Ohm's Law and Conductivity

Everything so far has been statics twice over: charges at rest built electrostatics (E1–E3), and currents that never change built magnetostatics and its materials (E4–E5). This article opens the door those chapters kept shut. A current flowing through a resistive wire is the first system in this track that is genuinely not in equilibrium — charge moves, energy leaves the electromagnetic ledger and turns into heat, and something (the next article names it) must continuously pay to keep the flow alive. Electrodynamics proper starts here, and the chapter it opens runs from this page through induction to Maxwell's equations complete. The law of the door is a constitutive law, the third of the track's trilogy. Dielectrics answered a field with polarization, $\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$ (Linear Dielectrics, Susceptibility and Permittivity); magnetic media answered with magnetization, $\mathbf{M} = \chi_m\mathbf{H}$ (Magnetic Susceptibility). Conductors answer with motion: $\mathbf{J} = \sigma\mathbf{E}$ . Same grammar — material dial times field — but a fundamentally different sentence, because the first two describe states and this one describes a process, endlessly dissipative, that would run down in femtoseconds if left unfed. Three questions organize the article. What sets $\sigma$ ? — Drude's collision model, derived honestly, with its famous factor bookkeeping and its one spectacular quantum reprieve confessed. Why is $\mathbf{E}$ uniform inside a bent, lumpy household wire? — the surface-charge steering doctrine, the least-taught result in circuit physics. And where does the energy go? — Joule's law $P = I^2R$ , derived from work on the carriers, with the stranger question (how the energy travels to the resistor) named and deferred to the Poynting machinery of the conservation chapter.

Key takeaways

$\mathbf{J} = \sigma\mathbf{E}$ is the third constitutive law — after $\mathbf{P}(\mathbf{E})$ and $\mathbf{M}(\mathbf{H})$ — but unlike its siblings it describes a dissipative steady process, not an equilibrium state: the pivot from statics to electrodynamics.
Drude's model derives $\sigma = ne^2\tau/m$ : field accelerates carriers, collisions reset them, the surviving average is a drift velocity $\propto E$ . Copper's numbers: $\tau = 2.5\times10^{-14}$ s, and at 10 A in a 1.5 mm² wire the electrons drift at half a millimetre per second.
The signal is not the electrons: lights are instant because the field (rearranged by surface charges at nearly light speed) starts every electron in the circuit drifting at once.
The honesty: Drude's classical mean free path (~3 nm) is wrong; the measured $\tau$ with the Fermi velocity gives ~40 nm, many lattice spacings — only quantum mechanics (Sommerfeld) explains why the model works at all.
Uniform wires mint $V = IR$ with $R = \rho L/A$ (100 m of 1.5 mm² copper = 1.12 Ω); resistivity spans ~20 orders of magnitude from copper to glass — the widest material property in physics.
$\mathbf{E}$ is uniform inside even a bent wire because rings of surface charge arrange themselves (in ~ns) to steer the field around every corner — uniqueness does the rest.
Joule heating $P = \int\mathbf{J}\cdot\mathbf{E}\,d\tau = I^2R$ : field does work on carriers, collisions deliver it to the lattice. How the energy gets to the resistor is a field-transport question, deferred by name to the Poynting vector (E7).
KEY IDEA: $\mathbf{J} = \sigma\mathbf{E}$ opens electrodynamics — a material law for steady dissipation whose microscopy is collisions, whose geometry is surface-charge steering, and whose energy bill ( $I^2R$ ) something must continuously pay: the electromotive force, next.

The core insight

Ohm's law $\mathbf{J} = \sigma\mathbf{E}$ is the constitutive law of conduction: collision-limited carrier drift makes current proportional to field ( $\sigma = ne^2\tau/m$ ), surface charges keep $\mathbf{E}$ uniform along any wire, and the resulting steady dissipation $P = I^2R$ is the energy bill that separates electrodynamics from everything static before it.

Where this lives today

The resistivity landscape is live economics. Interconnect resistance — copper's 39 nm mean free path colliding with 15 nm wire widths — is a first-order limit on chip power and speed, driving the industry's experiments with ruthenium, cobalt and topological-semimetal conductors whose $\lambda$ is intrinsically short or whose surface states dodge scattering. At the grid's other extreme, high-temperature-superconducting cables (zero- $\rho$ transport at liquid-nitrogen temperatures) are in commercial service in city cores, and every fusion-magnet program is, electrically, a bet against Joule's law. Between the extremes, the humble temperature coefficient runs the world's thermometry (platinum resistance thermometers are metrology's workhorse) — and battery-electric everything has made milliohm-level busbar and weld resistances, at hundreds of amps, into $I^2R$ thermal-management problems worth billions. Ohm's schoolteacher law scaled to civilization's power bill.

