Electric Current and Current Density

Flip a light switch and the lamp answers in a nanosecond-scale instant. Yet the electrons in the cord are creeping toward the bulb at less than a tenth of a millimetre per second — at that pace, the first electron to leave the switch would reach the lamp in about four hours. Both numbers are correct, both are measured, and holding them in one head without contradiction requires exactly the machinery this article builds: a clean, local description of charge in motion. Magnetostatics needs that machinery before it can compute anything. Charge at rest sources electric fields; charge in motion sources magnetic fields — and "charge in motion" must be sharpened into quantitative objects before the Biot–Savart law can eat them. The scalar current $I$ counts coulombs past a point per second; but a scalar attached to a wire cannot describe a lightning channel's core, the sheet of screening current on a superconductor, or the swirl inside a solid bus bar. The honest source variable is a vector field: the volume current density $\vec J$ , the moving-charge analogue of the charge density $\rho$ , with its sheet-limit sibling $\vec K$ and its thin-wire limit $I$ . These are not three kinds of physics — they are one flowing charge at three levels of geometric idealization, linked by a substitution rule that converts every line-current formula in this chapter into its surface and volume versions. The article builds the dictionary, exposes the microscopic content ( $\vec J = nq\vec v_d$ , whose tiny drift speed resolves the light-switch paradox), and delivers a promise made two articles ago: the carrier-by-carrier bookkeeping that turns the Lorentz force on individual charges into the macroscopic $I\,\mathrm{d}\vec\ell\times\vec B$ on a wire. The deepest constraint on $\vec J$ — charge conservation, and the steady-current condition $\nabla\cdot\vec J = 0$ that defines magnetostatics itself — gets the next article to itself.

Key takeaways

Electric current $I = \mathrm{d}Q/\mathrm{d}t$ counts charge crossing a surface per unit time, in amperes (1 A = 1 C/s).
Current comes in three geometric flavours: line current $I$ (thin wire), surface current density $\vec K$ (current per unit width on a sheet, A/m), and volume current density $\vec J$ (current per unit area, A/m²) — one quantity, three idealizations.
Microscopically $\vec J = nq\vec v_d = \rho_{\rm mob}\vec v_d$ : carrier density times drift velocity. In household copper the drift is $\sim 0.07$ mm/s — the signal travels near $c$ because the field does, not the electrons.
The total current through any surface is the flux $I = \int_S \vec J\cdot\mathrm{d}\vec a$ ; squeeze the distribution onto a sheet or a line and $\vec J\,\mathrm{d}\tau \to \vec K\,\mathrm{d}a \to I\,\mathrm{d}\vec\ell$ .
Summing the Lorentz force over carriers gives the force chain $\mathrm{d}\vec F = \vec J\times\vec B\,\mathrm{d}\tau = I\,\mathrm{d}\vec\ell\times\vec B$ — the substitution rule, owned here and spent by every following article.
Charge conservation constrains $\vec J$ through the continuity equation; its derivation and the steady-current condition $\nabla\cdot\vec J = 0$ are the business of the next article.
KEY IDEA: current is the flux of a vector field $\vec J = nq\vec v_d$ — line, sheet, and volume currents are the same flowing charge at three idealizations, and the substitution rule $I\,\mathrm{d}\vec\ell \leftrightarrow \vec K\,\mathrm{d}a \leftrightarrow \vec J\,\mathrm{d}\tau$ converts every formula between them.

The core insight

Current is the flux of a vector field $\vec J$ , the moving-charge analogue of charge density $\rho$ . Line current $I$ , surface current $\vec K$ , and volume current $\vec J$ are one physical quantity at three geometric idealizations, linked by the substitution rule $I\,\mathrm{d}\vec\ell \to \vec K\,\mathrm{d}a \to \vec J\,\mathrm{d}\tau$ , and all are bound to $\rho$ by charge conservation.

