The Hall Effect

In 1879, a graduate student at Johns Hopkins asked a question his textbook said was settled. Maxwell's Treatise claimed the magnetic force acts on the conductor carrying a current, not on the current itself — so a magnet should do nothing measurable inside the metal. Edwin Hall doubted the book, put a gold leaf in a magnet gap, ran a current along it, and found a steady sideways voltage across it — microvolts, but unmistakably there. The magnetic force acts on the moving charges, and they answer by piling up against one edge until their own electric field holds the line. That microvolt became one of the most informative small numbers in physics. Eighteen years before the electron was discovered, its sign announced that the mobile charges in most metals are negative; later, the same test exposed materials that conduct as if their carriers were positive — the "holes" semiconductor electronics is built on. Its size counts the carriers per cubic metre: a direct window into the interior census of a conductor. This closing article of the circuits chapter takes the chapter's workhorse quantities — current, drift velocity, resistivity — and crosses them with the magnetostatics chapter's force to produce a measurement no ohmmeter can make. Master it and you own the physics inside every phone compass, clamp meter, and brushless-motor sensor — and the doorway to the quantum version that now defines the ohm.

Key takeaways

Run current through a flat conductor in a perpendicular field: $q\mathbf{v}\times\mathbf{B}$ shoves the drifting carriers toward one edge, charge piles up, and a transverse Hall field grows until it cancels the shove: $E_H = v_d B$ .
The measurable is the Hall voltage $V_H = IB/(nqt)$ . The sample's width cancels; only the thickness along $\mathbf{B}$ survives.
The sign is the headline: both carrier signs get pushed to the same edge, so the polarity of that edge reveals what actually moves. Metals: negative. Some materials: positive — holes, the clue band theory later explained.
$R_H = 1/nq$ is a carrier-density meter: copper's $n = 8.5\times10^{28}\ \mathrm{m^{-3}}$ makes $V_H$ microvolts, a doped semiconductor's $n \sim 10^{22}$ makes millivolts — why every Hall sensor is a semiconductor.
Pair it with resistivity and you get the mobility, $\mu = |R_H|\,\sigma$ — the microscopic pair $(n, \mu)$ from two bench measurements.
KEY IDEA: a conductor in a magnetic field becomes its own voltmeter for the drift of its charges — reporting their sign, their density, and the field, in one transverse reading.

The core insight

In a current-carrying conductor, a perpendicular magnetic field deflects the drifting carriers until the piled-up edge charge produces a transverse field with $E_H = v_d B$ ; the resulting Hall voltage $V_H = IB/(nqt)$ measures the sign and density of the charge carriers — the interior census of a conductor, read from outside.

Where this lives today

You almost certainly carry several Hall elements right now. The compass in a phone is a trio of semiconductor Hall plates reading the geomagnetic field along three axes; every brushless motor — drone, e-bike, electric car — uses Hall switches to tell the controller where the rotor's magnets are; clamp meters read a wire's current without touching it by concentrating the wire's own $B$ onto a Hall element in a ferrite yoke. Fabs run Hall characterization on every process batch, because $(n, \mu)$ from Derivation C is still the fastest honest census of a doped film. And the effect's quantum edition became metrology's bedrock. Cool a two-dimensional electron gas to kelvin temperatures in a strong field and the Hall resistance locks onto exact plateaus at $R_{xy} = h/(\nu e^2)$ , insensitive to material, geometry, or dirt at parts-per-billion reproducibility. The von Klitzing constant $R_K = h/e^2 = 25{,}812.807\ \Omega$ is exact in the 2019 SI, so the ohm itself is maintained by Hall bars. Why the plateaus are exact — topology, Landau levels, edge states — is quantum-track physics; that a graduate student's microvolt of 1879 ended up defining electrical resistance is the kind of compound interest fundamental physics pays.

The physical picture

Take the chapter's most ordinary object — a flat strip carrying a steady current — and add one ingredient from magnetostatics: a uniform $\mathbf{B}$ through the strip's face. Every drifting carrier now feels $q\mathbf{v}\times\mathbf{B}$ (Magnetic Force on Moving Charges owns that law; we spend it here). The force is sideways — perpendicular to both drift and field — so the carrier river gets pushed toward one bank. For the first instants, that is exactly what happens: carriers land on one edge. But charge piling on an edge is a capacitor charging itself — a transverse field $E_H$ grows between the banks, pushing later carriers back. The pileup is self-limiting: it stops the instant the electric push balances the magnetic shove, and from then on the river runs straight again down a channel with a permanent sideways tilt in its potential. The transient settles within the material's charge-relaxation time — practically instantaneous — after which the strip presents a steady DC voltage between its edges, patiently reporting the balance inside.