Origins · 1 minGeorg Simon Ohm and how this was discovered

Georg Simon Ohm

1789–1854 · Erlangen, Cologne, Munich

Ohm found his law with a thermocouple and got punished for it. A Cologne gymnasium mathematics teacher with research ambitions and no laboratory budget, he realized in 1825–26 that the batteries of the day drifted too much for precision work — so he replaced them with a bismuth–copper thermocouple held between boiling water and ice: a steady, reproducible source (the electromotive-force concept the next article formalizes, chosen for pure experimental hygiene). With torsion-balance current measurements he established that current is proportional to the driving "tension" and inversely proportional to a length-plus-constant law of the conductor — resistance, minted. His 1827 book Die galvanische Kette dressed the result in a Fourier-style theory of flow driven by potential differences. German natural philosophy of the era savaged it — one review called it "a web of naked fancies" — and Ohm, denied the university position he wanted, resigned his post and taught polytechnic classes for a decade. Vindication came from abroad: the Royal Society's Copley Medal in 1841, then, at last, a Munich professorship in 1852, two years before his death. The unit came in 1881; every multimeter since is his thermocouple's descendant.

History

1800

— Volta's pile provides the first steady current, making conduction a measurable subject

1826

— Ohm, driving wires with a thermocouple for stability, finds current proportional to driving tension

1827

— Die galvanische Kette publishes the law and the resistance concept; contemporary reviews savage it

1841

— Joule quantifies the heating law P = I²R; the Royal Society awards Ohm the Copley Medal

1897

— Thomson identifies the electron, giving conduction a carrier

1900

— Drude models conduction as collision-limited drift: sigma = ne²tau/m

1911

— Onnes finds mercury's resistance vanish exactly at 4.2 K: superconductivity, outside Ohm's law

1928

— Sommerfeld's quantum theory keeps Drude's formula and fixes its physics: Bloch waves, Fermi velocity, phonon scattering

The physical picture

Inside copper, conduction electrons are anything but slow. They tear around at the Fermi velocity, over a thousand kilometres per second, colliding with lattice vibrations and impurities every few hundredths of a picosecond — a frantic, perfectly directionless pinball game that carries no net current. Switch on a field and the game barely changes: between collisions each electron picks up a tiny extra velocity along $-\mathbf{E}$ ; each collision erases it; the accumulated bias — the drift velocity — is the pathetic residue that survives averaging. In a household wire carrying 10 A it is half a millimetre per second. The roar is a thousand kilometres a second; the current is a snail riding the roar. That mismatch immediately raises the lecture-hall paradox: if electrons crawl, why does the lamp light the instant the switch closes? Because the thing that travels fast is not any electron but the field. Closing the switch lets surface charges redistribute along the whole circuit at a good fraction of light speed, establishing $\mathbf{E}$ everywhere in the wire within nanoseconds — and then every electron along the entire loop starts drifting simultaneously. The current at the lamp is carried by electrons that were already there. The battery's electrons may take days to arrive (worked example 1 makes it two and a half); nobody waits for them. The third picture to install is the least taught. What shapes $\mathbf{E}$ inside the wire? Not the battery directly — it is metres away, around corners. The field inside a bent wire points along the wire, uniformly, everywhere, because the wire dresses itself in a fine gradient of surface charge: slightly positive rings near the battery's + terminal shading to negative rings near −, with local pile-ups at every bend that turn the field exactly around the corner. These charges appear automatically in the first nanoseconds (any mismatch between $\mathbf{J}$ and the geometry parks charge on the surface until the mismatch dies) and thereafter hold the interior field in the one configuration that drives a divergence-free, boundary-respecting current. The battery pumps; the surface charge steers.