Where this lives today

Since 2019 the ampere is defined by counting: the SI fixes $e$ exactly, and single-electron pump standards literally shuttle electrons one at a time at gigahertz rates to realize the unit — current metrology as applied equation (1). At the opposite extreme of scale, power engineering lives in the gap between $I$ and $\vec J$ : skin effect crowds 50 Hz current into the outer centimetre of a transmission cable and shoves RF current into micrometres, which is why high-frequency conductors are woven from insulated strands (Litz wire) and busbars are hollow. Battery and semiconductor design track multiple carrier species with separate $n_iq_i\vec v_i$ ledgers — electrons and holes, lithium ions and counter-ions — all summed into one $\vec J$ . The sheet current $\vec K$ , introduced here as an idealization, is a load-bearing citizen of modern physics: superconductors screen magnetic fields with genuinely surface-confined currents (the Meissner effect), magnetized matter will shortly be replaced — exactly, not approximately — by bound surface currents $\vec K_b = \vec M\times\hat{\vec n}$ , and the quantum Hall effect's edge channels are nature's most perfect current sheets, quantized so sharply they anchored the resistance standard. When the Biot–Savart article computes fields "of a solenoid" it will really be computing the field of a $\vec K$ ; the dictionary built here is spent everywhere.

Origins · 1 minGeorg Simon Ohm and how this was discovered

Georg Simon Ohm

1789–1854 · Cologne, Nuremberg, Munich

Ohm did current science before current science was respectable — a Gymnasium mathematics teacher measuring, with a torsion balance and thermocouple sources of his own devising, how the "strength of the galvanic current" depended on the wire that carried it. His 1827 Die galvanische Kette, mathematisch bearbeitet laid out the proportionality between driving tension and current and the role of the conductor's length and section — the content of $\vec J = \sigma\vec E$ and everything circuit theory calls $V = IR$ . German academic opinion savaged the book (one reviewer called it "a web of naked fancies"), and Ohm resigned his post; vindication came from abroad, with the Royal Society's Copley Medal in 1841 and, at last, a Munich professorship in 1849. The unit of resistance carries his name; the microscopic form of his law, deferred from this article to the electrodynamics chapter, still opens every treatment of conduction.

History

1800

— Volta's pile delivers the first steady current, turning electricity from spark to flow and making current science possible.

1820

— Ørsted's compass twitches beside a wire: current, not static charge, is revealed as the source of magnetism.

1827

— Ohm publishes Die galvanische Kette: current responds linearly to driving tension through a conductor's geometry.

1879

— Hall's experiment probes the carriers inside the wire, showing the current's sign and density are measurable.

1897

— Thomson identifies the carrier itself: the electron, whose drift is the J = nqv of every metal.

1900

— Drude models conduction as electrons diffusing through a lattice, giving sigma its first microscopic pedigree.

2019

— The SI redefines the ampere by fixing e exactly; current standards count electrons one by one.

The physical picture

Picture a river. The total flow past a bridge — litres per second — is one number, the analogue of $I$ . But to understand the river you need the local story: how fast the water moves here, in what direction, how the flow crowds toward the deep channel and dies at the banks. That local information is the current density $\vec J$ : a vector at every point, aimed along the flow, whose magnitude is the charge crossing a tiny perpendicular window per second per unit area. Integrate $\vec J$ across the whole cross-section and the total discharge $I$ comes back. The hierarchy line → sheet → volume is nothing but a choice of which dimensions to stop caring about. A real wire is a volume current: $\vec J$ fills its interior, possibly unevenly (at high frequency it crowds toward the surface — the skin effect). Stand far enough away and the cross-section becomes invisible; only the total $I$ along a curve matters — a line current. In between sits the sheet: current flowing in a layer so thin you track only current per unit width, the surface density $\vec K$ . Solenoid windings smeared into a shell, the screening currents on a superconductor, the bound currents on a magnetized bar (two articles' worth of E5 material) — all live in the $\vec K$ column of the dictionary.

Figure

One flowing charge, three idealizations — the current dictionary

Now the microscopic content. A wire is not a pipe that fills with electrons when you flip the switch — it is a pipe already full, end to end. Switching on establishes an electric field along the whole circuit at nearly the speed of light (guided by surface charges on the wire itself), and that field starts all the electrons creeping at once, everywhere. The lamp lights when the electrons already inside it begin to move — not when electrons from the switch arrive. The hose was already full of water; the pressure wave, not the water, carries the message.

