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For sinusoidal current density $J(t)=J_{\mathrm{pk}}\cos (\omega t)$ and constant carrier density $n$, the electron drift velocity is $v_d(t)=-J_{\mathrm{pk}}\cos (\omega t)/(ne)$. Its displacement has amplitude $J_{\mathrm{pk}}/(ne\omega )$ and zero net change over a full cycle. The distance accumulated by this drift alone in one cycle is $4v_{\mathrm{pk}}/\omega =2v_{\mathrm{pk}}/(\pi f)$. With an RMS current, $v_{\mathrm{pk}}=\sqrt{2}{v}_{\mathrm{rms}}$. This does not measure the much larger microscopic path due to Fermi velocity and electron-phonon scattering.

Charge carrier mobility is the magnitude of steady drift velocity per unit weak electric field in a scalar transport model. In the Drude model, ${\mathbf{v}}_d=\mathrm{sgn}(q)\mu \mathbf{E}$ and $\sigma =n|q|\mu$. The positive charge carrier mobility separates ease of motion from the sign of electric charge; its SI unit is ${\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$.

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]

The current density $\mathbf{J}(\mathbf{r},t)$ describes the amount and direction of local electric charge transport per area per time. Its SI unit is $A/{\mathrm{m}}^2$. Through a small area element with unit normal $\mathbf{n}$, the contribution to electric current is $dI=\mathbf{J}\cdot \mathbf{n}dA$: the dot product selects transport through the surface. A current density tangent to the surface contributes zero. If current density is uniform and perpendicular to a wire's transverse slice, its magnitude $j=|\mathbf{J}|$ satisfiesHere $A$ is the wire cross-sectional area. For nonuniform current density, $I/A$ is only the average normal component, and the surface integral is needed. ^[ref].

Consider two separate uniform-flow cases, each carrying electric current $I=2\mathrm{A}$. A wire cross-sectional area of $1\mathrm{m}{\mathrm{m}}^2=10^{-6}{\mathrm{m}}^2$ gives $j=2\times 10^{6}A/{\mathrm{m}}^2$; doubling the wire cross-sectional area to $2\mathrm{m}{\mathrm{m}}^2$ gives $j=10^{6}A/{\mathrm{m}}^2$. Equal electric current can therefore mean different local transport intensity. This distinction matters for Joule heating, whose local power per volume depends on current density and electric field.

For a toy electron transport model, stipulate mobile electron number density $n=8\times 10^{28}{\mathrm{m}}^{-3}$ and drift velocity ${\mathbf{v}}_d=-10^{-4}m/s\hat{\mathbf{x}}$, with the positive unit normal $\hat{\mathbf{x}}$. Each electron has electric charge $-e$, where the elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$ is exact in SI. These are defined model inputs, rather than measured parameters of a particular material. ^[ref] The current density and its magnitude areFor wire cross-sectional area $A=1\mathrm{m}{\mathrm{m}}^2$ and constant transport over time $10\mathrm{s}$, this givesThe two minus signs make conventional current point opposite to the electron drift velocity. A small drift velocity can carry an appreciable electric current because many charge carriers contribute; it does not specify the propagation speed of an electrical disturbance.

For constant diffusion coefficient $D$ and drift velocity $v\ne 0$, the balance $|v|t\sim \sqrt{Dt}$ gives the characteristic length $D/|v|$ and time $D/v^2$. The fundamental Gaussian distribution of the advection-diffusion equation has a fixed-position exponential factor $\exp [-v^{2}t/(4D)]$. Total probability is still conserved: this describes probability moving away from a fixed observation region, rather than disappearance of the particle. At zero drift, this finite exponential time scale is absent.

The drift velocity is a charge carrier population's mean velocity relative to its material's rest frame. It describes the net transport superposed on microscopic motion. For a single species, current density is $\mathbf{J}=nq{\mathbf{v}}_d$; for electrons, $q=-e$ reverses the direction. It is distinct from the propagation speed of an electrical disturbance. ^[ref].

