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 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 . Try the drift velocity calculator to see how to compute it. The drift velocity depends on the voltage difference . A universal quantity is the electrical mobility defined as the ratio of the two:
The Einstein-Smoluchowski relation connects the diffusion constant with electrical mobility as follows:
- – Diffusion constant;
- – Electrical mobility;
- – Boltzmann constant;
- – Temperature; and
- – 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 is equal to the charge of the electron. The electron mobility in cooper at room temperature is about . The resulting diffusion constant is .
As a second example, consider the sodium ions (Na⁺) in water. The electrical mobility is now , which gives a much smaller diffusion constant of .
💡 You might also be interested in our number density calculator to calculate the number density of charge carriers.