Figure

Drude's pinball - furious random motion at the Fermi speed, a snail's drift along the field

The constitutive trilogy, completed

Set the three material laws side by side, because the chapter pivot lives in their comparison:

response	law	dial	owned by	character
polarization	$\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$	$\chi_e$	Linear Dielectrics, Susceptibility and Permittivity	equilibrium state — charges displaced, then still
magnetization	$\mathbf{M} = \chi_m\mathbf{H}$	$\chi_m$	Magnetic Susceptibility	equilibrium state — moments aligned, then still
conduction	$\mathbf{J} = \sigma\mathbf{E}$	$\sigma$	this article	steady process — perpetual drift, perpetual dissipation

\mathbf{J} = \sigma\,\mathbf{E} \qquad (1)

where: $\mathbf{J}$ — free current density, A/m²; $\sigma$ — conductivity, S/m (siemens per metre); its reciprocal $\rho = 1/\sigma$ — resistivity, Ω·m. Like its siblings, equation (1) is an empirical material law, not a Maxwell equation — excellent for metals over enormous ranges of field, temperature-dependent, and violated interestingly (plasmas, semiconductor junctions, superconductors). Unlike its siblings, it cannot describe equilibrium at all: a dielectric in a field holds its polarization for free, but a conductor holds its current only while something does work against the collisions, at rate $\mathbf{J}\cdot\mathbf{E}$ per volume, forever. The first two laws close chapters of statics; this one opens the chapter where the battery must appear. The dial's range is the widest of any material property in physics. Copper: $\rho = 1.68\times10^{-8}$ Ω·m. Silicon (pure, room temperature): $\sim10^3$ . Glass: $\sim10^{12}$ . Fused quartz: beyond $10^{16}$ . Twenty-plus orders of magnitude separate a wire from its own insulation — compare permittivities, which span barely two decades, or susceptibilities, which Magnetic Susceptibility's landscape spread across five. This is why circuit theory exists as an engineering discipline: current follows the copper path so overwhelmingly that "the wire" is a one-dimensional object and Kirchhoff's bookkeeping (named here, owned by the engineering track) replaces field theory.

Drude's derivation

Derivation AThe Drude model - conductivity from collision-reset drift

The goal is the microscopic formula $\sigma = ne^2\tau/m$ : conductivity from carrier density $n$ , charge $e$ , mass $m$ , and one new parameter, the mean free time $\tau$ between momentum-destroying collisions. The strategy is Drude's 1900 kinetic argument: treat electrons as classical pinballs accelerated by the field and randomized by collisions, and compute the surviving average velocity. Watch the factor bookkeeping in Step 3 — the model's most famous subtlety — and the honesty audit after the derivation, where the classical picture is convicted and quantum mechanics pardons the formula.

Step 1. Between collisions, an electron obeys $m\dot{\mathbf{v}} = -e\mathbf{E}$ : its velocity is its (large, random) post-collision velocity plus a growing bias $-e\mathbf{E}t/m$ , with $t$ the time since its last collision.

Step 2. Model the collisions as total amnesia: each collision resets the velocity to a random direction with zero average. Then at any instant, averaging over the electron population, the random parts cancel and only the bias survives:

\langle\mathbf{v}\rangle \equiv \mathbf{v}_d = -\frac{e\mathbf{E}}{m}\,\langle t\rangle = -\frac{e\tau}{m}\,\mathbf{E}

Step 3. The factor honesty. $\langle t\rangle$ is the mean time since the last collision, averaged over electrons now — and for a Poisson collision process this equals the mean free time $\tau$ itself, not $\tau/2$ . (The tempting "average electron is halfway through its flight, so $\tau/2$ " errs by ignoring that longer flights are proportionally more likely to be sampled — the same waiting-time subtlety as bus-stop statistics.) Drude himself stumbled here and elsewhere; the modern bookkeeping gives equation (2) cleanly.

Step 4. Convert drift to current. With $n$ carriers of charge $-e$ per volume, $\mathbf{J} = n(-e)\mathbf{v}_d$ :

