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An Alfvén wave is a magnetohydrodynamic wave in which magnetic tension restores a sideways disturbance of a conducting fluid. In the small-amplitude, uniform-background ideal magnetohydrodynamics model, its shear polarization of waves has displacement perpendicular to both the background magnetic field and the wavevector. It is an incompressible flow disturbance: the mass density and gas pressure have no first-order fluctuation. Its dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{v}}_A{)}^2$, where ${\mathbf{v}}_A$ is the background Alfvén velocity.

The mechanism resembles a plucked taut string: magnetic flux freezing makes sideways fluid flow bend magnetic field lines, and their magnetic tension accelerates neighboring fluid. The fluid supplies inertia and the bent magnetic field supplies the restoring stress. This permits energy transport without a first-order compression, unlike an acoustic wave. Hannes Alfvén's 1942 paper, Existence of Electromagnetic-Hydrodynamic Waves, considered precisely a uniform axial magnetic field, infinite electrical conductivity and sideways velocity depending on axial position and time; it proposed a solar application. This was a theoretical prediction, followed by the distinct Lundquist liquid-metal Alfvén wave experiment.^[ref]

For oblique propagation, the shear Alfvén wave has velocity normal to the plane containing the wavevector and background magnetic field. The fast magnetosonic wave and slow magnetosonic wave instead have velocity in that plane and generally involve compression, with gas pressure and magnetic pressure contributing to their restoring forces. For exactly parallel propagation the plane is not unique: either transverse polarization of waves is allowed, and some modal speeds coincide. For $k_{\parallel }=0$ the shear branch has $\omega =0$, while the fast magnetosonic wave can still propagate perpendicular to the magnetic field. One should identify the polarization of waves as well as the phase velocity. The statement that the shear mode is incompressible concerns this linear mode; a finite linearly polarized disturbance can drive a quadratic compressive response, as in the magnetic-pressure obstruction to a linearly polarized Alfvén wave.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

Introducing excess mobile charge carriers into a semiconductor, for example through a contact under biasing. Their subsequent motion and electron-hole recombination can change the electrical conductivity and an output electric current.

Consider isotropic electrons and electron holes with equal density $n$, charges $\mp e$, equal positive charge carrier mobility $\mu$, and single-species zero-field electrical conductivity ${\sigma }_0=ne\mu$. In a transverse magnetic field, opposite Hall conductivity components cancel, while diagonal components add to $2{\sigma }_0/[1+(\mu B{)}^2]$. Inverting gives ${\rho }_{xy}=0$ and ${\rho }_{xx}={\rho }_{yy}=[1+(\mu B{)}^2]/(2{\sigma }_0)$. Its large-field growth is ${\sigma }_0{B}^2/(2n^2{e}^2)$. The total zero-field resistivity is $1/(2{\sigma }_0)$, half the single-species resistivity; confusing these quantities introduces a factor-of-two normalization error.

A conduction current is transport of electric charge by mobile carriers through matter. For a local isotropic ohmic medium, its current density is ${\mathbf{J}}_c=\sigma \mathbf{E}$, with electrical conductivity $\sigma$. It differs from displacement current $\partial \mathbf{D}/\partial t$, which can be nonzero in a capacitor's insulating gap without carriers crossing the gap.

An electron occupying an extended state in a conduction band. Its response to an applied electric field depends on its band group velocity, occupation and scattering. Merely counting such electrons does not by itself determine electrical conductivity.

How can a clear chemical solution reveal otherwise invisible mobile ions? The documented Conductivity of Solutions by Harvard Natural Sciences Lecture Demonstrations demonstration uses metal electrodes enclosed in tubing, an electrical supply, and a lamp in the circuit. The protocol specifies about $150\mathrm{mL}$ samples in labeled $250\mathrm{mL}$ beakers and a separate deionized-water rinse station. The recorded readout is lamp brightness after the electrodes enter each sample; the simplified variant tests water before and after adding sodium chloride. These are apparatus and procedure specifications, not new measurements reported here.^[ref]

The useful comparison is water, a sucrose chemical solution, and a sodium chloride chemical solution with the same probe geometry. A documented university account reports no visible light with water, a glowing lamp with dissolved sodium chloride, and a weak filament glow with its sucrose sample. The diagram summarizes the latter as little or no visible light; the precise detection threshold depends on the tester. Sucrose is a control for the idea that any dissolved material must conduct: its dispersed neutral molecules do not provide the abundant mobile ions supplied by the salt. This contrast is a documented teaching observation, not a numerical calibration of lamp brightness.^[ref][ref]

Rinsing the electrodes between samples controls carryover of dissolved ions, and the water blank checks the solvent background. Repeating the blank after the salt sample would test whether that rinse was effective. Additional methodological controls are fixed immersed area, electrode separation, supply setting, temperature, and comparison at stated concentrations. These additional controls are proposed checks, not undocumented claims about the source demonstration.

