How is the electrical conductivity of a metallic conductor derived in terms of drift velocity and mobility?

In one line

In a metal the current density is \(J = nev_d = ne\mu E\), so conductivity is \(\sigma = ne\mu = ne^2\tau/m\), set by the free-electron density and how often electrons collide.

Explained simply

A metal is like a hall packed with free electrons. Apply a voltage and they all drift slowly; the current depends on how many there are (\(n\)) and how freely they move (\(\mu\)). Copper has a huge number of free electrons, so even with a modest mobility it conducts extremely well.

How it works

1. Current through area \(A\): \(I = neAv_d\), so \(J = I/A = nev_d\).
2. Drift velocity: \(v_d = \mu E\), so \(J = ne\mu E\).
3. Ohm's law in microscopic form: \(J = \sigma E\). Comparing, \(\sigma = ne\mu\).
4. Since \(\mu = e\tau/m\), \(\sigma = \dfrac{ne^2\tau}{m}\), and resistivity \(\rho = \dfrac{m}{ne^2\tau}\).
5. Temperature effect: in metals \(n\) is essentially fixed, but rising temperature makes ions vibrate more, so \(\tau\) and \(\mu\) fall and resistivity rises.
6. Contrast with semiconductors: in semiconductors, \(n\) rises sharply with temperature and outweighs the mobility drop, so conductivity increases.

Key formula

\(\sigma = ne\mu = \dfrac{ne^2\tau}{m}\), \(\rho = \dfrac{1}{\sigma}\), \(J = \sigma E\). \(\sigma\): conductivity (S/m), \(n\): free electron density (m⁻³), \(e\): electron charge (C), \(\mu\): mobility (m²/V s), \(\tau\): relaxation time (s), \(m\): electron mass (kg), \(J\): current density (A/m²), \(E\): field (V/m).

Quick example

Find the relaxation time in copper: \(\tau = m\sigma/(ne^2) = 9.11\times10^{-31}\times5.8\times10^7/(8.5\times10^{28}\times(1.6\times10^{-19})^2) \approx 2.4\times10^{-14}\) s. Electrons collide roughly every \(10^{-14}\) s, which explains why drift is so slow.

Common mistakes

• Using \(\sigma = ne\tau/m\) (forgetting \(e^2\)).
• Saying metals conduct better when heated.
• Confusing current density (A/m²) with current (A).

How MU asks it

Asked as "Derive an expression for conductivity of a conductor in terms of mobility" (3 to 5 marks) and numericals on \(\sigma\), \(\mu\) or \(\tau\). Start from \(I = neAv_d\), go to \(J = ne\mu E\), compare with \(J = \sigma E\), and add the \(\tau\) form and the temperature comment.