# 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.

## 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** $D$ 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.

## 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 $u$. 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:

## Einstein-Smoluchowski relation

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 $\small D = 0.001333\ \rm mm^2/s$.

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