The inference is that mobile electric charge carriers complete the circuit. In the chemical solution, an electric field drives positive ions along the field and negative ions against it; multiplying each drift velocity by its electric charge makes both contributions to conventional current point the same way. With an alternating supply the direction reverses each half-cycle. The external wires instead carry electric current through electrons. The lamp alone identifies neither the species nor an exact electrical conductivity.

Important limits remain. Water has a small intrinsic population of hydronium and hydroxide, so a dark lamp does not prove absence of ions. Changes in probe geometry affect measured electrical conductance; electrolysis, electrode interfaces, contamination, and heating can also affect a real circuit. The comparison supports a transport model but does not independently measure a coordination number, a partitioned partial atomic charge, or a lattice energy.

Nor does poor conduction by ordinary dry sodium chloride establish that all ionic crystals are electrical insulators. Crystallographic defects can allow transport, and solid electrolytes can have substantial ionic electrical conductivity without melting. The authored abstract of A lithium superionic conductor by Noriaki Kamaya and others reports room-temperature ionic conduction in a solid electrolyte; the full experimental article was not accessible here.^[ref]

For scalar electrical conductivity $\sigma (\mathbf{x})$ and constant permeability ${\mu }_0$, the moving-fluid Ohm law and the quasistatic Ampere-Maxwell law give $\mathbf{E}=-\mathbf{u}\times \mathbf{B}+\eta \nabla \times \mathbf{B}$, where $\eta =1/({\mu }_0\sigma )$ is the magnetic diffusivity. Taking minus the curl and using Faraday's law of induction gives ${\partial }_t\mathbf{B}=\nabla \times (\mathbf{u}\times \mathbf{B})-\nabla \times (\eta \nabla \times \mathbf{B})$. Expanding this expression and imposing $\nabla \cdot \mathbf{B}=0$ yields the title equation. The familiar constant-diffusivity resistive induction equation omits the last term because $\nabla \eta =0$; nonrelativistic motion alone does not establish that simplification.

An intrinsic semiconductor has electron and hole densities $n=p=n_i$, so ${\sigma }_i=e{n}_i({\mu }_e+{\mu }_h)$. In a fully ionised, uncompensated n-type sample with $N_D\gg n_i$, $n\simeq N_D$, $p\simeq n_i^2/N_D$, andThus a carrier-density ratio alone is not generally the electrical conductivity ratio. If electron mobility is unchanged and electron transport dominates the intrinsic conductivity, it reduces to $N_D/n_i$; equal electron and hole mobilities instead give $N_D/(2n_i)$. Ionised-impurity scattering can change the doped mobility.

A concrete laboratory route to an Arrhenius plot is to infer a rate constant at each controlled temperature from a changing electrical conductivity. The Technical University of Munich's Activation Energy of a First-Order Reaction supplies a teaching protocol, rather than a published set of measured rate constants.

The manual follows tertiary amylbromide conversion, which produces conducting ions. A measuring cell connects to a thermostat; an immersed electrical conductivity probe also measures solution temperature, and a magnetic stirrer mixes it. Use $80\mathrm{mL}$ of $80\%$ by volume ethanol in water, equilibrate, then add $0.5\mathrm{mL}$ of tertiary amylbromide. Record the baseline before addition and set time zero at addition. Repeat with fresh mixtures at $25$, $30$, and $35{}^{\circ }\mathrm{C}$, each within $\pm 0.5{}^{\circ }\mathrm{C}$; record for $30$ minutes at the first two temperatures and until completion at the last. Clean the cell between runs and keep the added volume fixed. The completed $35{}^{\circ }\mathrm{C}$ mixture supplies endpoint electrical conductivity measurements at the other temperatures: one endpoint reading cannot be used unchanged at every temperature.^[ref]

To see what is inferred from the readout, let ${\sigma }_0$, $\sigma (t)$, and ${\sigma }_{\infty }$ be the baseline, current, and completed-mixture electrical conductivities at the same temperature. Assume their baseline-subtracted changes are proportional to converted concentration. Combining that measurement model with a first-order reaction givesThus the electrical conductivity increases toward an endpoint, while the remaining reactant fraction decreases. Fit $-\ln f$ against time to obtain $k$; then fit $\ln (k/k_{\mathrm{ref}})$ against $1/T$ in kelvin to estimate an apparent activation energy. This baseline-aware derivation states the measurement assumptions explicitly; it does not infer a rate constant from the steepness of a raw conductivity trace alone.