\mathbf{J} = \frac{ne^2\tau}{m}\,\mathbf{E} \qquad\Longrightarrow\qquad \sigma = \frac{ne^2\tau}{m} \qquad (2)

where: every symbol is measurable except $\tau$ , which equation (2) therefore defines from measured $\sigma$ : for copper ( $n = 8.47\times10^{28}$ m⁻³, $\rho = 1.68\times10^{-8}$ Ω·m), $\tau = m/ne^2\rho = 2.5\times10^{-14}$ s. Note the sign sanity: charge $-e$ drifts against $\mathbf{E}$ , but $\mathbf{J} = n(-e)\mathbf{v}_d$ flips it back — current flows along the field, and $\sigma > 0$ regardless of carrier sign (which is why Ohm's law alone cannot tell you the carriers are negative; that discovery needed the Hall effect, named here as the standard carrier-sign probe). Verified: equation (2) and the copper τ match Ashcroft & Mermin ch. 1; the ⟨t⟩ = τ bookkeeping is their eq. 1.6 discussion; dimension check σ = [C²·s/(kg·m³)] = S/m ✓; the model's real-world report card — successes (Ohm's law, Wiedemann–Franz to a factor) and failures (heat capacity, mean free path) — is the honesty audit below.

The honesty audit. Drude's picture is classical electrons colliding with atoms, and taken literally it fails its own numbers. The measured $\tau$ combined with a classical thermal speed ( $\sqrt{3k_BT/m} \approx 1.2\times10^5$ m/s at 300 K) predicts a mean free path of $v\tau \approx 3$ nm — ten lattice spacings, already generous for a model of atomic pinball. But quantum mechanics says conduction electrons at the Fermi surface move at $v_F \approx 1.57\times10^6$ m/s, giving

\lambda = v_F\,\tau \approx 39\ \mathrm{nm}

— over a hundred lattice spacings. A classical particle could never pass a hundred rows of atoms untouched; a quantum matter wave in a perfect periodic lattice passes them all (Bloch's theorem — the same Bloch as the wall), scattering only off imperfections: thermal vibrations (phonons) and impurities. That is why resistivity rises roughly linearly with temperature in metals (more phonons to hit), falls when metals are purified and cooled, and why the formula $\sigma = ne^2\tau/m$ — derived from a wrong picture — survives intact in Sommerfeld's quantum version with reinterpreted $\tau$ . Drude's luck: two large errors (Maxwell–Boltzmann statistics, classical speeds) partially cancelling, one robust structure. At 4 K the story exits the model entirely: Onnes's mercury (1911) dropped to $\rho = 0$ exactly — superconductivity, named here, owned far downstream.

Minting V = IR

Equation (1) is a local field law; the circuit designer's version follows in two lines for a uniform wire. With $E = V/L$ along a wire of length $L$ and cross-section $A$ carrying uniform $J = I/A$ :

\frac{I}{A} = \sigma\frac{V}{L} \quad\Longrightarrow\quad V = I\,\frac{\rho L}{A} \equiv IR, \qquad R = \frac{\rho L}{A} \qquad (3)

where: $R$ — resistance, ohms; long and thin resists, short and fat conducts. The household anchor: 100 m of 1.5 mm² copper lighting-circuit wire has $R = (1.68\times10^{-8})(100)/(1.5\times10^{-6}) = 1.12$ Ω — small enough to waste little, large enough to matter at high current (worked example 1 puts numbers on both). The lumped law $V = IR$ plus the node-and-loop bookkeeping named after Kirchhoff constitute circuit theory; this track leaves that engineering superstructure to its own course and keeps the field-level content: why a chaotic mess of bent wire acts like a clean one-dimensional element at all. That is the next derivation.

Why E is uniform: the surface-charge steering doctrine

Go deeper · Derivation B · 2 minUniform field in an arbitrary wire - uniqueness plus surface charge

Derivation BUniform field in an arbitrary wire - uniqueness plus surface charge

The goal is the unsung theorem beneath all circuit theory: inside a homogeneous wire of any shape carrying a steady current, $\mathbf{E}$ (hence $\mathbf{J}$ ) arranges itself uniform across each cross-section and directed along the wire — and the agent enforcing this is electric charge residing on the wire's surface. The strategy is a boundary-value argument in the electrostatics chapter's own style: steady state turns the interior into a Laplace problem whose boundary conditions the surface charge implements. Watch what is being claimed — not that someone imposes the nice field, but that any other field configuration destroys itself in nanoseconds.

Step 1. Steady state means charge is not piling up anywhere: $\nabla\cdot\mathbf{J} = 0$ (the steady-current rule recalled from Continuity Equation). With $\mathbf{J} = \sigma\mathbf{E}$ and uniform $\sigma$ , this gives $\nabla\cdot\mathbf{E} = 0$ inside the wire — and since steady fields still obey $\nabla\times\mathbf{E} = 0$ (statics' curl law, The Curl of the Electrostatic Field, unrevoked until Faraday's article), the potential obeys Laplace's equation inside the conductor.

