Formulas/physics/Current Electricity/Resistivity from Microscopic Parameters

Resistivity from Microscopic Parameters

Resistivity in terms of electron mass m, carrier density n, charge e, and mean relaxation time τ. Directly derivable from the drift velocity expression and Ohm's law.

Class 12

Derivation

From the drift velocity: $v_d = eE\tau/m$ .

Current density: $J = nev_d = \dfrac{ne^2\tau}{m}E$

From Ohm's law in microscopic form $J = \sigma E$ :

\sigma = \frac{ne^2\tau}{m}

Since $\rho = 1/\sigma$ :

\boxed{\rho = \frac{m}{ne^2\tau}}

Physical interpretation of each factor

Factor	Role	Effect on ρ
$m$ ↑	heavier electrons accelerate less	ρ ↑
$n$ ↑	more carriers	ρ ↓
$e$ ↑	stronger force and more charge	ρ ↓
$\tau$ ↑	fewer collisions, longer free path	ρ ↓

Temperature dependence recovered

For metals, $n$ is nearly constant. As $T$ rises, more intense lattice vibrations reduce $\tau$ , so $\rho \propto 1/\tau$ increases — consistent with the empirical formula $R_T = R_0(1 + \alpha T)$ .

Note

This derivation assumes the Drude (free electron) model. It gives the correct functional form but predicts $\tau$ values that are sometimes off by an order of magnitude. Quantum mechanics (Fermi-Dirac statistics) gives a more accurate picture.

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