Switch on a constant electric field ${\mathbf{E}}_0$ at $t=0$. For prescribed initial current density $\mathbf{J}(0)$, the exact Drude model solution is the displayed formula. For electrons, the corresponding drift velocity isStarting from zero drift velocity, the current density reaches $1-e^{-1}\approx 0.632$ of its limiting value after one momentum relaxation time. Switching the electric field off at $T$ instead gives $\mathbf{J}(t)=\mathbf{J}(T)e^{-(t-T)/\tau }$ for $t\ge T$. A finite step of electric field changes the acceleration immediately but leaves drift velocity and current density continuous.

More generally, multiplying the current density equation by an integrating factor givesThis is the causal response of a first-order relaxation equation: only earlier electric fields enter the integral. It also shows why instantaneous Ohm's law is an approximation when the electric field changes appreciably during a momentum relaxation time.

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]

An electrolyte is a conducting medium in which mobile ions carry electric current, according to IUPAC Gold Book: electrolyte. In introductory chemistry, a substance that yields ions when dissolved in water is also called an electrolyte. Both usages point to the same mechanism: an electric field gives mobile cations and anions directed drift velocities.

Mobility matters as much as the presence of electric charge. In a simple ideal model of solid sodium chloride, ions are confined to their sites and do not carry a sustained transport electric current. Molten sodium chloride has mobile ions; dissolved sodium chloride gives mobile ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$ ions in water. Real solids can have some ionic transport through imperfections; the immobile-solid picture is an introductory limit, rather than a claim that all solids are perfect insulators.

A strong electrolyte yields essentially all ions from the dissolved substance, whereas a weak electrolyte yields only a fraction under the specified conditions. Dissolved sodium chloride is a strong electrolyte. Sucrose is a nonelectrolyte: it dissolves as neutral molecules, so dissolving something does not automatically create mobile ions. Water's own small population of hydronium and hydroxide remains possible. See 11.2 Electrolytes (§11.2: strong, weak, and nonelectrolytes) and Flowers et al., Chemistry 2e, §11.4, Table 11.3: sucrose and sodium chloride.

Acetic acid is a weak electrolyte and a weak acid. In water, some neutral acetic acid molecules transfer a proton to water, producing hydronium and the negatively charged partner:Both sides have net electric charge zero. At chemical equilibrium, neutral acetic acid coexists with these ions; the fraction converted depends on the conditions. Electrolyte strength describes this extent of ion production, while concentration describes how much substance is present per specified quantity of chemical solution. A dilute strong electrolyte can contain fewer ions per volume than a concentrated weak electrolyte. Neither the words strong and weak nor a conductivity comparison alone specifies concentration.

Now specify an electrolytic cell with a positive anode on the left, a negative cathode on the right, and an approximately uniform electric field pointing right in the bulk electrolyte. With a steady external source and neglecting diffusion and convection, positive cations drift right toward the negative cathode; negative anions drift left toward the positive anode. Yet both populations contribute a rightward conventional current! For number densities $n_{+},n_{-}$ and drift velocities ${\mathbf{v}}_{+},{\mathbf{v}}_{-}$, the transport current density isThe product of electric charge and velocity determines conventional current: for the anion, both the charge and its rightward velocity component are negative, making its rightward contribution positive. This is the same sign rule used for electron conventional current. It follows directly from the existing charge carrier formula.

For an independent numerical example, stipulate $n_{+}=n_{-}=10^{25}{\mathrm{m}}^{-3}$, $v_{+,x}=2\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-1}$, and $v_{-,x}=-3\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-1}$. With the exact elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$, the contributions areEqual number densities of singly charged cations and anions give zero bulk charge density, while their motion gives nonzero current density. This is why a neutral electrolyte can conduct. These numbers are chosen toy inputs, not measurements of sodium chloride; maintaining sustained cell electric current also requires suitable electrode reactions. Blocking electrodes can instead lead to charge accumulation and decay of the electric current.