Figure

The self-balancing act: magnetic shove piles charge on one bank until the Hall field holds the river straight

Feel the two punchlines coming. First, the equilibrium field must satisfy $E_H = v_d B$ — and the drift velocity in copper is sub-millimetre per second (Current, Resistance, and Resistivity's famous shock). Crossing something that slow with even a tesla gives a tiny field: metallic Hall voltages are microvolts, and Hall's 1879 achievement was partly instrumental nerve. Second, positive and negative carriers making the same current pile onto the same bank with opposite signs of accumulated charge — so the polarity of that bank is an honest witness of what actually moves.

The balance, made quantitative

Derivation AFrom force balance to the Hall voltage

We derive the equilibrium Hall field and then the measurable voltage $V_H = IB/(nqt)$ . The strategy is the standard steady-state one: impose that in equilibrium the net transverse force on a drifting carrier vanishes, then translate carrier language into ammeter language with $I = nqAv_d$ . Two things are worth watching along the way. First, how the microscopic drift velocity gets traded away for laboratory quantities ( $I$ , geometry, $n$ ); second, how the strip's width — seemingly the most relevant dimension, since the voltage is measured across it — cancels out of the final answer entirely.

Step 1. Set the geometry: current $I$ along $x$ through a strip of width $w$ (along $y$ ) and thickness $t$ (along $z$ ), with $\mathbf{B} = B\hat{\mathbf{z}}$ through the face. Carriers of charge $q$ drift at $\mathbf{v}_d$ along the strip; the magnetic force on each is $q\mathbf{v}_d\times\mathbf{B}$ — magnitude $qv_dB$ , directed across the strip.

Step 2. Demand equilibrium. Once the edge charge has built the transverse field $\mathbf{E}_H$ , a drifting carrier feels sideways force $q(E_H - v_dB)$ ; steady straight-line flow requires it to vanish:

E_H = v_d\,B

where: $E_H$ is the Hall field and $v_d$ the drift speed. Read the smallness: $v_d \sim 10^{-4}\ \mathrm{m/s}$ in household copper crossed with a full tesla gives $E_H \sim 10^{-4}\ \mathrm{V/m}$ — real but feeble, because the drift it protects is glacial. Why does equilibrium happen at all, rather than carriers piling up forever? Each arriving carrier strengthens $E_H$ and weakens the net sideways force on the next — textbook negative feedback, relaxing exponentially fast to exact cancellation; the same self-limiting logic as the surface charges of Ohm's Law and Conductivity's steering doctrine.

Step 3. Convert to the voltmeter's reading. The Hall field is uniform across the width, so $V_H = E_H\,w = v_dBw$ . Eliminate the microscopic $v_d$ via the current census $I = nqA\,v_d$ , $A = wt$ (Current, Resistance, and Resistivity owns this counting), so $v_d = I/(nqwt)$ :

V_H = \frac{IB}{nq\,t}

where: $n$ is the carrier number density, $q$ the carrier charge, and $t$ the strip thickness along $\mathbf{B}$ . The width $w$ has cancelled — it entered once through the voltage lever arm and once, inversely, through the drift speed at fixed current. Only the dimension parallel to $\mathbf{B}$ survives: thin samples give big signals, which is why Hall used gold leaf and why sensor chips use films a micrometre thin. Why is the cancellation physical rather than algebraic luck? Widen the strip and the same current spreads over more carriers moving more slowly — a weaker $E_H$ integrated over a longer path. The two effects are one fact, fixed total current, seen twice. Verified: copper, $I = 10$ A, $B = 1$ T, $t = 0.1$ mm, $n = 8.5\times10^{28}$ : $V_H = 10\times1/(8.5\times10^{28}\times1.6\times10^{-19}\times10^{-4}) = 7.3\ \mathrm{\mu V}$ ; the same numbers via $E_H = v_dB$ with $v_d = 7.3\times10^{-4}$ m/s and $w = 1$ cm agree to the digit.

The sign verdict

Here is the part that made the effect historic. Naively, reversing the carrier sign should reverse the drift, reverse the force, and move the pileup to the other bank — leaving the voltage unchanged and the carrier's sign invisible. The geometry says otherwise, and it is worth being slow about why.