The controls distinguish a faster chemical reaction from a larger instrumental response. Measure the actual solution temperature, retain the same solvent composition and initial reactant concentration, and compare endpoint and baseline at that temperature. As further diagnostic checks, repeat runs and inspect regression residuals for departure from a first-order reaction. These checks follow from the model: electrical conductivity also depends on how ions move, a baseline drift can mimic conversion, and a mixing delay can distort early points. Near completion, a small endpoint error produces a large relative error in $f$ and its natural logarithm. A successful fit supports an apparent rate model for this solvent and interval; it does not directly image a transition state or identify a unique reaction mechanism. No measured $k$ or $E_a$ is supplied here.

How can an invisible change in concentration produce a readable signal? The saponification of ethyl acetate by hydroxide in water gives ethanol and an acetate ion:The rate law used for this assay is $r=k_2[\mathrm{ethyl}\mathrm{acetate}][\mathrm{O}{\mathrm{H}}^{-}]$. Each reactant has coefficient one, so $r$ equals either disappearance rate. If the two initial concentrations are equal, they remain equal under this stoichiometry, giving $-dc/dt=k_2{c}^2$ and $k_{\mathrm{obs}}=k_2$, without a factor of two.

Manual for Advanced Physical Chemistry Laboratory-II by Nasima Arshad (Unit 7, Experiment 2, printed pp. 66–67) specifies a conductivity cell holding about $60\mathrm{mL}$ in constant-temperature baths, with $0.040\mathrm{mol}{\mathrm{L}}^{-1}$ ethyl acetate and sodium hydroxide stocks equilibrated separately. Mixing $30\mathrm{mL}$ of each initiates the run. The readout is electrical resistance: every minute for ten minutes, every two minutes for ten more, then every five minutes for twenty-five. The cell is rinsed with distilled water. A matched sodium hydroxide–water dilution estimates the initial resistance; a potassium chloride standard determines the cell constant. The manual asks for comparison with the extrapolated initial signal.^[ref]

The post-mixing initial concentration is $c_0=0.040(30/60)=0.020\mathrm{mol}{\mathrm{L}}^{-1}$ for each reactant, assuming additive volumes. This matches our illustrative starting concentration, but the prescribed $k_{\mathrm{obs}}=0.50$ is not a measured value from the manual. As hydroxide is replaced by the less mobile acetate ion, electrical conductance decreases. The preceding experiment describes two platinum electrodes and an alternating-voltage Wheatstone bridge to limit polarization at the electrodes. For measured electrical resistance $R(t)$, electrical conductance is $G(t)=1/R(t)$; electrical conductivity is $\kappa (t)=K_{\mathrm{cell}}G(t)$. Keep the probe geometry fixed and use one signal convention throughout.^[ref]

For the equal-starting-concentration model, assume an affine calibration between electrical conductance and remaining hydroxide concentration, with endpoints $G_0$ and $G_{\infty }$ at the same temperature. This givesThe last plot has slope $k_2{c}_0$, with units ${\mathrm{s}}^{-1}$, whereas plotting $1/c$ against time has slope $k_2$, with units $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$. These are different slopes. This endpoint transformation also shows why raw resistance must not be substituted for conductance.

Useful additional controls are repeat runs, fixed immersion depth, an independently checked final signal, and comparisons of several starting concentrations. Mixing delay obscures early conversion; temperature drift changes both the rate constant and the signal; contamination adds mobile ions. The affine calibration can fail when ionic mobilities change with composition. Near completion, $G-G_{\infty }$ becomes small, so endpoint error is strongly amplified. Those are methodological limits and proposed checks, not new experimental observations. A conductivity trace supports a calibrated rate law over its tested range; it does not independently reveal the full reaction mechanism.