An electrolyte connects the microscopic motion of ions to observable electrical conductivity. A counterion-only model keeps just the mobile counterions needed to oppose a fixed electric charge; Poisson-Boltzmann equations use ideal Boltzmann distributions to model their equilibrium distributions, rather than the drift example above.

The positive electron mobility relates the drift velocity of conduction electrons to an electric field: ${\mathbf{v}}_d=-{\mu }_e\mathbf{E}$. In the isotropic Drude model, ${\mu }_e=e\tau /m^{\ast }$, where $\tau$ is the momentum relaxation time and $m^{\ast }$ the band effective mass. The minus sign describes negative charge; the electron mobility itself is positive.

The positive hole mobility is the magnitude of an electron hole's weak-field drift velocity per unit electric field. In an isotropic relaxation-time model, ${\mu }_h=e{\tau }_h/m_h^{\ast }$, and its contribution to electrical conductivity is $ep{\mu }_h$. The sign is opposite to an electron's drift because the electron hole has positive charge.

A conceptual single-molecule measurement follows a labelled bound molecule along an independently mapped DNA contour. Project fitted positions onto contour distance and estimate the mean-squared displacement at several lag times. For unbiased one-dimensional diffusion,Independent localization errors of variance ${\sigma }_{\mathrm{loc}}^2$ add an intercept $2{\sigma }_{\mathrm{loc}}^2$. Finite exposure adds motion blur, and drift, binding transitions and the finite DNA contour must be distinguished from free sliding. A slope fit with those limitations can estimate $D_1$; simply fitting all observed displacements to a noiseless line can bias it.

Retain the printed coefficient. The original PDF really prints $D_1=5\times 10^{10}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, not a negative exponent. In sliding diffusivity with per-step dissociation, for a basepair step $a\simeq 0.34\mathrm{nm}$, an unbiased nearest-neighbour walk has step interval $a^2/(2D_1)$. If the dissociation probability per step is $P$, the mean number of steps is approximately $1/P$, givingFor the printed values, ${\tau }_{\mathrm{bound}}\simeq 1.16\times 10^{-24}\mathrm{s}$ and ${\ell }_{\mathrm{rms}}\simeq 0.34\mu \mathrm{m}$. The latter is a random-walk displacement, not the contour distance travelled in a million steps. The step interval is about $1.16\times 10^{-30}\mathrm{s}$, even shorter than the light-travel time across a basepair by many orders of magnitude. This is an unphysical extrapolation for a protein, not a meaningful experimentally measurable diffusivity.

Consequently no fluorescence camera could temporally resolve trajectories under the stated formal values, regardless of the spatial excursion's length. An actual corrected exponent cannot be inferred uniquely from the paper. If a physically plausible $D_1$ were supplied instead, resolving sliding would require exposures substantially shorter than ${\tau }_{\mathrm{bound}}$ and diffusion displacements larger than localization error, with enough detected photons. The dependence on $D_1$ is explicit above, so no guessed source correction is needed.

A single fluorophore creates a diffraction-broadened point-spread function, but its centre can be fitted much more precisely than that function's width. With Gaussian width $s$ and $N_{\gamma }$ detected independent photons, the shot-noise limit is ${\sigma }_{\mathrm{loc}}\simeq s/\sqrt{N_{\gamma }}$. Pixelation, background photons, dye orientation, motion blur, drift and label-to-target geometry impose additional uncertainty and systematic error. Localization precision is therefore not identical to either optical resolution or position accuracy.^[ref]

In myosin-V tracking, localizing a label on one head or its lever arm distinguishes the paths of individual motor parts from the average motion of the whole motor. A trailing head that becomes the leading head travels roughly twice the stalk advance: about $74\mathrm{nm}$ versus a roughly $37\mathrm{nm}$ motor stride. A near-head label can therefore alternate a large movement with a pause while the partner head advances. Other lever-arm label positions give unequal alternating steps whose sum is approximately twice the stride. Those patterns support hand-over-hand walking rather than a simple inchworm model in which the same head remains in front. The label geometry is needed when interpreting individual step sizes.^[ref]

Super-resolution localization microscopy extends the same principle to a map containing many molecules by separating otherwise overlapping emitters in time. Sparse activation, blinking or transient binding yields individually fitted positions, which are accumulated. Dense sampling, registration and labelling specificity determine the final structural resolution; merely taking more photons from an unresolved simultaneous pair does not make it a single isolated emitter.