Figure

Why the lamp is instant while electrons crawl — the full-hose picture

The dictionary, quantified

Start with the number every meter reads. Pick a surface — the cross-section of a wire, say — and count the net charge crossing it:

I = \frac{\mathrm{d}Q}{\mathrm{d}t} \qquad (1)

where: $I$ — current in amperes ( $1\ \mathrm{A} = 1\ \mathrm{C/s}$ ); $Q$ — net charge that has crossed the chosen surface. Signs matter twice: positive charge moving left-to-right and negative charge moving right-to-left both count as current left-to-right. Since 2019 the SI defines the ampere by fixing $e = 1.602176634\times10^{-19}\ \mathrm{C}$ exactly — an ampere is a prescribed number of elementary charges per second. The local upgrade replaces the count with a field. At each point, let $\vec J$ point along the local flow with magnitude equal to current per unit area through a window held perpendicular to it; then the count through any surface $S$ is the flux

I = \int_S \vec J\cdot\mathrm{d}\vec a \qquad (2)

where: $\mathrm{d}\vec a$ — oriented area element. The dot product does the geometric bookkeeping: a window held obliquely to the flow catches proportionally less. Equation (2) is the exact analogue of "charge = volume integral of $\rho$ " — $\vec J$ is to moving charge what $\rho$ is to sitting charge, and in Maxwell's equations they will appear side by side as the two source fields. What is $\vec J$ made of? Carriers. If $n$ carriers per unit volume, each of charge $q$ , share an average (drift) velocity $\vec v_d$ , then

\vec J = nq\,\vec v_d = \rho_{\rm mob}\,\vec v_d \qquad (3)

where: $\rho_{\rm mob} = nq$ — the mobile charge density (a neutral wire has $\rho = 0$ overall but $\rho_{\rm mob} \neq 0$ ; the lattice ions cancel the charge, not the current). Read the structure: current is charge density in flight. Two very different rivers can carry the same $\vec J$ — few fast carriers or many slow ones — and metals sit at the extreme "many and slow" end, as the worked example quantifies. When several species move at once (an electrolyte's positive and negative ions drifting opposite ways), each contributes $n_iq_i\vec v_i$ , and opposite charges moving oppositely add. The sheet version replaces volume density with surface density: charge $\sigma_{\rm mob}$ per unit area, moving at $\vec v$ , gives

\vec K = \sigma_{\rm mob}\,\vec v, \qquad I = \int K\,\mathrm{d}\ell_\perp \qquad (4)

where: $\vec K$ — surface current density in $\mathrm{A/m}$ : current per unit width measured across the flow. The units are the perennial trap — $\vec K$ divides by a width, $\vec J$ by an area — and the limiting-cases table below shows each is the squeezed limit of the one above. The three source elements that will feed the Biot–Savart law and the force law line up as the substitution rule:

I\,\mathrm{d}\vec\ell \;\longleftrightarrow\; \vec K\,\mathrm{d}a \;\longleftrightarrow\; \vec J\,\mathrm{d}\tau \qquad (5)

where: each element carries the units A·m and plays the identical role in every magnetostatic formula: write the physics once for the line element, translate freely. The rule is not a mnemonic but a theorem, and its proof is the Lorentz-force assembly below — the derivation Lorentz Force Law deferred to this article by name. One more law belongs in the dictionary by association: in ohmic conductors the drift responds linearly to the driving field, $\vec J = \sigma\vec E$ , with $\sigma$ the conductivity. That constitutive relation — Ohm's law in its microscopic, geometry-free form — is named here so you recognize $\vec J$ 's most common employer, and deferred in full (mechanism, Drude picture, validity limits) to Ohm's Law and Conductivity in the electrodynamics chapter, where it properly lives.

Derivation AJ equals nqv — counting carriers through a window

The goal is the microscopic formula $\vec J = nq\vec v_d$ , the bridge between the field-level description and the carriers that actually do the flowing — and with it the widely used scalar form $I = nqAv_d$ . The strategy is the swept-volume count: freeze a small window of area $A$ perpendicular to the drift, wait a short time $\mathrm{d}t$ , and ask which carriers made it through. The one subtlety worth watching: only the drift velocity counts, because the enormous random thermal velocities sweep as many carriers backward as forward and cancel from the net.

Step 1. Identify who crosses. In time $\mathrm{d}t$ every carrier moves $v_d\,\mathrm{d}t$ downstream (on average), so the carriers that cross the window are exactly those currently inside the slab of volume

\mathrm{d}V = A\,v_d\,\mathrm{d}t.

Step 2. Count their charge. With $n$ carriers per unit volume, each of charge $q$ :

\mathrm{d}Q = nq\,\mathrm{d}V = nq\,A\,v_d\,\mathrm{d}t.