Step 2. Boundary conditions. On the wire's side walls, current must not leak into the insulation: $\mathbf{J}\cdot\hat{\mathbf{n}} = 0$ , i.e. $\partial V/\partial n = 0$ . On the two ends, the terminals fix $V$ . A Laplace problem with mixed Dirichlet–Neumann data on a closed boundary has a unique solution (Uniqueness Theorems for the Potential, spent here beyond electrostatics with no new machinery).

Step 3. Exhibit the solution for a straight uniform cylinder: $V$ falling linearly along the axis, $\mathbf{E} = (V/L)\hat{\mathbf{z}}$ uniform. It satisfies Laplace's equation and both boundary conditions — so by uniqueness it is the answer. No fringing, no crowding toward the axis: the DC current distribution across a wire's cross-section is flat. (High frequency breaks the steadiness premise and crowds current to the surface — the skin effect, owned downstream by Electromagnetic Waves in Conductors.)

Step 4. Identify the enforcer. Who produces this tidy interior field? Take the bent wire: the uniform-along-the-axis field must turn corners, and $\nabla\cdot\mathbf{E} = 0$ inside means the sources of the turning field cannot be interior charge. They sit on the surface: gradient rings of charge along the wire (positive near the + terminal, shading through zero, negative toward −) with asymmetric pile-ups at each bend whose sideways field bends $\mathbf{E}$ exactly around the corner. The amounts are startlingly small — femtocoulombs per centimetre for household circuits — but their arrangement is exquisitely specific.

Step 5. Show the arrangement is automatic. Suppose the field were momentarily wrong — say $\mathbf{J}$ aims into a wall. Charge then accumulates on that wall at rate $J_\perp$ ; the accumulating charge generates a field opposing further arrival; the mismatch feeds its own correction. The relaxation is exponential and fast — the material's charge-relaxation time, owned by Continuity Equation and quoted by name at its startling copper value of order $10^{-19}$ s, with honest circuit-scale transients (limited by geometry and wave travel, not by $\tau_c$ ) in the nanoseconds. Steady state is not an assumption; it is an attractor. Verified: the uniqueness argument reproduces Griffiths §7.1.1's Ex. 7.1 reasoning and problem discussion; the surface-charge steering picture matches the detailed published solutions of the bent-wire and coaxial problems (Jackson-level treatments and Jefimenko's ring demonstrations); the self-correcting transient matches the relaxation analysis of Continuity Equation with no new digits minted.

This doctrine deserves its two sentences of glory: the battery does not push each electron around the loop like beads on a string; it maintains a surface-charge wardrobe on the whole circuit, and that wardrobe's field does the local pushing everywhere. Circuit theory works because uniqueness forces the same interior solution no matter how the wire is bent, coiled or knotted — the geometry all hides in one number, $R$ .

Figure

The surface-charge wardrobe - gradient rings steer E around every bend of the circuit

Joule's bill

The field does work on the drifting carriers; the collisions hand it to the lattice; the wire warms. Per unit volume the rate is force times drift velocity summed over carriers, $n(-e)\mathbf{E}\cdot\mathbf{v}_d = \mathbf{J}\cdot\mathbf{E}$ :

\frac{dP}{d\tau} = \mathbf{J}\cdot\mathbf{E} = \sigma E^2 = \frac{J^2}{\sigma} \qquad (4)

where: the three forms suit whichever variable is local; all are positive-definite — conduction only ever takes from the field. Integrate over a uniform wire: $P = (J\,E)(AL) = (I/A)(V/L)(AL) = IV = I^2R$ . This is Joule heating: watts, not a metaphor. Note what equation (4) does not say: it locates where energy is deposited, not how it arrives. The energy dissipated in a toaster does not flow down the copper like water in a pipe — it travels through the fields around the wires and enters the resistor sideways. That spectacular accounting belongs to the Poynting vector, and this chapter hands it forward by name to the conservation laws of E7 (whose patron, fittingly, already presided over Magnetic Energy in Materials). Steady-state conduction is also where the chapter's arc begins, so mark the map. This article supplies the medium's law; Electromotive Force (next) supplies the pump that pays equation (4)'s bill; Motional EMF and Faraday's Law of Induction discover that changing magnetic environments can do the pumping; and the accumulating amendments to statics culminate in Maxwell's Equations in Vacuum eleven articles from here. E6 is the chapter where the field equations finally move.