Go deeper · Derivation B · 2 minWhy the Hall polarity reveals the sign of the carriers

Derivation BWhy the Hall polarity reveals the sign of the carriers

We show that carriers of opposite sign, arranged to produce the same conventional current, are deflected toward the same edge — so the edge accumulates charge of the carrier's own sign, and the Hall polarity flips with it. The strategy is careful bookkeeping of two sign flips: reverse $q$ and you must reverse $\mathbf{v}_d$ to keep $I$ fixed, and $q\mathbf{v}_d\times\mathbf{B}$ is even under the double flip. Watch the force stay put while the accumulated charge changes sign — that asymmetry is the whole measurement.

Step 1. Fix the laboratory facts: conventional current $I$ along $+\hat{\mathbf{x}}$ , field $\mathbf{B} = B\hat{\mathbf{z}}$ . Case one, positive carriers drifting along $+\hat{\mathbf{x}}$ : the force $q\,v_d\,(\hat{\mathbf{x}}\times\hat{\mathbf{z}}) = -qv_dB\,\hat{\mathbf{y}}$ deflects toward the $-y$ bank; positive charge accumulates there, and that bank sits at higher potential.

Step 2. Case two, negative carriers ( $q = -e$ ) making the same current must drift along $-\hat{\mathbf{x}}$ : the force $(-e)(-v_d)(\hat{\mathbf{x}}\times\hat{\mathbf{z}}) = -ev_dB\,\hat{\mathbf{y}}$ points also toward the $-y$ bank. The two sign flips cancel in the product: the magnetic shove is a property of the current, not the carrier.

Step 3. But the cargo differs. In case two it is negative charge that lands on the $-y$ bank, which now sits at lower potential: the voltmeter reading between the same terminals has reversed. The working rule: the deflected edge is fixed by $I$ and $B$ ; the sign of what piles up there — hence the polarity of $V_H$ — is the sign of the mobile carriers. In the formula, $V_H = IB/(nqt)$ simply carries $q$ with its sign. Why was this a bombshell in 1879? Because no other DC measurement can tell the difference: positive charge flowing right is electrically identical, terminal to terminal, to negative charge flowing left — same $I$ , same $R$ , same heating. The Hall polarity is the rare bulk, room-temperature observable that breaks the tie, and Hall found the metallic sign negative — eighteen years before J.J. Thomson showed the electron existed. Verified: $\hat{\mathbf{x}}\times\hat{\mathbf{z}} = -\hat{\mathbf{y}}$ in a right-handed frame; both cases give force along $-\hat{\mathbf{y}}$ , and the potential assignments follow from which sign of charge terminates the Hall field lines.

And then the plot twisted. Run the same test on zinc, cadmium, or a p-doped semiconductor and the polarity comes out positive — these materials conduct as though their mobile charges were positive, though only electrons are present. Classical physics has no honest way out; the resolution belongs to quantum band theory, where a nearly full band of electrons behaves, in every transport respect, like a dilute gas of positive holes. Within this track we flag the fact and point the road: the Hall sign is the standard bench test for carrier type (n-type versus p-type), and why holes act positive is band-theory business outside this track. What matters here is the integrity of the instrument: the Hall polarity faithfully reports the effective carrier sign, and it was the first instrument that could.

Figure

Same current, same shove, opposite verdict - the deflected edge is fixed, but the sign of what lands there is the carriers own

The carrier census: Hall coefficient and mobility

Go deeper · Derivation C · 2 minR_H = 1/nq, and the two-measurement census of a conductor

Derivation CR_H = 1/nq, and the two-measurement census of a conductor

We package the Hall response into the material constant $R_H$ , then show that pairing it with a resistivity measurement yields both members of the microscopic transport pair $(n, \mu)$ — the census that no amount of pure resistance measurement can perform. The strategy has three moves: define $R_H$ so that geometry drops out, evaluate it for copper and for a doped semiconductor to see the six-orders-of-magnitude landscape, and close the loop through the Drude relation. Watch the division of labour: the derivation of $\sigma = nq\mu$ belongs to Ohm's Law and Conductivity, and we only spend it here.

Step 1. Define the Hall coefficient $R_H \equiv E_H/(JB)$ — transverse field per unit current density per unit field. Substituting $E_H = v_dB$ and $J = nqv_d$ collapses everything microscopic except the census:

R_H = \frac{1}{nq}

where: the sign of $R_H$ is the carrier sign (Derivation B's verdict, now a number) and its magnitude is the reciprocal of charge per unit volume of mobile carriers. In terms of bench quantities, $V_H = R_H\,IB/t$ — measure the slope of $V_H$ against $B$ at fixed $I$ and the geometry hands you $R_H$ .