Step 3. Divide by the time to get the current, and by the area to get the density:

I = \frac{\mathrm{d}Q}{\mathrm{d}t} = nqAv_d, \qquad J = \frac{I}{A} = nq\,v_d.

Promote to vectors along the drift: $\vec J = nq\vec v_d$ . For multiple species, sum: $\vec J = \sum_i n_iq_i\vec v_i$ . Why the thermal chaos drops out: each carrier's true velocity is $\vec v_{\rm thermal} + \vec v_d$ with $|\vec v_{\rm thermal}|$ up to $10^6$ m/s in copper — but $\langle\vec v_{\rm thermal}\rangle = 0$ by isotropy, so the average that (3) needs is the drift alone. The current is a whisper of order on top of a hurricane of noise. *Verified: dimensions $[\mathrm{m^{-3}}][\mathrm{C}][\mathrm{m/s}] = \mathrm{A/m^2}$ ✓; the formula reproduces the standard Griffiths §5.1.3 / Purcell Ch. 4 result and feeds the drift-speed estimate of worked example 1.*

Go deeper · Derivation B · 1 minRecovering I from a nonuniform J — the flux integral at work

Derivation BRecovering I from a nonuniform J — the flux integral at work

The goal is practice with equation (2) in the case real conductors present: a current density that varies across the section. Take a cylindrical wire of radius $R$ carrying $\vec J = J_0\,(s/R)\,\hat{\vec z}$ — zero on the axis, maximal at the surface, the qualitative profile of high-frequency (skin-effect) conduction. The strategy is straight integration in polar coordinates; the point to absorb is that $I$ and $\vec J$ carry genuinely different information, related only through the geometry of the section.

Step 1. Set up the flux with the polar area element $\mathrm{d}a = s\,\mathrm{d}s\,\mathrm{d}\phi$ and $\vec J \parallel \mathrm{d}\vec a$ :

I = \int_S \vec J\cdot\mathrm{d}\vec a = \int_0^{2\pi}\!\!\int_0^{R} J_0\,\frac{s}{R}\;s\,\mathrm{d}s\,\mathrm{d}\phi.

Step 2. The angular integral contributes $2\pi$ (nothing depends on $\phi$ ):

I = \frac{2\pi J_0}{R}\int_0^R s^2\,\mathrm{d}s.

Step 3. Do the radial integral and collect:

I = \frac{2\pi J_0}{R}\cdot\frac{R^3}{3} = \frac{2\pi}{3}\,J_0R^2.

Why the comparison matters: a uniform density $J_0$ through the same section would carry $\pi R^2 J_0$ — our profile carries only $2/3$ of that, because the $s$ -weighting starves the small-radius region twice over (little density there, and little area). Numbers: $J_0 = 2\times10^6\ \mathrm{A/m^2}$ in a 1-mm-radius wire gives $I = 4.2$ A versus 6.3 A uniform. Ammeters read the integral; the field inside the wire, the heating profile, and the skin-effect design brief all need the density. *Verified: direct integration ✓; uniform-limit cross-check $\pi R^2J_0$ recovered by dropping the $s/R$ factor; numbers recomputed for this edition.*

Go deeper · Derivation C · 1 minThe substitution rule from the Lorentz force — force on a current element

Derivation CThe substitution rule from the Lorentz force — force on a current element

The goal is the theorem promised in Lorentz Force Law: summing the magnetic force $q\vec v\times\vec B$ over the carriers in a conductor produces the macroscopic chain $\mathrm{d}\vec F = \vec J\times\vec B\,\mathrm{d}\tau \to I\,\mathrm{d}\vec\ell\times\vec B$ , which simultaneously proves the substitution rule (5). The strategy is two applications of the previous derivations: convert carrier count to $\vec J$ , then collapse the geometry. Nothing new enters — that is the point; the rule is bookkeeping, exact whenever the idealization is.

Step 1. Force per unit volume. Each of the $n$ carriers per unit volume feels $q\vec v_d\times\vec B$ (the drift is what survives averaging, as in Derivation A), so

\vec f = nq\,\vec v_d\times\vec B = \vec J\times\vec B.

Why this line is the whole continuum Lorentz force: it is the magnetic half of $\vec f = \rho\vec E + \vec J\times\vec B$ stated in the chapter opener, now with its microscopic pedigree shown.