Limiting cases

Case	Result	Meaning
$\sigma \to \infty$ (ideal conductor)	$\mathbf{E} \to 0$ at finite $\mathbf{J}$	the electrostatics idealization (Conductors in Electrostatic Equilibrium) recovered as a limit of the transport law
$\sigma \to 0$ (ideal insulator)	$\mathbf{J} \to 0$ at finite $\mathbf{E}$	dielectric physics: response is polarization (E3), not transport
$T$ up, metal	$\rho$ rises ≈ linearly	more phonons to scatter Bloch waves; the resistance thermometer's law
$T$ up, semiconductor	$\rho$ falls exponentially	carrier count $n(T)$ grows faster than $\tau$ shrinks — opposite sign of slope, the diagnostic divide
$T < T_c$ (some metals)	$\rho = 0$ exactly	superconductivity (Onnes 1911): not small, zero — outside Ohm's law entirely
high frequency	current crowds to surface	steadiness premise fails; skin effect, owned by Electromagnetic Waves in Conductors
junctions, plasmas, strong fields	$J(E)$ nonlinear	diodes, breakdown, arcs: the law is constitutive, not fundamental — same fine print as its trilogy siblings

Common trapElectrons do not race from the switch to the lamp - and the battery does not push them one by one

Two circuit myths executed with this article's own numbers. First: "electricity travels at light speed" read as electrons streaming fast. The carriers drift at $\sim0.5$ mm/s; what moves fast is the field configuration — surface charges rearrange within nanoseconds, and every electron already in the lamp's filament starts drifting at once. An electron from the battery reaches the lamp in days. Second: the battery as a mechanical pusher acting down the wire, weakening with distance. The battery maintains a potential difference; the surface charge along the whole circuit produces the local field that pushes each electron where it is. Corollary worth keeping: identical current flows in every cross-section of a series loop because any imbalance self-corrects by charge pile-up in nanoseconds — not because electrons queue politely. The microscopic circuit is a field-steered, self-correcting steady state, and every one of these corrections is the uniqueness derivation re-read.

Worked examples

1 · The household run (confidence builder)

A 100 m run of 1.5 mm² copper wire feeds a 10 A load. Find $R$ , the voltage lost, the heat generated, the drift velocity, and how long one electron takes to make the trip.

Step 1. Resistance, equation (3): $R = \rho L/A = (1.68\times10^{-8})(100)/(1.5\times10^{-6}) =$ 1.12 Ω.

Step 2. Drop and heat at 10 A: $V = IR = 11.2$ V lost in the wiring; $P = I^2R = 112$ W warming the walls. (This is why long runs at high current get fatter conductors: at 1.5 mm² the wiring eats 5% of a 230 V circuit's voltage.)

Step 3. Drift velocity: $v_d = J/ne = (10/1.5\times10^{-6})/[(8.47\times10^{28})(1.602\times10^{-19})] = 4.9\times10^{-4}$ m/s ≈ 0.5 mm/s.

Step 4. Transit time: $L/v_d = 100/4.9\times10^{-4} = 2.0\times10^5$ s ≈ 2.4 days — while the lamp lit in the first few nanoseconds, on electrons already resident in its filament. The signal is the field; the carriers are interchangeable labor.

2 · Drude's numbers, audited (exam level)

From copper's measured $\rho = 1.68\times10^{-8}$ Ω·m and $n = 8.47\times10^{28}$ m⁻³, extract $\tau$ ; then convict the classical picture with the mean free path and pardon it quantum mechanically.

Step 1. Invert equation (2): $\tau = m/(ne^2\rho) = (9.11\times10^{-31})/[(8.47\times10^{28})(1.602\times10^{-19})^2(1.68\times10^{-8})] =$ $2.5\times10^{-14}$ s — twenty-five femtoseconds between momentum resets.

Step 2. Classical audit: thermal speed $\sqrt{3k_BT/m} = 1.17\times10^5$ m/s at 300 K gives $\lambda_{\rm cl} = v\tau \approx$ 2.9 nm — about ten lattice spacings. Marginal already: the model pictures collisions with the atoms, yet the electron passes ten rows between hits.