Step 2. Put in copper: $n = 8.5\times10^{28}\ \mathrm{m^{-3}}$ (one mobile electron per atom — the census Current, Resistance, and Resistivity used for drift velocity), $q = -e$ : $|R_H| = 1/(8.5\times10^{28}\times1.6\times10^{-19}) = 7.3\times10^{-11}\ \mathrm{m^3/C}$ , negative sign. A doped semiconductor at $n = 10^{22}\ \mathrm{m^{-3}}$ gives $|R_H| = 6.2\times10^{-4}\ \mathrm{m^3/C}$ : same field, same current density, a signal millions of times larger. Fewer carriers must drift faster to carry the current, and the Hall voltage measures the drift — the entire economics of the sensor industry in one comparison: metals for wiring, semiconductors for sensing.

Step 3. Close the loop with mobility. Conductivity factorizes as $\sigma = nq\mu$ (Drude bookkeeping owned by Ohm's Law and Conductivity); multiply by $|R_H| = 1/nq$ and the census cancels the other way:

\mu = |R_H|\,\sigma

where: both factors on the right are bench measurements. Copper: $\mu = 7.3\times10^{-11} \times 5.96\times10^{7} = 4.4\times10^{-3}\ \mathrm{m^2/(V\,s)}$ — 44 cm²/Vs, extracted without ever seeing an electron. The pair $(n, \mu)$ from ( $R_H$ , $\sigma$ ) is Hall characterization, daily routine in every semiconductor fab. Why can't resistivity alone deliver this? Because $\sigma = nq\mu$ is one equation in two unknowns — many slow carriers and few fast ones can share the same $\sigma$ ; the Hall measurement supplies the second, independent equation. (Honesty for metals: free-electron $R_H$ matches copper, silver, gold well, but band effects push zinc and cadmium to the wrong sign entirely — the measurement is honest, the free-carrier interpretation fails, and holes are the repair.) Verified: $1/(8.5\times10^{28}\times1.602\times10^{-19}) = 7.34\times10^{-11}$ ; $7.34\times10^{-11}\times5.96\times10^{7} = 4.38\times10^{-3}\ \mathrm{m^2/Vs}$ ; semiconductor $R_H$ recomputes to $6.24\times10^{-4}\ \mathrm{m^3/C}$ .

One compact way to say how feeble the effect is in metals: the Hall angle — the tilt of the total electric field away from the current direction — satisfies $\tan\theta_H = \mu B$ . Copper at a full tesla: $\theta_H \approx 0.25°$ ; a semiconductor with $\mu = 0.1\ \mathrm{m^2/Vs}$ manages $5.7°$ ; mobility-engineered materials (InSb, graphene) reach tens of degrees, where magnetoresistance — the longitudinal cousin — becomes a usable signal too.

Limiting cases

case	result	meaning
$B \to 0$	$V_H \to 0$ linearly	the linear regime — basis of Hall magnetometry
$t \to$ thin film	$V_H \propto 1/t$ grows	why sensors are micrometre films and Hall used gold leaf
$n$ huge (metal)	$V_H$ microvolts	metals: great wires, poor sensors
$n$ small (doped semiconductor)	$V_H$ millivolts	the sensor regime; sign gives n-type vs p-type
carriers positive	$V_H$ polarity flips	the 1879 verdict; the p-type bench test
two carrier types together	$R_H$ from a mobility-weighted competition	partial cancellation; sign can flip with temperature in semimetals
quantum limit: 2D electrons, low T, high B	$R_{xy} = h/(\nu e^2)$ in exact plateaus	quantized Hall resistance, $h/e^2 = 25{,}812.807\ \Omega$ — the resistance standard

Common trapThe magnetic force does the pushing, so it must be doing the work

It never does — $q\mathbf{v}\times\mathbf{B}$ is always perpendicular to $\mathbf{v}$ , so its power $\mathbf{F}\cdot\mathbf{v}$ is identically zero, in the Hall geometry as everywhere else (Magnetic Force on Moving Charges proves the theorem; the Hall strip is a clean place to believe it). During the brief transient the work of separating charges is paid by the source driving the current; in steady state the transverse forces balance and nothing moves sideways at all. A related trap: the Hall voltage is not an EMF — an ideal voltmeter reads $V_H$ precisely because no transverse current flows, and drawing real current from the Hall terminals disturbs the very balance being measured (sensors use high input impedance for exactly this reason). The strip is a reporter of the interior balance, not a sideways battery.