Step 2. Force on a volume element: $\mathrm{d}\vec F = \vec J\times\vec B\,\mathrm{d}\tau$ . Integrated over a thick conductor, this is the working formula for bus bars, plasmas, and coil stresses.

Step 3. Collapse to a sheet. For current in a thin layer of thickness $t$ , write $\mathrm{d}\tau = t\,\mathrm{d}a$ and let $t \to 0$ with $\vec K = \vec J t$ held fixed: $\mathrm{d}\vec F = \vec K\times\vec B\,\mathrm{d}a$ .

Step 4. Collapse to a wire. For a thin wire of section $A$ , $\vec J$ runs along the wire, so $\vec J\,\mathrm{d}\tau = \vec J\,(A\,\mathrm{d}\ell) = (JA)\,\mathrm{d}\vec\ell = I\,\mathrm{d}\vec\ell$ , and

\mathrm{d}\vec F = I\,\mathrm{d}\vec\ell\times\vec B, \qquad \vec F = I\!\int\mathrm{d}\vec\ell\times\vec B.

Why the same three lines prove the general rule: nowhere did the force matter — Steps 3 and 4 only manipulated the source element $\vec J\,\mathrm{d}\tau$ itself. Any magnetostatic formula built on $I\,\mathrm{d}\vec\ell$ (the Biot–Savart law, next-but-one) converts to sheets and volumes by the same substitutions, which is how the field of a solenoid, a rotating charged sphere, or a solid conductor will be computed in the articles ahead. *Verified: reduces to $\vec F = I\vec L\times\vec B$ for a straight segment in uniform field (Griffiths §5.1.2–5.1.3); dimensional chain $\mathrm{A\cdot m\cdot T} = \mathrm{N}$ ✓; applications of this formula (parallel wires, torques on loops) are owned by* Magnetic Force on Current-Carrying Wires.

Limiting cases

Case	Result	Meaning
volume → sheet	$\vec K = \lim_{t\to0}\vec J\,t$	slab squeezed at fixed $\vec Jt$ ; units drop A/m² → A/m
sheet → line	$I = \lim_{w\to0}Kw$	strip squeezed at fixed $Kw$ ; the thin-wire fiction
single species at rest	$\vec J = 0$ , $\rho \neq 0$	electrostatics — charge without current
neutral conductor carrying current	$\rho = 0$ , $\vec J \neq 0$	current without charge — the magnetostatic workhorse
two species counter-drifting	contributions add	electrolytes, plasmas: opposite charge × opposite velocity = same-sign current
steady state	$\nabla\cdot\vec J = 0$	no charge piles up; current lines close — derived next article
ohmic response	$\vec J = \sigma\vec E$	constitutive, not fundamental — owned by the electrodynamics chapter

Common trapDrift speed is not signal speed — and K is not J

The two classic stumbles of this topic. First: electrons in a lamp cord drift at $\sim 0.07$ mm/s (worked example 1), yet the lamp is instant — because the field, guided by surface charges on the wires, establishes itself around the circuit at an appreciable fraction of $c$ and sets every carrier moving at once. The wire was already full. Saying "electrons rush from the switch to the bulb" gets the physics wrong by twelve orders of magnitude. Second: the densities divide by different geometry — $\vec J$ by an area (A/m²), $\vec K$ by a width (A/m) — and plugging one where the other belongs is the most common dimensional error in magnetostatics. Check units before integrating: flux of $\vec J$ over an area, flux of $\vec K$ across a width, both landing in amperes.

Worked examples

1 · How fast do the electrons actually go? (confidence builder)

A copper wire of cross-section $A = 1.0\ \mathrm{mm^2}$ carries $I = 1.0\ \mathrm{A}$ . Copper: density $8.96\times10^{3}\ \mathrm{kg/m^3}$ , molar mass $63.5\ \mathrm{g/mol}$ , one conduction electron per atom. Find the carrier density, the drift speed, and the time for an electron to travel the 1.0 m to a lamp.

Step 1. Carrier density from the atom count: $n = \dfrac{\rho_m N_A}{M} = \dfrac{(8.96\times10^{3})(6.022\times10^{23})}{0.0635} = 8.5\times10^{28}\ \mathrm{m^{-3}}$ — a hundred million times denser than the air's molecules; this astronomical $n$ is the secret of the ampere.

