Electrical Mobility Calculator

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.

Electrical mobility0.00029406 m²/(V·s)2.94064 cm²/(V·s) — mobility is drift velocity per unit field, and it is a property of the material, not of the current

Drift velocity29.4064 µm/sI/(nAq) — 105.8631 mm per hour. Electrons crawl; it is the FIELD that propagates at nearly light speed, which is why the lamp lights instantly

Current density4.0e+5 A/m²0.4 A/mm² — above about 5 A/mm² a copper conductor needs deliberate cooling

Carrier density8.49e+28 per m³copper has 8.49×10²⁸, roughly one free electron per atom

Charge passing per second1 C/s6.2415e+18 electrons a second

Conductivity4.0e+6 S/mnqµ — copper is 5.96×10⁷ S/m

Resistivity250 nΩ·m

Resistance of that length1 Ω

Time to traverse it at drift speed3.936 daysagainst 50 ns for the signal itself — a ratio of about 6.8e+12 to one

Hall coefficient7.3516e-11 m³/C1/(nq) — its SIGN tells you whether the carriers are electrons or holes, which no other simple measurement reveals

Hall voltage367.58 nVIB/(nqt) — thinner samples give bigger signals, which is why Hall sensors are made as thin films

Hall field1.4703e-5 V/mv_d × B, exactly balancing the magnetic force on the carriers once the transverse charge has built up

Why the Hall effect mattersit counts the carriersconductivity alone gives only the product nµ. The Hall coefficient gives n by itself, so the two together separate how MANY carriers there are from how freely each one moves

The formula

v_d = I/(nAq) = µE; σ = nqµ; R_H = 1/(nq); V_H = IB/(nqt)

Electrons crawl, signals race

A single amp through a 2.5 mm² copper wire moves its electrons at a few hundredths of a millimetre per second — slower than a growing fingernail. Yet the lamp lights the instant the switch closes, because the electric field propagates through the conductor at a good fraction of the speed of light and starts every electron in the circuit moving at once. Nothing has to travel from the switch to the lamp.

The two speeds differ by about twelve orders of magnitude, and conflating them is the most common misconception in basic electricity. Drift velocity explains how much charge arrives; field propagation explains when it starts.

The Hall effect counts the carriers

Conductivity is nqµ — a single number containing both how many carriers there are and how freely each moves, with no way to separate them. The Hall coefficient is 1/nq, which depends on the count alone. Measuring both therefore pulls the two apart, and it is how carrier concentrations in semiconductors are actually determined.

Its sign carries information nothing else does: a positive Hall coefficient means the moving charges are positive. That is how holes in p-type semiconductors were confirmed to behave as genuine positive carriers rather than as a bookkeeping convenience.