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In 1986, Johannes Georg Bednorz and Karl Alexander Müller precipitated mixed oxalates from aqueous barium, lanthanum and copper nitrates using oxalic acid. Heating at $900{}^{\circ }\mathrm{C}$ for five hours preceded pressing at $4\mathrm{kbar}$ and reheating for sintering. X-ray diffraction resolved three phases. Rectangular pieces carried gold electrodes and indium wires; four-terminal sensing in a continuous-flow cryostat measured electrical resistance versus temperature. Annealing atmosphere and measuring electric current density were varied; larger electric current densities suppressed the drop. An onset near $35\mathrm{K}$ differed from the much lower temperature where the drop approached three orders of magnitude.^[ref]

This was evidence for possible granular superconductivity, with phase identification unfinished. The paper deferred Meissner effect and heat capacity tests; its proof note corrected bulk lanthanum/barium ratios. A resistive onset did not establish a uniform bulk superconducting transition temperature.^[ref]

Their 1987 authored lecture describes later magnetometer checks on lead, followed by magnetic susceptibility measurements: samples with a resistive drop developed diamagnetism at somewhat lower temperatures. Combining magnetic, transport and X-ray diffraction results identified barium-substituted $\mathrm{L}{\mathrm{a}}_2\mathrm{Cu}{\mathrm{O}}_4$ as the superconducting phase. These later results strengthened the case beyond the initial mixed-phase transport evidence.^[ref]

A motivating laboratory realization is The Hall Effect and Conductivity in Germanium by Rex Tayloe. The handout specifies a germanium sample holder with temperature and magnetic field control, a low-voltage supply, digital meters, and a current-reversing switch. Measure the sample dimensions, map the contacts, and establish the directions of electric current, magnetic field, and transverse voltage. At fixed temperature and zero magnetic field, sweep positive and negative electric currents and record longitudinal electric potential differences. A straight voltage-versus-electric current relation tests Ohm's law; sample geometry converts its slope to electrical resistivity. For the Hall effect, minimize the zero-field transverse offset and measure transverse voltage at all four combinations of $\pm I$ and $\pm B$, repeating for several electric currents. The readouts test longitudinal proportionality, transverse polarity, and transverse proportionality to electric current. These are protocol instructions, not reported measurement outcomes.^[ref]

For interpreting such measurements, an ideal uniform bar of wire cross-sectional area $A$ with longitudinal sensing contacts separated by $L$ gives $\sigma =LI/(A{V}_L)$. Four-terminal sensing suppresses voltage drops in current contacts when the voltage meter draws negligible electric current; it does not fix uncertain dimensions or a nonuniform current density. Reverse electric current and use $[V_L(+I)-V_L(-I)]/2$ to reject a constant voltage offset. Keep temperature stable, allow the current density to settle, and reduce electric current to test for Joule heating. Those are controls for interpreting the material coefficient, rather than reasons to assume every measured curve is linear.

For the transverse measurement define $\mathbf{J}=J_x\hat{\mathbf{x}}$, $\mathbf{B}=B_z\hat{\mathbf{z}}$, width $w$ along $y$, and thickness $d$ along $z$. Define $V_H=V(y=-w/2)-V(y=+w/2)=E_{y}w$, so $J_x=I/(wd)$ givesExtract the part odd in both electric current and magnetic field usingThis combination rejects a constant offset and longitudinal pickup even in magnetic field. It cannot reject another signal with the same two reversal symmetries. Check field polarity and magnitude independently; a sign error in the contact map changes the inferred Hall coefficient.

For one Drude model species in transverse open circuit, the steady Lorentz force balance is $0=q(\mathbf{E}+{\mathbf{v}}_d\times \mathbf{B})-m{\mathbf{v}}_d/\tau$ with $v_{d,y}=0$. Its $y$ component gives $E_y=v_{d,x}{B}_z$. ConsequentlyLongitudinal electrical conductivity and the Hall coefficient therefore separate number density from charge carrier mobility in this single-species model. Inferring a momentum relaxation time additionally requires an appropriate mass. A constant-field longitudinal measurement alone cannot supply all three.