Step 2. Drift speed from $I = nqAv_d$ : $v_d = \dfrac{I}{nqA} = \dfrac{1.0}{(8.5\times10^{28})(1.60\times10^{-19})(10^{-6})} = \boxed{7.4\times10^{-5}\ \mathrm{m/s}}$ — seven-hundredths of a millimetre per second, slower than a garden snail by a factor of ten.

Step 3. Time to the lamp: $t = \dfrac{1.0}{7.4\times10^{-5}} = 1.4\times10^{4}\ \mathrm{s} \approx \boxed{3.8\ \mathrm{hours}}$ . The lamp, of course, lit in nanoseconds — the resolution is the full-hose picture, not fast electrons.

Step 4. Perspective check: the same electrons' thermal speeds are $\sim 10^{6}\ \mathrm{m/s}$ — the drift is a part-in- $10^{10}$ bias on the chaos, yet it carries every ampere civilization runs on.

2 · A spinning charged disk — current without a wire (exam level)

A plastic disk of radius $R = 10\ \mathrm{cm}$ carries uniform surface charge $\sigma = 10\ \mathrm{\mu C/m^2}$ and spins at $\omega = 100\ \mathrm{rad/s}$ about its axis. Find the surface current density $\vec K(s)$ and the total current passing any radial line.

Step 1. Identify the flow: charge at radius $s$ moves azimuthally at speed $v = \omega s$ , so by (4)

\vec K(s) = \sigma\,\omega s\,\hat{\boldsymbol\phi}

growing linearly from zero at the center to $K(R) = (10^{-5})(100)(0.10) = \boxed{1.0\times10^{-4}\ \mathrm{A/m}}$ at the rim. No battery, no wire — bodily transported charge is current, full stop.

Step 2. Total current across a radial cut (each ring of width $\mathrm{d}s$ is a loop of current $\mathrm{d}I = K\,\mathrm{d}s$ ):

I = \int_0^R K\,\mathrm{d}s = \sigma\omega\int_0^R s\,\mathrm{d}s = \frac{\sigma\omega R^2}{2} = \frac{(10^{-5})(100)(0.10)^2}{2} = \boxed{5.0\ \mathrm{\mu A}}.

Step 3. Sanity checks: dimensions $[\sigma][\omega][R^2] = (\mathrm{C/m^2})(\mathrm{s^{-1}})(\mathrm{m^2}) = \mathrm{A}$ ✓; and each element of the disk crosses the radial line once per period $T = 2\pi/\omega$ , so $I = Q_{\rm total}/T = \sigma\pi R^2\,\omega/2\pi = \sigma\omega R^2/2$ ✓ — the same answer by pure counting.

Step 4. Why this matters downstream: exactly this $\vec K(s)$ , fed through the substitution rule into the Biot–Savart law, is how the field of rotating charged bodies (Griffiths's spinning shell, planetary dynamo cartoons) will be computed in the articles ahead.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.1.3 (Currents) · Jackson, Classical Electrodynamics, 3rd ed., §5.1–5.3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 4 · Zangwill, Modern Electrodynamics, §9.1–9.3.

Provenance. Definitions and the $\mathbf{J}=nq\mathbf{v}_d$ relation follow Griffiths §5.1.3 and Purcell Ch. 4. Constants used: elementary charge $e=1.602\times10^{-19}$ C, Avogadro $N_A=6.022\times10^{23}$ mol⁻¹, copper density $8.96\times10^3$ kg/m³, molar mass $63.5$ g/mol — all standard. A verifier should recompute the drift-speed example ( $n\approx8.5\times10^{28}$ m⁻³, $v_d\approx7\times10^{-5}$ m/s) and the cross-section integral $I=\tfrac{2}{3}\pi R^2 J_0$ . Continuity derivation cross-checks against Jackson §5.1 and Zangwill §9.1. (In this edition the continuity derivation itself is owned by the next article, Continuity Equation; this article states the constraint and defers by name.) Worked numbers recomputed: $v_d = 7.4\times10^{-5}$ m/s, 3.8 h per metre; spinning disk $K(R) = 1.0\times10^{-4}$ A/m, $I = \sigma\omega R^2/2 = 5.0\ \mathrm{\mu A}$ , cross-checked by the period-counting route; nonuniform- $J$ wire 4.2 A vs 6.3 A uniform.

Builds on: Lorentz Force Law · Magnetic Force on Moving Charges

Leads to: Continuity Equation · Biot-Savart Law · Magnetic Force on Current-Carrying Wires · Ampère's Law · Magnetic Field of Steady Current Configurations

If the drift speed is a snail's pace, why are electrical signals and radio waves so fast?