Worked examples

1 · Why nobody notices the Hall effect in their extension cord (confidence builder)

A copper strip 1 cm wide and 0.1 mm thick carries $I = 10$ A across the face of a strong lab magnet, $B = 1$ T. Find the drift speed, the Hall field, and the Hall voltage. Copper: $n = 8.5\times10^{28}\ \mathrm{m^{-3}}$ .

Step 1. Drift speed from the current census: $v_d = I/(nqwt) = 10/(8.5\times10^{28}\times1.6\times10^{-19}\times0.01\times10^{-4}) = 7.3\times10^{-4}\ \mathrm{m/s}$ — under a millimetre per second, the chapter's recurring shock.

Step 2. Hall field from the balance: $E_H = v_dB = 7.3\times10^{-4}\ \mathrm{V/m}$ .

Step 3. Voltage across the width: $V_H = E_H w = 7.3\times10^{-4}\times0.01 = \mathbf{7.3\ \mu V}$ — or in one step, $V_H = IB/(nqt)$ , same digits. Ten amps, a tesla, and the answer is microvolts: detecting this in 1879 with a mirror galvanometer and no amplifier was the experimental feat. The measured polarity corresponds to negative carriers.

2 · Designing a Hall probe, then reading a mystery wafer (exam level)

(a) A Hall sensor is a doped-silicon film, $n = 1.0\times10^{22}\ \mathrm{m^{-3}}$ (n-type), thickness $t = 10\ \mathrm{\mu m}$ , biased at $I = 1.0$ mA. Find its sensitivity in volts per tesla, and its output in the Earth's field ( $50\ \mathrm{\mu T}$ ). (b) A different wafer, $t = 25\ \mathrm{\mu m}$ , biased at 2.0 mA in $B = 0.50$ T, shows $V_H = +8.0$ mV with the polarity of positive carriers. Find its carrier type and density.

Step 1. Sensitivity: $V_H/B = I/(nqt) = 10^{-3}/(10^{22}\times1.6\times10^{-19}\times10^{-5}) = \mathbf{62\ mV/T}$ — Derivation C's $1/n$ economics in action: at one ten-thousandth of Example 1's current, vastly more signal.

Step 2. Earth's field: $V_H = 0.062\times50\times10^{-6} = \mathbf{3.1\ \mu V}$ . Compass chips must amplify and chop to dig microvolts out of offset and noise — Hall physics sets the signal, electronics does the rescue.

Step 3. The mystery wafer: positive polarity means holes — p-type. Density: $n = IB/(qtV_H) = (2.0\times10^{-3}\times0.50)/(1.6\times10^{-19}\times25\times10^{-6}\times8.0\times10^{-3}) = \mathbf{3.1\times10^{22}\ \mathrm{m^{-3}}}$ . Two terminals, one magnet — and you have read the doping of the crystal. (Check: running it backwards returns $V_H = 8.1$ mV, consistent within rounding.)

The graduate student who out-measured Maxwell

Origins · 1 minEdwin Herbert Hall and how this was discovered

Edwin Herbert Hall

1855–1938 · Johns Hopkins, then Harvard — the student who checked the Treatise's footnote

Hall was 24, a graduate student in Henry Rowland's Baltimore laboratory, when he read Maxwell's assertion that the magnetic force acts "not on the electric current, but on the conductor which carries it." Rowland suspected otherwise and set his student the test. Hall's first tries on solid wires showed nothing — the effect scales as $1/(nt)$ , and thick metal buries it. The decisive move was experimental taste: switch to gold leaf, where the tiny thickness multiplies the signal, and in October 1879 the galvanometer moved — a steady transverse potential, proportional to both current and field. It was a bulk measurement of something nobody had ever isolated: the sign of electricity's moving part, reported negative for most metals eighteen years before Thomson's electron. Hall spent his Harvard career on metallic transport, including the "anomalous" positive-sign metals that would wait half a century for band theory's holes; the quantized version of his thesis voltage, found by Klaus von Klitzing in 1980, now anchors the SI ohm.