A Hall coefficient is not universally the reciprocal of signed number density. For example, adding independent electron and electron hole contributions with densities $n,p$ and positive charge carrier mobilities ${\mu }_e,{\mu }_h$ in the weak-magnetic field limit givesThe Hall coefficient can vanish even while electrical conductivity is positive, and its sign weights charge carrier mobility as well as number density. The germanium handout explicitly asks students to track possible transverse sign changes with temperature; it does not establish a universal single-carrier interpretation. Real electronic band structure and unequal scattering also limit that interpretation.^[ref]

An electrical calorimetry experiment asks whether a measured electrical work input accounts for a measured warming. The University of Iowa's demonstration uses an insulated flask containing water, an immersed heater, a thermometer, electrical meters and a timer. Its procedure compares the duration, electrical readings and temperature rise. The following low-power measurement design and synthetic readout develop that idea; they are not measurements reported by the demonstration. ^[ref]

Use a covered calorimeter with weighed water, an immersed resistor, a calibrated thermometer and gentle stirring. Put an ammeter in series with the heater branch and a high-input-electrical resistance voltmeter across the immersed resistor's terminals. Record $V(t)$, $I(t)$, temperature $T(t)$ and room temperature $T_a(t)$ before, during and after switching the heater on. Measure the electric potential difference at the heater, rather than at the supply: lead and contact electrical resistance outside the receiving boundary can consume power that never reaches the water. Account for voltmeter loading if the ammeter also includes the sensing branch's electric current.^[ref]

The electrical readout gives operating electrical resistance $R(t)=V(t)/I(t)$ when $I(t)\ne 0$, and power $P(t)=V(t)I(t)$. To test Ohm's law, obtain several pairs at the same temperature; one ratio alone cannot establish a linear response. For small electrical resistance, use four-terminal sensing to exclude lead drops and reverse electric current to test stable offsets from the thermoelectric effect. Use brief low-power resistance measurements if the intended result is the unheated value: the test itself causes Joule heating.

For a covered assembly at approximately constant pressure, with negligible evaporation, negligible endpoint mechanical energy changes and no phase transition, takeHere $c_w$ is the water's specific heat capacity at constant pressure, $C_{\mathrm{cal}}$ is the participating vessel, heater and sensor heat capacity, $Q_{\mathrm{loss},\mathrm{out}}$ is signed outward heat leakage, and $W_{\mathrm{stir},\mathrm{on}}$ is stirring work entering the assembly. The storage term approximates its enthalpy change; for a rigid fixed-volume receiver use the corresponding heat capacity at constant volume. Calibrate $C_{\mathrm{eff}}$ independently if the purpose is to test electrical input against thermal storage. Using the same heating run to define $C_{\mathrm{eff}}=E_{\mathrm{el}}/\Delta T$ and then claiming it verifies the equality would be circular.

For the synthetic trace, stipulate $V=5\mathrm{V}$, $I=1\mathrm{A}$, $R=5\Omega$, $C_{\mathrm{eff}}=500J/K$, room temperature $20{}^{\circ }\mathrm{C}$, and loss coefficient $G=0.5W/K$, with negligible stirring input. The heater runs for $120\mathrm{s}$, beginning at room temperature. The calorimeter heat-loss correction model isIt predicts $E_{\mathrm{el}}=600\mathrm{J}$, $\Delta T\approx 1.13080\mathrm{K}$, storage $565.398\mathrm{J}$ and outward loss $34.602\mathrm{J}$. Neglecting leakage undercounts deposited energy by about $5.77\%$; assuming no loss predicts $1.20\mathrm{K}$. After switching off, the excess temperature undergoes exponential decay with time constant $C_{\mathrm{eff}}/G=1000\mathrm{s}$. A cooling slope can estimate $G$ only with known $C_{\mathrm{eff}}$ and a credible single-temperature model.

Controls separate causes of a temperature rise. Check the electrical meters' zeros and thermometer calibration; keep water mass, immersion, lid and stirring fixed between runs. Record a heater-off stirred baseline to estimate sensor drift and stirring input. Repeat cooling runs over the same temperature range to estimate environmental heat transfer; a baseline taken only at room temperature cannot measure warming-dependent leakage. Vary electric current while keeping $R$ and initial thermal conditions comparable: corrected deposited energy should scale as $I^2$ for equal durations. An independent calorimeter calibration or known reference heat capacity makes this a quantitative test rather than a qualitative observation.

Finite sensor response, incomplete thermal equilibrium, changing $R(T)$ or heat capacity, evaporation, and imperfect loss corrections limit the inference. A final temperature alone cannot distinguish smaller electrical input from larger heat leakage or a larger participating heat capacity. Agreement tests the stated assembly and its energy accounting; it does not show that all electrical work in arbitrary devices becomes heat.