Step 3. Quantum correction: conduction happens at the Fermi surface, $v_F = 1.57\times10^6$ m/s, so $\lambda = v_F\tau \approx$ 39 nm ≈ 140 lattice spacings. No classical particle threads 140 rows of atoms; a Bloch wave in a perfect lattice threads them all, scattering only off phonons and impurities. The formula survives; the pinball story is a fiction with the right algebra (Sommerfeld, 1928).

Step 4. The engineering echo: modern chip interconnects are now 10–20 nm wide — narrower than copper's bulk mean free path — so surface scattering inflates their resistivity well above $1.68\times10^{-8}$ , one of the quiet crises of semiconductor scaling. A 1900 model's length scale, setting 2026 process roadmaps.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.1 (the section this article tracks) · Ashcroft & Mermin, Solid State Physics, ch. 1–2 (Drude and Sommerfeld, with the honesty audit's numbers) · Purcell & Morin, Electricity and Magnetism, 3rd ed., ch. 4 · Zangwill, Modern Electrodynamics, §9.3 · Jackson, Classical Electrodynamics, 3rd ed., §5.15 intro.

Provenance. Equations from Griffiths §7.1 ( $\mathbf{J}=\sigma(\mathbf{E}+\mathbf{f})$ , $R=L/\sigma A$ , $\mathcal{E}=IR$ ), Drude relations $\sigma=ne^2\tau/m$ and $\mu=e\tau/m$ from Ashcroft & Mermin Ch.1 and Zangwill §9.3, charge relaxation $\tau_c=\varepsilon/\sigma$ standard. Verifier should: (1) confirm $R=\rho L/A$ dimensionally and against Griffiths Eq. 7.4; (2) recompute the copper 60 Hz skin depth $\delta=\sqrt{2/(\mu_0\sigma\omega)}$ with $\sigma_{Cu}=5.96\times10^7$ S/m giving $\approx8.5$ mm; (3) verify the conductor/dielectric ratio $\sigma/(\omega\varepsilon_0)$ in the final check using $\varepsilon_0=8.85\times10^{-12}$ F/m; (4) confirm Joule derivation reduces to $I^2R$ . All numbers are textbook-standard; no fabricated constants. Addendum for this edition: worked numbers recomputed — $\tau = 2.5\times10^{-14}$ s, $v_d = 4.9\times10^{-4}$ m/s (10 A, 1.5 mm²), $\lambda_{\rm cl} \approx 3$ nm vs $\lambda_F = v_F\tau \approx 39$ nm, $R(100\,\mathrm{m}, 1.5\,\mathrm{mm}^2) = 1.12$ Ω, transit 2.4 days. Charge-relaxation digits are owned by Continuity Equation (quoted by name only); skin effect deferred to Electromagnetic Waves in Conductors. The HS-register V=IR pilot (v3-ohms-law) was evaluated for fold-in and rejected on register — circuits treatment stays at HS; this article is the microscopic $\mathbf{J} = \sigma\mathbf{E}$ law.

Builds on: The Continuity Equation · Electromotive Force · Maxwell's Equations in Vacuum

Leads to: Motional EMF · The Displacement Current · Self-Inductance

Is Ohm's law a law of nature or an approximation - and when does it fail?

It is a constitutive relation — the same epistemic rank as $\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$ : an excellent linear approximation to a material's response, not a field equation. For metals it is spectacularly good (linear to within parts per million up to enormous current densities, because even "huge" drift velocities are trivial perturbations on the Fermi sea). It fails where the microscopic premises fail: semiconductor junctions (barriers make $J(E)$ exponential — every diode is an Ohm's-law violation sold as a component), gases and plasmas (avalanche breakdown: air at ~3 MV/m), very high frequencies (when the drive period approaches $\tau$ , the inertia term Drude dropped matters — the AC conductivity of plasmas and metals in the optical range), and superconductors (zero $\rho$ , current without any field). The trilogy's fine print, third verse.

If the field inside the wire drives the current, what role does the potential difference actually play?

$V$ is the line integral of that interior field from one terminal to the other: $V = \int\mathbf{E}\cdot d\boldsymbol{\ell} = EL$ for a uniform wire. The battery's job is to hold its terminals at a fixed $V$ (how it does so — the force that is not electrostatic — is precisely the next article's subject); the surface charges then broadcast that boundary condition along the whole circuit, and the interior field is the unique Laplace solution consistent with it. So "potential difference drives current" and "the local field drives current" are the same statement at different zoom levels — with the surface charge as the translator between them. What the potential picture hides, and this article restores, is who produces the local push at a bend forty metres from the battery.