History

1879

— Edwin Hall, following Rowland's hunch against Maxwell's Treatise, measures a transverse voltage in gold leaf

1880

— Hall documents the "anomalous" positive sign in some metals — the puzzle that waits for band theory

1897

— J.J. Thomson identifies the electron; the metallic Hall sign acquires its carrier

1929

— Peierls explains positive Hall coefficients via nearly filled bands: holes

1948

— the transistor era begins; Hall characterization becomes the fab's standard doping census

1980

— von Klitzing finds the Hall resistance of a 2D electron gas quantized in exact steps of h/e² — since 1990 the world's resistance standard

Going further

This is the track's final article (EMAG serial 0685): with this circuits chapter's fourteen articles joining the 239 of the main sequence, the SuperSTEM electromagnetism track stands complete at 253 — from Scalar and Vector Fields to here, the on-ramp connecting Halliday-level circuit craft to the track's Griffiths-and-beyond spine.

Back-references spent here: Magnetic Force on Moving Charges (the force law and its zero-work theorem) · Current, Resistance, and Resistivity (drift velocity and the current census $I = nqAv_d$ ) · Ohm's Law and Conductivity (Drude's $\sigma = nq\mu$ and the surface-charge steering doctrine).

Beyond the track (plain labels): Quantum Hall Effect · Semiconductor Band Theory (holes, doping) · Spin and Anomalous Hall Effects · Magnetoresistance and Sensor Engineering. Provenance: Authored fresh for the SuperSTEM E&M track's circuits chapter (recon-§d gap heal); sources: v3 circuit pilots (rewritten to UG register) + legacy wiki circuit articles (narrative) — no AllQFT base exists for this topic.

If the carriers in metals are electrons moving backwards, why do we still draw conventional current forwards?

Because for every measurement except the Hall effect (and a few relatives), the two descriptions are exactly equivalent — negative charge flowing left is the same current, field, and heating as positive charge flowing right. Franklin's convention predates the electron by a century and a half, and rewiring the world's diagrams would buy nothing: circuit analysis is sign-convention-blind. The Hall effect is precisely the exception that lets you check the convention against reality — which is why it was historic.

Why did Hall need gold leaf when a thick copper bar carries much more current?

Because thickness sits in the denominator of V_H = IB/(nqt): a thick bar spreads the same current over more, slower carriers, and the balance settles at a feebler field. Halving t doubles V_H at fixed current, so the winning strategy is the thinnest film that still carries your current — gold leaf in 1879, micrometre epilayers today. (Raising I helps too, but ohmic heating in a thin film caps that lever quickly.)

Is the Hall voltage an EMF? Could I use the strip as a battery for a sideways circuit?

No — and the distinction is worth owning. An EMF is a seat of energy conversion that can drive charge around a closed loop (EMF and Internal Resistance's pump concept); the Hall voltage is an electrostatic potential difference maintained by fixed edge charge in balance with the magnetic shove. Draw transverse current and you upset that balance: the "battery" sags immediately, and what little energy you extract is stolen from the longitudinal source driving I, the magnetic force acting only as a broker (zero work — the warning above). Hall sensors are therefore read with high-impedance electronics, as voltmeters, never as sources.

Semiconductors have both electrons AND holes — what does the Hall effect read then?

A mobility-weighted competition: each species tries to impose its own Hall field, and the measured R_H is the two-band combination R_H = (p μ_h² − n μ_e²)/(e (p μ_h + n μ_e)²). Doped material is dominated by its majority carrier, so the bench test reads n-type or p-type cleanly; in near-intrinsic or compensated samples the terms partially cancel, and R_H can pass through zero and flip sign with temperature as populations shift — not a malfunction but a thermometer of the bands. The one-carrier R_H = 1/nq is the honest limit, not the general law.

Why is the quantum Hall resistance exact when every real sample is dirty and irregular?

That exactness is the astonishing part, and its full explanation (topological invariants, Landau levels, edge channels) belongs to the quantum track. The honest classical-track statement: in a two-dimensional electron gas at high B and low T, the Hall conductance stops depending on sample details and locks to integer multiples of e²/h, reproducible to parts per billion across materials and laboratories. Since 1990 the ohm has been maintained via R_K = h/e² = 25,812.807 Ω, made exact by definition in the 2019 SI — the rare case of a transport measurement promoted to a constant of nature.

My magnet sticks to the fridge through B alone — why does a Hall sensor need a current too?

Because B exerts no net force on stationary charge and creates no voltage in a dead conductor: the Hall effect is a cross effect, needing a drift (supplied by I) and a field to cross it with. That is a feature — the output is proportional to the product IB, so a stable bias current makes a magnetometer, while a known B makes a clamp meter. Same chip, two instruments, depending on which factor you hold fixed.