Historically, the Royal Society's archive description of James Prescott Joule's manuscript dated 13 October 1840 records a coil of wire in a jar of water, a thermometer and a galvanometer, and summarizes his inference that warming depends on electrical resistance and the square of electric current intensity. This electrical experiment precedes Joule's paddle-wheel experiment; the input here is electrical work, whereas the paddle supplies mechanical work. The catalogue, rather than the inaccessible manuscript images, supports this account. ^[ref]

The 1988 study grew alternating iron/chromium layers by molecular beam epitaxy on gallium arsenide. Magnetization and torque measurements tested the magnetic configuration. An AC electrical resistance measurement passed electric current along the layers while the applied magnetic field changed. Comparing field directions and spacer thicknesses linked the resistance decrease to magnetic saturation. Electron microscopy and surface analysis checked layer formation, while leaving atomic-scale interface roughness unresolved.^[ref]

The independent Jülich study used four-terminal sensing on epitaxial Fe/Cr/Fe strips. A single iron film supplied an anisotropic magnetoresistance comparison. Magneto-optic Kerr effect signals and Brillouin light scattering from spin waves identified the layer alignments, rather than inferring them from resistance alone. Field scans along different magnetic axes connected the optical signatures to the resistance changes.^[ref]

Peter Andreas Grünberg's retrospective explains why the preceding interlayer exchange coupling measurements mattered: coupled spin wave modes supplied signatures of magnetic alignment before that alignment became an electrical signal.^[ref]

A resistance curve alone does not reveal the microscopic direction of every magnetic layer. The optical measurements add a different observable, while the reference film tests an orientation-dependent response that can occur without a multilayer. Sample preparation and cross-checks therefore belong to the physical explanation.

Four-terminal sensing uses one pair of contacts to carry electric current and a separate pair to sense electric potential difference. The sensing instrument draws negligible electric current, so the voltage drops along its own leads are negligible, and the current-contact drops outside the sensing interval are excluded. A measurement using just two shared leads includes their electrical resistance and contact drops, which can overwhelm the small sample signal in a resistance-versus-temperature experiment for superconductivity. Four contacts do not automatically remove offsets from the thermoelectric effect. ^[ref]

For a stable offset $V_0$, let $I>0$ denote the magnitude of the reversed electric current. The measured electric potential differences are $V(+I)=IR+V_0$ and $V(-I)=-IR+V_0$, so subtraction gives the displayed electrical resistance. For example, with stipulated $I=1\mathrm{mA}$, $R=0.2\mathrm{m\Omega }$ and $V_0=3\mu \mathrm{V}$, the two readings are $3.2$ and $2.8\mu \mathrm{V}$; their difference correctly recovers $R$. If the offset changes between readings, its change leaves a residual measurement error; the sample and measuring circuit must remain sufficiently stable during reversal.

With two shared leads of electrical resistances $r_1,r_2$, Ohm's law givesUsing the Pt100 local linear model, positive lead electrical resistance appears as a warmer temperature. At true $25{}^{\circ }\mathrm{C}$, stipulated leads of $0.50\Omega$ each change $109.625\Omega$ into $110.625\Omega$, producing $27.597\dots {}^{\circ }\mathrm{C}$: a calculated bias of about $2.60{}^{\circ }\mathrm{C}$. Subtracting a measured $1.00\Omega$ loop correction works only while that loop electrical resistance remains appropriate; cable temperature changes can invalidate it.

Four-terminal sensing separates the electric current and electric potential difference leads. Negligible sensing electric current suppresses sensing-lead drops; force-lead drops lie outside the interval. Real voltmeter loading, connections, and leakage still need checking.^[ref]

For a stable offset $E$ from the thermoelectric effect, two signed electric potential differences obey $V_{+}=IR+E$ and $V_{-}=-IR+E$. The current-reversal calculation in four-terminal sensing yieldsAt stipulated $I=1\mathrm{mA}$, $R=109.625\Omega$ and $E=3\mu \mathrm{V}$, $V_{+}=109.628\mathrm{mV}$ and $V_{-}=-109.622\mathrm{mV}$ recover $R$ exactly in this model. If the offset or sample temperature changes between readings, cancellation is incomplete. Reversal also leaves Joule heating unchanged because it depends on $I^2$.

The motivating observation is an abrupt disappearance of detectable electrical resistance on cooling, rather than a smooth improvement in conduction. The diagram pairs the measuring arrangement with an idealized transition. Its curves are deliberately synthetic; neither their scales nor their abrupt step are a reconstruction of measured mercury data.