For unequal starting concentrations $a=[\mathrm{A}{]}_0$ and $b=[\mathrm{B}{]}_0$ in a one-to-one chemical reaction, let $x$ be the consumed concentration. Then $[\mathrm{A}]=a-x$, $[\mathrm{B}]=b-x$, andThe natural logarithm acts on a dimensionless quantity; differentiating the left side with respect to $x$ gives $1/((a-x)(b-x))$, verifying the relation. Mixing $20\mathrm{mL}$ of the stated ethyl acetate stock with $40\mathrm{mL}$ of the sodium hydroxide stock would give $a=0.01333\dots$ and $b=0.02666\dots \mathrm{mol}{\mathrm{L}}^{-1}$. The difference $[\mathrm{B}]-[\mathrm{A}]=b-a$ stays constant, rather than their ratio. Neither the equal-concentration reciprocal formula nor a pseudo-first-order reaction approximation is justified solely by that twofold excess.

A crystallographic defect is a departure from ideal repeated structure. In an ionic crystal, a missing ion can provide an available site for another ion to move into; movement still requires overcoming an activation energy. This is why an ideal immobile-site model is a limit rather than a universal law for solid electrical conductivity.^[ref]

The manufacturer-issued Protein Preparation Handbook reports a paired dialysis comparison: $2\mathrm{mL}$ of bovine serum albumin, $0.25mg/mL$ in $1\mathrm{M}$ sodium chloride, was held behind a $10\mathrm{kDa}$ MWCO membrane against $45\mathrm{mL}$ of water in a conical tube, shaken at room temperature. The bath was replaced after two hours. A conventional stirred-beaker treatment against $2\mathrm{L}$ of water supplied the comparison control. Residual salt was followed by sample electrical conductivity; results averaged two samples. This readout tests small-ion removal, not protein retention.^[ref]

Its separate retention experiment used individual proteins at $1mg/mL$ in saline or carbonate-bicarbonate buffer solution, pH $9.4$, with membranes rated $2$, $3.5$, $7$, $10$ and $20\mathrm{kDa}$. After seventeen hours at $4{}^{\circ }\mathrm{C}$, retained protein was estimated by a bicinchoninic acid assay. The size series and membrane series compare retention; they do not isolate shape independently of mass. The figure caption supplies neither a no-membrane recovery control nor a complete uncertainty analysis. These are manufacturer comparisons, not proof of universal recovery or unchanged enzyme function.^[ref]

The two experiments answer complementary questions: did the small solute leave, and did the desired macromolecule remain? In a new experiment, a bath blank and a starting salt standard would help interpret electrical conductivity, while a matched undialyzed protein aliquot and recovered volume would help distinguish handling loss from dilution. Those are proposed additional controls, not controls claimed for the published figures. Other ions also contribute to electrical conductivity, so the readout cannot generally identify sodium chloride alone in an unfamiliar buffer solution.

A dielectric is a material in which an electric field produces electric polarization while transport of electric charge is sufficiently small for the intended use. An insulating solid in a capacitor is an example; real dielectrics can still have dielectric loss and nonzero electrical conductivity.

The important distinction is between rearranging bound electric charge and transporting mobile charge carriers through the specimen. Bound electric charges can move a little within a chemical atom or molecule, or an existing electric dipole can turn. Their averaged electric dipole moments produce electric polarization even if there is negligible steady electric current between the electrodes. A changing electric polarization does give a bound current; the absence of steady conduction does not mean that the material is electrically unresponsive.^[ref]

For uniform electric polarization $\mathbf{P}=P\hat{\mathbf{x}}$ in a rectangular slab, polarization density gives zero bound volume charge density in the interior and opposite bound surface charge densities $-P,+P$ on the left and right faces. The electric charge has been redistributed, not created. More generally, the divergence theorem makes the cancellation explicit:Here $\mathbf{n}$ is the outward normal vector on the material boundary. This accounts for both bound volume charge density and bound surface charge density; omitting either can falsely suggest that an initially neutral dielectric acquired net electric charge.

An electrical insulator between capacitor plates supports a useful combination: appreciable electric polarization, little leakage electric current, and tolerance of the intended electric field. This lets a small parallel-plate capacitor store separated electric charge and recoverable capacitor energy. The comparison in dielectric capacitor response explains why specifying fixed electric charge or fixed electric potential difference changes the answer. High relative permittivity alone does not establish good insulation or a large usable energy density; dielectric loss and dielectric breakdown matter as well.

Conversion of electrical energy into heat in a dielectric driven by an electric field. In the stated harmonic convention, a passive scalar response has ${ϵ}^{\prime \prime }\ge 0$ for positive angular frequency. A dielectric capacitance measurement measures a dissipative component as well as capacitance; leakage from electrical conductivity can contribute to the inferred loss.

