# Electrical Mobility Calculator

By Miłosz Panfil, PhD
Last updated: Aug 07, 2018

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 a 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 certain area. The diffusion constant is the velocity of growth in 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 `V`. A universal quantity is the electrical mobility `μ` defined as the ratio of the two,

`μ = u / V`.

## Einstein-Smoluchowski relation

The Einstein-Smoluchowski relation connects the diffusion constant with the electrical mobility,

`D = μ * kB * T / q`,

where

• `D [m²/s]` is the diffusion constant,
• `μ [m²/(V * s)]` is the electical mobility,
• `kB = 1.3806503 * 10^(-23) J/K` is the Boltzmann constant,
• `T [K]` is the temperature,
• `q [C]` is the charge of the carriers.

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 `μ = 3000 mm²/(V * s)`. The resulting diffusion constant is `D = 77.08 m²/s`. As a second example, consider the sodium ions (Na⁺) in water. The electrical mobility is now `μ = 0.0519 mm²/(V * s)`, which gives much smaller diffusion constant `D = 0.001333 mm²/s`.

Miłosz Panfil, PhD
Electrical mobility
mm² / (V * s)
Temperature
°F
Charge
e
Diffusion constant
mm² / s
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