# Electrical Mobility Calculator (Einstein Relation)

URL: https://speedycalculator.com/calculators/physics/electrical-mobility/
Last updated: 2026-10-30

> The electrical mobility calculator finds the diffusion coefficient of charged particles from their electrical mobility and temperature using the Einstein relation, or the mobility from the diffusion coefficient.

## Inputs

- Find
- Electrical mobility μ (cm²/(V·s), m²/(V·s))
- Diffusion coefficient D (cm²/s, m²/s)
- Temperature T (K, °C, °F)
- Charge number z

## Results

- Diffusion coefficient D
- Electrical mobility μ
- Thermal voltage k_BT/q

## Formula

```
D = μ·k_B·T / q
```

## Explanation

Charged particles both drift in electric fields and spread by diffusion. Einstein showed in 1905 that the two are linked through temperature – a deep result connecting random motion and response to forces.

### How to use the Electrical Mobility Calculator

- **Find**
- **Electrical mobility μ**
- **Diffusion coefficient D**
- **Temperature T**
- **Charge number z**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose what to find and enter the values.

### Tips

- Use absolute temperature (K) in gas laws and efficiency formulas.
- Temperature differences are the same in K and °C.
- Heat = mass × specific heat × temperature change when no phase change occurs.
- Phase changes need latent heat at a constant temperature.

### Common mistakes to avoid

- Using °C in pV = nRT.
- Forgetting latent heat during melting or boiling.
- Confusing heat capacity (J/K) with specific heat (J/(kg·K)).

### Inputs explained

FieldWhat to enterFindChoose one: Diffusion coefficient D, Mobility μElectrical mobility μUnits: cm²/(V·s), m²/(V·s)Diffusion coefficient DUnits: cm²/s, m²/sTemperature TUnits: K, °C, °FCharge number zA number

### What each result means

- **Diffusion coefficient D** = [Electrical mobility μ]×1.380649e-23×[Temperature T]÷([Charge number z]×1.602176634e-19)
- **Electrical mobility μ** = [Diffusion coefficient D]×[Charge number z]×1.602176634e-19÷(1.380649e-23×[Temperature T])
- **Thermal voltage k_BT/q** = 1.380649e-23×[Temperature T]÷1.602176634e-19×1000

### Example values

How the main result changes when **Electrical mobility μ** changes and everything else stays at the example values above:

Electrical mobility μ (cm²/(V·s))Diffusion coefficient D (cm²/s)70018.09641,05027.14461,40036.19281,75045.2412,10054.28922,80072.3856

## Worked example

Inputs — Find: Diffusion coefficient D; Electrical mobility μ: 1,400 cm²/(V·s); Temperature T: 300 K; Charge number z: 1.

Results — **Diffusion coefficient D = 36.1928 cm²/s**; Thermal voltage k_BT/q = 25.852 mV.

## Key terms

- **Mobility**: Drift velocity per unit electric field.

## FAQ

**What is electron mobility in silicon?**
About 1,400 cm²/(V·s) at room temperature.

**What does the Einstein relation connect?**
Drift (mobility) and diffusion of the same particles.

**Does it hold for degenerate semiconductors?**
It needs corrections at very high doping.

**Who discovered it?**
Einstein (1905) and Smoluchowski (1906).

**What formula does the Electrical Mobility Calculator use?**
D = μ·k_B·T / q.


## Sources

- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)
- [OpenStax – University Physics Volume 2](https://openstax.org/details/books/university-physics-volume-2)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)