Physics · Electricity and magnetism · Grade 8

Electric current in metals

If electrons creep along a wire at less than a millimetre per second, why does the lamp light instantly?

The situation

You plug a kettle into a socket — and the current appears instantly. But the orderly drift speed of electrons in a copper wire is a fraction of a millimetre per second: they take hours to cover 1 metre of wire! An incandescent lamp, at the moment it is switched on, draws a current ten or more times its working current, and it often burns out just then. What actually moves inside a metal, where does resistance come from, and why does it depend on temperature?

Topic

Electric current in metals. Drift of electrons in a copper wire and the rise in a lamp's resistance when it heats up

Key ideas

Drift velocity
R(T) and superconductivity
Hall effect

Formulas

j = \frac{I}{S} = n e v_{d}

Current density: $n$ is the concentration of free electrons, $v_{d}$ is the orderly (drift) speed.

j = σ E, σ = \frac{1}{ρ} = \frac{n e^{2} τ}{m}

Ohm's law in differential form (Drude model): $τ$ is the mean time between collisions.

R = ρ \frac{l}{S}, R = R_{0} (1 + α t)

Resistance of a conductor and its dependence on temperature ( $t$ in °C, $R_{0}$ is the resistance at 0 °C).

U_{H} = \frac{I B}{n q d}

Hall voltage: $d$ is the thickness of the plate; its sign shows the sign of the charge carriers.

\frac{κ}{σ} = L T, L \approx 2.44 \cdot 10^{- 8} Vt \cdot Ω / K^{2}

Wiedemann–Franz law: the same electrons provide both thermal ( $κ$ ) and electrical ( $σ$ ) conductivity.

Theory

What moves in a metal?

In a metal crystal the positive ions form a lattice, while some of the outer electrons move freely throughout the crystal. Experiments have confirmed this. Riecke (1901) passed a current for a year through Cu–Al–Cu cylinders: a charge of about $3.5 \cdot 10^{6}$ C passed. If the current had been carried by copper ions, then by $m = M Q / (z F)$ at least 1.15 kg of copper would have been transferred (even if each ion carried two charges as in electrolysis, Cu²⁺; if they were Cu⁺ ions in the lattice — 2.3 kg). But the mass did not change and the metals did not mix — so the current is carried not by ions but by particles that are the same in all metals. Tolman and Stewart (1916) suddenly stopped a rapidly rotating coil: the charges that carried on forward by inertia gave a short current pulse, and from its charge $q / m$ turned out to equal that of the electron. See both experiments in “Evidence” mode.

Drift speed and Ohm's law (Drude)

Electrons in chaotic motion collide with the vibrating ions and with defects. The field $E$ accelerates them between collisions, so an average drift speed $v_{d} = e E τ / m$ arises. From this $j = n e v_{d} = (n e^{2} τ / m) E = σ E$ — Ohm's law in differential form; summing it over the length and cross-section of a wire gives $U = I R$ . The Joule–Lenz law follows in the same way: $w = σ E^{2}$ (per unit volume), $Q = I^{2} R t$ . In a copper wire at 10 A, $v_{d}$ is only ≈ 0.7 mm/s. A lamp lights instantly because the field spreads along the wire at a speed close to the speed of light and all the electrons start to move at once.

The Sommerfeld model and the difficulties of classical theory

Classical (Drude) theory treats the electrons as an ideal gas: $v_{T} = \sqrt{3 k T / m} \approx 115$ km/s. But it cannot explain two things: the contribution of the electrons to the heat capacity is far smaller than expected, and $ρ \propto T$ (classical theory gives $\sqrt{T}$ ). In the Sommerfeld model the electrons obey the Pauli principle and fill the energy levels up to the Fermi energy $E_{F}$ (≈ 7 eV in copper). Only the electrons near the Fermi surface take part in current and heat; their speed is $v_{F} \approx 1.6 \cdot 10^{6}$ m/s, and the mean free path is ≈ 40 nm — a hundred times longer than the lattice spacing: a perfect lattice does not scatter the electron wave, only vibrations and defects do.

Resistance and temperature, superconductivity

When heated, the ions vibrate more strongly, electrons are scattered more often and $τ$ decreases — in pure metals the resistance rises almost linearly: $R = R_{0} (1 + α t)$ , $α \approx 1 / 273$ K⁻¹. In alloys such as constantan and nickelin $α$ is very small — standard resistors are made from them. A platinum thermometer (Pt100) measures temperature precisely from $R (t)$ . The resistance of a cold lamp filament is about 15 times smaller than in the working state — hence the large current at the moment of switching on. Some substances lose their resistance completely below a critical temperature: mercury at 4.2 K, YBCO ceramic at ≈ 92 K (warmer than liquid nitrogen!). A superconductor expels the magnetic field — a magnet floats above it.

Hall effect

When a current flows through a plate in a magnetic field, the Lorentz force pushes the charge carriers to one edge. Charge builds up at the edges and creates a transverse field $E_{H}$ , which balances the Lorentz force: $q E_{H} = q v_{d} B$ . From this $U_{H} = I B / (n q d)$ . The sign of the voltage gives the sign of the carriers: negative (electrons) in copper and n-type semiconductors, positive (holes) in p-type semiconductors. In metals $n$ is large, so $U_{H}$ is in microvolts; in semiconductors it is in millivolts — and magnetic field sensors, contactless current meters and rotation-speed sensors are made from them. In metals such as zinc and beryllium the sign of $U_{H}$ is “wrong” — only band theory explains this.

The Wiedemann–Franz law and the model

In metals heat is also carried mainly by free electrons, so a good electrical conductor is also a good heat conductor: $κ = L σ T$ . For copper $L σ T \approx 425$ W/(m·K), and the table gives 401 — close! In alloys lattice vibrations carry heat too and the law is less accurate. Model: the lattice window is schematic (not to scale); the drift is exaggerated by a factor of about $4 \cdot 10^{8}$ relative to the thermal motion; $ρ (T)$ is linear; in the lamp only radiation is taken into account, $ρ_{W} \propto T^{1.2}$ , and the filament mass is about 0.03 g. For more precise measurement of resistance, see the Wheatstone bridge in the “Circuits and Kirchhoff's rules” experiment.

Did you know?

In 1911 the Dutch physicist Heike Kamerlingh Onnes found that when mercury is cooled to about 4.2 K its resistance suddenly disappears — this is how superconductivity was discovered.
The Hall effect was discovered in 1879 by the American physicist Edwin Hall, while he was a graduate student at Johns Hopkins University.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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