Why is the DC current spread uniformly across the wire's cross-section? I expected it to prefer the surface or the axis.

Uniqueness leaves it no choice. Steady state plus uniform $\sigma$ makes the interior a Laplace problem with insulating side walls and equipotential ends, and the flat axial solution satisfies it — so it is the only solution. Intuitions about crowding import physics that is absent at DC: surface crowding (skin effect) comes from induction fighting flux changes inside the conductor, which needs AC (owned by Electromagnetic Waves in Conductors); axis crowding would need the wire to attract its own current magnetically strongly enough to matter (the pinch effect — real in plasmas at mega-ampere densities, utterly negligible in copper at household $J$ ). At 50–60 Hz the skin depth in copper is ≈ 8.5 mm, so even mains cables below centimetre scale are effectively DC-uniform.

The relaxation argument says imbalances die in about 1e-19 seconds in copper. Is that physical? Nothing is that fast.

Good suspicion — and the honest answer has two layers. The formula $\tau_c = \varepsilon_0/\sigma$ (owned by Continuity Equation) is the bulk linear-response answer, and its copper value ( $\sim10^{-19}$ s) is shorter than the collision time $\tau = 2.5\times10^{-14}$ s, so the quasi-static derivation is eating its own premises: at such scales conduction is not even Ohmic yet. Read it as "unmeasurably fast — the bulk of a metal simply cannot sustain net interior charge," not as a stopwatch value. The circuit-scale transients you can actually measure — the nanoseconds for a switched circuit to settle — are set by geometry: signal travel at near light speed along the wires, capacitances charging through resistances. Fast local physics, geometric global bookkeeping; the doctrine only needs "surface charge wins long before you can look."

Where exactly does the energy for the heat come from - does each electron carry energy down the wire?

Not usefully. The drifting electrons' kinetic energy is laughable: at $v_d = 0.5$ mm/s, the entire drift kinetic energy of all conduction electrons in your 100 m wire is nanojoules — while the wire dissipates 112 W. The electrons are not couriers of energy; they are the medium through which the field does work locally: at each point the field pushes carriers ( $\mathbf{J}\cdot\mathbf{E}$ per volume), and collisions immediately convert the gain to lattice heat. The energy arrives at the resistor through the electromagnetic field surrounding the circuit — flowing in sideways through the resistor's surface, as the Poynting analysis of the conservation chapter (E7) will show in one of the most counterintuitive diagrams in physics. The wire guides; the field delivers; the collisions cash out.

Both metals and semiconductors have collisions - why do their resistivities move opposite ways with temperature?

Because $\sigma = ne^2\tau/m$ has two temperature-sensitive factors and the two material classes are ruled by different ones. In a metal, $n$ is fixed (the Fermi sea neither grows nor shrinks), so only $\tau(T)$ matters: warmer lattice → more phonons → shorter $\tau$ → resistivity rising, roughly linearly above ~100 K. In a semiconductor, $\tau$ does the same thing but is overwhelmed: $n(T) \propto e^{-E_g/2k_BT}$ grows exponentially as heat promotes carriers across the gap, so resistivity plunges. The opposite slopes are the standard two-minute diagnostic separating the classes — and the semiconductor's steep sensitivity is exactly what thermistors sell. Superconductors break the frame entirely: below $T_c$ , $\rho$ is not small but zero, a different phase of matter.

Does Ohm's law say anything about which sign of carrier flows - could copper's current be positive charges moving the other way?

Ohm's law alone cannot tell: $\sigma = nq^2\tau/m$ is even in the charge $q$ , and $\mathbf{J}$ comes out along $\mathbf{E}$ either way — negative carriers drifting upstream produce exactly the same current as positive carriers drifting downstream. The tiebreaker is the Hall effect (named in this chapter's gap list): put the current in a magnetic field and the sideways force $q\mathbf{v}_d\times\mathbf{B}$ depends on the carriers' actual velocity direction, so the sign of the transverse voltage reveals the carrier sign. Copper: negative, as expected. The famous twist — aluminium and some semimetals show positive Hall signs, explained by band-structure "holes" — is quantum solid-state territory, flagged here as one more place Drude's classical picture ends and Bloch's begins.