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]

The Drude model explains finite electrical conductivity by balancing charge carrier acceleration in an electric field against loss of directed momentum through collisions. Its useful surprise is that continuous microscopic motion can produce a constant macroscopic current density: the applied electric field continually restores the directed momentum lost to the material. The simplest version has one mobile species, constant number density $n$, electric charge $q$, mass $m>0$, and momentum relaxation time $\tau >0$, in a stationary, homogeneous, isotropic material. At zero magnetic field, its mean drift velocity and current density obeyFor electrons, $q=-e$ with positive elementary charge $e$, so ${\sigma }_0=n{e}^2\tau /m$. The electron drift velocity points against a constant electric field, whereas conventional current points along it. Random microscopic velocities can have zero mean without being zero individually; drift velocity is this population mean, rather than each particle's speed.^[ref]

The same first-order linear differential equation predicts both Drude current relaxation after switching an electric field and the Drude frequency response to a sinusoidal electric field. Together they let a measured electrical conductivity be interpreted in terms of a density, a mass, and a time scale; a single constant-field measurement determines only their combination $n{q}^2\tau /m$.

The original 1900 paper Zur Elektronentheorie der Metalle by Paul Drude sought a common account of electrical transport and heat transfer using mobile charged particles and ideas from the kinetic theory of gases. Its introduction, especially printed pp. 568–569, explains that comparing different transport phenomena could constrain the populations of free and bound charged particles. The modern momentum relaxation time formulation above is a useful presentation of the model, not a verbatim translation of the original notation or its historical assumptions about mass.^[ref]

For an ohmic element with finite nonzero electrical resistance $R$, its electrical conductance is $G=1/R$: an electric potential difference $\Delta V$ drives electric current $I=G\Delta V$. Greater positive electrical conductance means a greater electric current for the same positive electric potential difference. Electrical conductivity is instead a bulk material coefficient relating current density to electric field.

Electrical conductivity $\sigma$ measures the local response of current density to electric field. In the rest frame of an isotropic Ohmic conductor, the local form of Ohm's law isLarger positive electrical conductivity gives larger current density at the same electric field. It is a bulk material coefficient, distinct from the geometry-dependent electrical conductance of a whole component and from a sheet's surface conductivity. Electrical resistivity is ${\rho }_{\mathrm{el}}=1/\sigma$ in this scalar model. ^[ref].

The local and whole-wire forms fit together particularly neatly. For uniform current in a straight homogeneous wire, with length $L$, wire cross-sectional area $A$, constant electrical conductivity and terminal electric potential difference $V$, the axial electric field has magnitude $E=V/L$. ThusFor the defined electrical resistance example $\sigma =5\times 10^5(\Omega \mathrm{m}{)}^{-1}$, $L=1\mathrm{m}$, $A=10^{-6}{\mathrm{m}}^2$ and $V=5\mathrm{V}$, the local values are $E=5V/m$ and $j=2.5\times 10^{6}A/{\mathrm{m}}^2$; integrating gives $I=2.5\mathrm{A}$. One material law therefore supplies both a spatial description and a terminal relation.

The scalar form requires isotropy. In an anisotropic material the response can instead be a tensor, $J_i={\sum }_j{\sigma }_{ij}{E}_j$, and current density need not be parallel to electric field; uniaxial conductivity tensor is one existing example. These statements concern the material's rest frame. A moving electrical conductor instead uses the moving-conductor Ohm law.

An electrical insulator has sufficiently small transport of electric charge for its operating conditions. Insulation is a practical requirement involving electric potential difference, temperature, geometry and duration, rather than a universal claim of zero electrical conductivity. Electric polarization can be substantial in an electrical insulator; dielectric breakdown can end the insulating regime.

For an idealized uniform slab with small constant electrical conductivity $\sigma$, the Ohmic conductor model predicts leakage electrical resistance $R=d/(\sigma A)$ in parallel with capacitance $C=ϵA/d$. An isolated capacitor then hasThis is a derived model of slow discharge, not an assertion that every leakage mechanism obeys Ohm's law. Its time scale matches charge relaxation. The fixed-charge comparison in dielectric capacitor response needs an observation time short compared with $RC$, but long enough for electric polarization to settle. During a maintained voltage, leakage consumes power $V^2/R$ as Joule heating; delayed electric polarization can add dielectric loss under changing electric fields.^[ref]

Large relative permittivity alone does not rule out either mechanism.^[ref]