Formula and Method for Electrical Mobility
Electrical mobility describes how readily a charged particle — an ion, an electron, or a charged aerosol droplet — drifts through a gas or fluid when pushed by an electric field. It is defined as the ratio of drift velocity to field strength, mu = v_d / E, and it is the working principle behind ion mobility spectrometers, differential mobility analyzers (DMAs) used to size and sort aerosol particles, and drift-velocity measurements of charge carriers. This calculator computes mobility directly from a measured drift velocity and field strength, then uses the particle's charge and the fluid's viscosity to back out a drag (friction) coefficient and an equivalent spherical diameter via Stokes' law.
Finding mobility from drift velocity and field
At equilibrium, a charged particle moving through a viscous gas accelerates until the electric force qE exactly balances the drag force, after which it drifts at a constant terminal velocity v_d. Dividing that velocity by the field strength that produced it gives the mobility: mu = v_d / E. Because both v_d and E are directly measurable — v_d from a time-of-flight measurement or particle-tracking image, E from the applied voltage divided by electrode spacing — this ratio is usually how mobility is measured experimentally, whether for gas-phase ions, electrons in a semiconductor, or aerosol particles in a classifier.
From mobility to a drag coefficient and particle size
Because mobility also equals charge divided by the friction (drag) coefficient, mu = q/f, rearranging gives f = q/mu. For a rigid sphere moving slowly through a continuous fluid, Stokes' law states that the drag coefficient is f = 3πηd, where η is the fluid's dynamic viscosity and d is the sphere's diameter. Combining the two relationships lets you solve for an equivalent diameter, d = q / (3πηmu) — the size a spherical particle would need to have, in that fluid, to produce the observed mobility. This estimate skips the Cunningham slip correction, so it is most accurate for particles larger than roughly 1 micrometer in air; smaller aerosol particles will have a true diameter smaller than the plain-Stokes value reported here.
When the plain Stokes model breaks down
This calculator's diameter estimate assumes a spherical particle in slow (low Reynolds number, non-turbulent), continuum-regime flow — valid for particles from about 1 to 100 micrometers in air. Below roughly 1 micrometer, the mean free path of gas molecules becomes comparable to the particle size, so drag is lower than plain Stokes predicts and a Cunningham slip correction factor is required for accurate sizing (this is why real differential mobility analyzers apply it). For electrons or holes moving through a semiconductor lattice, mobility is instead governed by quantum-mechanical scattering off phonons, impurities, and lattice defects rather than viscous drag, so the Stokes diameter relationship in this calculator does not apply to solid-state carrier mobility — only the underlying definition mu = v_d/E still holds.