm m^2/s$.

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