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 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 ΔVΔV. A universal quantity is the electrical mobility μμ 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 [m2/s]D [m2/s] – Diffusion constant;
- μ [m2/(V⋅s)]μ [m2/(V⋅s)] – Electrical mobility;
- kB=1.3806503×10−23 J/KkB=1.3806503×10−23 J/K – Boltzmann constant;
- T [K]T [K] – Temperature; and
- q [C]q [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)μ=3000 mm2/(V⋅s). The resulting diffusion constant is D=77.08 m2/sD=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)μ=0.0519 mm2/(V⋅s), which gives a much smaller diffusion constant of D=0.001333 mm2/sD=0.001333 mm2/s.
💡 You might also be interested in our number density calculator to calculate the number density of charge carriers.
FAQs
- What does the Electrical Mobility Calculator do?
- The electrical mobility calculator explores the Einstein-Smoluchowski relation connecting the random motion of electrons in a wire to their mobility in the presence of a voltage difference. Use the calculator above for instant results in your browser.
- Is the Electrical Mobility Calculator free to use?
- Yes. All Try To Calculator tools are free and do not require an account.
- Are my inputs stored or sent to a server?
- No. Calculations run locally in your browser. We do not collect the numbers you enter or the results shown.
- Can I use the Electrical Mobility Calculator for professional decisions?
- This tool is for education and quick estimates. For medical, legal, tax, or financial decisions, verify results with a qualified professional.
- Where can I find related calculators?
- Browse more Physics tools on Try To Calculator at /physics, or use the related calculators section on this page.
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