In a report communicated in November 1911, Heike Kamerlingh Onnes described a mercury thread in seven connected glass U-tubes immersed in a stirred liquid-helium bath. Expansion spaces kept freezing mercury from breaking the glass or interrupting the thread. Separate leads supplied electric current and sensed electric potential difference. He compensated a troublesome thermoelectric effect and reversed the main electric current to extract the resistive signal. He reported a rapid fall between $4.21$ and $4.19\mathrm{K}$ on his stated historical temperature scale, with electrical resistance disappearing at the lower value. The apparatus and control of spurious electric potential differences are part of the discovery, not incidental details. ^[ref]

A modern version places a contacted sample and a calibrated thermometer close together in a cooling environment. Use four-terminal sensing, record the small applied electric current $I$, the sensed electric potential difference $V$, and the sample temperature $T$, then plot $R(T)=V/I$. Repeat on warming as well as cooling, and allow readings to stabilize near thermal equilibrium. Disagreement between warming and cooling is a reason to check sample–thermometer lag and reproducibility before quoting a superconducting transition temperature. Mikhailov and Kordosky's undergraduate experiment uses this strategy with liquid-nitrogen cooling. ^[ref]

Choose the measuring electric current small enough that Joule heating does not appreciably change the sample temperature, and repeat with a smaller electric current to check that the transition persists. Cancel stable offsets using the reversal described in four-terminal sensing. A reading indistinguishable from zero supplies an experimental upper bound, not a proof of exact zero electrical resistance. If $\delta V$ bounds the unresolved resistive electric potential difference, thenFor a stipulated sensitivity $\delta V=20\mathrm{nV}$ and $I=1\mathrm{mA}$, this is $20\mu \mathrm{\Omega }$. The bound applies between the sensing contacts; it is not the electrical resistance of the complete wiring. A conducting path through part of an inhomogeneous sample can also make $V$ negligible, so the Field-cooled Meissner experiment adds independent evidence about the material's magnetic response.

PHYWE's documented experiment uses a one-metre slide wire, a connection box, an unknown resistor, a known comparison resistor, a supply, and a digital multimeter in steady-current mA mode as the null detector. Its two yellow detector leads join the fixed-resistor junction and the slider. Move the slider until the indicated electric current vanishes; read the two wire lengths on the ruler and infer the unknown electrical resistance from their ratio. The guide advises changing the comparison resistor if no null is reachable or the slider lies near an end.^[ref]

The inference requires uniform electrical resistivity and wire cross-sectional area, so the two wire electrical resistances have the ratio of their lengths, by $R={\rho }_{\mathrm{el}}L/A$. Retaining PHYWE's labeling, its equation (6) is $R_x=R_C{\ell }_1/{\ell }_2$, where $R_C$ is the comparison electrical resistance. This is a different arm placement from our schematic. The guide's example table reports larger discrepancies near wire ends.^[ref]

Useful additional controls distinguish a true null measurement from an insensitive or disconnected detector. Check the detector zero with the supply off, then move the adjustable arm slightly to either side of the candidate balance and verify opposite detector signs. Reconnect the unknown with short, secure leads and repeat the balance. Any lead or contact electrical resistance included in that arm contributes to the inferred value, even though a detector lead carries no electric current at exact balance. For very small unknown electrical resistances, four-terminal sensing addresses this separate connection problem; a detector null alone does not remove it.

Check for Joule heating by repeating at lower supply electric potential difference and with shorter powered intervals, holding ambient temperature fixed. At balance the unknown still dissipates powerHalving $V_s$ quarters this power while halving ideal detector sensitivity. A balance that moves with excitation or elapsed time suggests changing electrical resistance, including self-heating; it is not explained by passive detector loading alone. These are proposed controls, not claimed outcomes of PHYWE's experiment. Standard-resistor accuracy, wire uniformity, contact reproducibility and finite detector resolution still limit the electrical measurement. Large electrical resistances also make unwanted leakage paths consequential.

Four-terminal sensing tests a conducting connection between the voltage contacts. A magnetometer instead tests the sample's magnetic response. These measurements provide complementary constraints: the first is sensitive to a continuous path, while the second also depends on screening geometry and preparation history.

For transport, define $V_{ab}$ between the inner contacts and use the reversed-electric current estimator from four-terminal sensing. Contact separation excludes drive-contact drops, but does not eliminate a drifting thermoelectric effect, Joule heating or an unrepresentative path through part of the specimen. Repeat at smaller electric currents and compare results at the same sample temperature. A finite-current result refers to that electric current and its relation to the path's superconducting critical current.

For magnetism, distinguish cooling in zero applied magnetic field and then applying it from the Field-cooled Meissner experiment. Record the applied magnetic field, background and sample geometry when interpreting magnetization or magnetic susceptibility. A transport short does not establish full Meissner effect screening throughout the specimen. Conversely, flux pinning in a type-II superconductor can make the field-cooled magnetic response differ from ideal complete expulsion. Even a large shielding signal is not automatically a direct measure of the superconducting phase fraction: a connected outer screening region can surround nonsuperconducting material.