Electrical Mobility
Calculate the diffusion constant from electrical mobility using the Einstein-Smoluchowski relation D = μ · kB · T / q. Free online physics calculator for charge carriers and semiconductors with interactive charts.
About This Calculator
The Electrical Mobility Calculator applies the Einstein-Smoluchowski relation (also called the Einstein relation) to compute the diffusion constant D from the electrical mobility μ, temperature T, and charge q of charge carriers. This fundamental relationship in solid-state physics connects random thermal motion (diffusion) to directed drift under an electric field.
The core formula is D = μ · kB · T / q, where kB = 1.380649 × 10⁻²³ J/K is the Boltzmann constant. Electrical mobility μ measures how quickly a charged particle moves through a medium under an applied electric field, with units of m²/(V·s). The diffusion constant D (m²/s) quantifies how rapidly charge carriers spread due to thermal motion. This relation is essential for understanding charge transport in semiconductors, metals, electrolytes, and biological systems.
Regional Notes
This calculator uses the internationally standardized Boltzmann constant (kB = 1.380649 × 10⁻²³ J/K, CODATA 2018) and SI units throughout. The physics is universal — the same formulas apply in India, the United States, the United Kingdom, and worldwide. The defaults use the electron charge (1.6 × 10⁻¹⁹ C) and room temperature (300 K), which are standard reference values in any region.
In India, this calculator is useful for students in IIT JEE, GATE, and university-level solid-state physics courses. In the United States, it supports AP Physics and graduate-level semiconductor device courses. In the United Kingdom, it aligns with A-Level Physics and undergraduate materials science curricula.
Frequently Asked Questions
What is the Einstein-Smoluchowski relation?
The Einstein-Smoluchowski relation (also called the Einstein relation) connects the diffusion constant D to electrical mobility μ, temperature T, and charge q of charge carriers. The formula is D = μ · kB · T / q where kB is the Boltzmann constant.
What is electrical mobility?
Electrical mobility (μ) quantifies how quickly a charged particle moves through a medium under an applied electric field. It is defined as the drift velocity per unit electric field, measured in m²/(V·s). Higher mobility means faster charge transport.
How does temperature affect the diffusion constant?
The diffusion constant D is directly proportional to temperature T according to D = μ · kB · T / q. Higher temperatures increase thermal motion, causing charge carriers to diffuse more rapidly through the material.
What units are used for the diffusion constant?
The diffusion constant D is measured in square meters per second (m²/s). For typical solids and semiconductors, values range from 10⁻⁴ m²/s for high-mobility materials like copper to 10⁻⁹ m²/s or lower for insulating materials.
How do you calculate electrical mobility from the diffusion constant?
The Einstein relation can be rearranged to μ = D · q / (kB · T). If you know the diffusion constant, temperature, and charge, you can compute the electrical mobility using this inverse formula.
What is the typical electrical mobility of electrons in copper?
At room temperature (300 K), the electron mobility in copper is approximately 0.003 m²/(V·s) (or 3,000 mm²/(V·s)). This high mobility makes copper an excellent conductor, with a corresponding diffusion constant of about 77 m²/s.
How does electrical mobility differ in semiconductors vs metals?
Electron mobility in silicon (~0.15 m²/(V·s)) is actually much higher than in copper (~0.003 m²/(V·s)). However, metals like copper have vastly higher charge carrier concentrations (≈10²³ cm⁻³ vs ≈10¹⁰ cm⁻³ in intrinsic silicon), resulting in much higher electrical conductivity overall. Mobility in semiconductors is more sensitive to temperature, doping levels, and crystal defects than in metals.
What is the Boltzmann constant value used in this calculator?
This calculator uses the CODATA 2018 recommended value of the Boltzmann constant: kB = 1.380649 × 10⁻²³ J/K. This is the internationally accepted standard and has been fixed as an exact value since the 2019 redefinition of SI base units.