Because the signal never was the carriers. What travels is the electromagnetic field — configured and guided by minute surface charges that arrange themselves on the conductors within nanoseconds — and fields in and around cables propagate at a good fraction of $c$ (typically $0.6$ – $0.9c$ , set by the insulation's permittivity). The electrons everywhere along the line respond to the arriving field locally and at once. The hose analogy is exact in structure: pressure wave fast, water slow; field fast, electrons slow.

A current-carrying wire has moving charge in it — why isn't it electrically charged?

Because the lattice of positive ions sits still and cancels the conduction electrons' charge point for point: $\rho = 0$ even though $\vec J = nq\vec v_d \neq 0$ . Charge density and current density are independent source fields — you can have either without the other (a charged balloon: $\rho$ without $\vec J$ ; a neutral wire: $\vec J$ without $\rho$ ). This independence is why magnetostatics is a separate subject from electrostatics, and (a relativistic teaser) why a boost can trade one for the other: a wire neutral in the lab is charged in the frame of a moving test particle, which is precisely how magnetism reveals itself as relativistic electricity.

Is the drift velocity the same for every carrier, as the formula seems to assume?

No — individual carriers have wildly different instantaneous velocities, dominated by thermal chaos at $\sim10^6$ m/s in a metal. $\vec v_d$ in $\vec J = nq\vec v_d$ is the ensemble average velocity, and the derivation only needs that average: the random parts sweep equal charge both ways through any window and cancel. When the average itself varies across the conductor (skin effect, temperature gradients), $\vec J$ simply becomes position-dependent, which is exactly why the field description beats a single scalar.

Why do we need three descriptions — wouldn't J alone, being the most general, suffice?

In principle yes: $I$ and $\vec K$ are limits of $\vec J$ (a genuinely line-like $\vec J$ is a delta function on a curve). In practice the idealizations earn their keep by matching the physics: a metre from a 1-mm wire, the cross-sectional distribution of $\vec J$ affects $\vec B$ at the part-per-thousand level, and dragging delta functions through every integral to honor it is pain without payoff. Conversely, some currents are fundamentally sheet-like (superconductor screening, magnetization bound currents) and their natural variable is $\vec K$ . The substitution rule is the guarantee that no physics is lost moving between levels — one theorem, three notations.

Do I use the total charge density or only the mobile part in J = ρv?

Only the movers, weighted each by their own velocity: $\vec J = \sum_i \rho_i\vec v_i$ over species. In a metal wire, the lattice's $\rho_+$ has $\vec v = 0$ and contributes nothing; only $\rho_{\rm mob} = -ne$ drifts. The distinction bites in electrolytes and plasmas, where several species move at once — and note the sign structure: negative charge drifting left is current to the right, so counter-drifting opposite charges reinforce rather than cancel. Writing $\vec J = \rho\vec v$ with the total $\rho$ is correct only for convection currents where all the charge moves bodily together, like the spinning disk of worked example 2.

Does J = σE contradict J = nqv_d?

They answer different questions. $\vec J = nq\vec v_d$ is a definition-level identity — true always, it just says what current density is made of. $\vec J = \sigma\vec E$ is a constitutive claim about a class of materials: in an ohmic conductor, collisions make the drift settle at a velocity proportional to the driving field, so the identity acquires the linear form with $\sigma = nq\mu$ ( $\mu$ the carrier mobility). Superconductors, vacuum tubes, and strongly driven semiconductors carry perfectly good $nq\vec v_d$ currents while flouting $\sigma\vec E$ entirely. The full story — Drude's model, what sets $\sigma$ , where linearity fails — belongs to Ohm's Law and Conductivity in the electrodynamics chapter.

Why is the steady-current condition div J = 0 such a big deal for this chapter?

Because every formula of magnetostatics silently assumes it. Steady currents mean no charge piles up anywhere ( $\partial\rho/\partial t = 0$ ), which by charge conservation forces $\nabla\cdot\vec J = 0$ : current lines close on themselves, circuits are loops, and Kirchhoff's node law holds. The Biot–Savart law and Ampère's law are only consistent for such divergence-free sources — feed them a current that dead-ends and they contradict each other. The next article derives the continuity equation properly, extracts the steady condition, and shows exactly where the magnetostatic approximation lives and dies.