In PAINT, fluorescent probes bind transiently so that separated binding events can be localized and accumulated. DNA-PAINT uses reversible complementary-strand recognition at docking sites on a DNA nanostructure. The bound probe supplies a localized spot while freely diffusing probes contribute background; repeated visits build up a positional map, with replenishment rather than permanent reliance on one fluorophore. This is an imaging principle, not a requirement for a particular target sequence or preparation protocol.^[ref]

The free electron model replaces the microscopic ionic potential and interacting electronic motion by independent mobile electrons in a uniform neutralizing background, with approximately parabolic dispersion $ϵ={\hslash }^2{k}^2/(2m^{\ast })$. Its quantum version fills states according to Fermi-Dirac statistics and the Pauli exclusion principle. Scattering is represented by a momentum relaxation time $\tau$, rather than derived from the model's ideal free trajectories. This model explains a Fermi energy much larger than room-temperature thermal energy, an electronic heat capacity proportional to temperature at low temperature, and a Drude electrical conductivity. With appropriate scattering assumptions it also yields the Wiedemann-Franz law. It cannot by itself explain band gaps, detailed band anisotropy, strong correlations, or the microscopic temperature dependence of $\tau$.

The inertial term is mass times the rate of change of mean drift velocity. The electric force is $eE$, taking the printed $e$ as the signed electron charge, and $-\gamma ⟨v⟩=-m^{\ast }⟨v⟩/\tau$ describes loss of directed momentum to scattering. electrons retain large microscopic velocities; this damping is of their mean directed motion. In a steady DC field,Both charge factors are negative for electrons, so the conductivity is positive. If $e$ instead denotes a positive charge magnitude, the force and current equations each require a minus sign and give the same result.

For AC response choose physical fields as real parts of phasors proportional to $e^{-i\omega t}$. Substitution givesThusThe real part decreases from ${\sigma }_0$ to zero. The imaginary part starts at zero, reaches ${\sigma }_0/2$ at $\omega \tau =1$, and falls as $1/\omega$. The real part falls as $1/{\omega }^2$ at high frequency. With an $e^{+i\omega t}$ convention, only the sign of the imaginary part changes. Positive $\mathrm{Re}\sigma$ gives positive cycle-averaged dissipation $⟨jE⟩=\mathrm{Re}\sigma |\hat{E}{|}^2/2$.

The supplied DC conductivity determines the transport time:Inertia matters when $\omega \tau \sim 1$, giving ${\omega }_c\simeq 4.02\times 10^{13}\mathrm{rad}{\mathrm{s}}^{-1}$, or ${\nu }_c={\omega }_c/(2\pi )\simeq 6.40\times 10^{12}\mathrm{Hz}$. Quoting the former number as hertz would introduce an erroneous factor of $2\pi$.

Let $e>0$ denote the elementary charge, so an electron's charge is $-e$. In the relaxation-time Drude model with positive transport effective mass,The steady drift velocity is ${\mathbf{v}}_d=-e\tau \mathbf{E}/m^{\ast }=-\mu \mathbf{E}$, defining positive electron mobility $\mu =e\tau /m^{\ast }$. The current density is opposite to the electron drift:By the definition of electrical conductivity, $\mathbf{J}=\sigma \mathbf{E}$, soIts units are $({\mathrm{m}}^{-3})(\mathrm{C})({\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1})=\mathrm{S}{\mathrm{m}}^{-1}$. The mobility definition alone establishes the result in the scalar linear-response regime; the relaxation-time equation supplies a microscopic approximation for $\mu$.

Interpret “3 mm wire” as a circular wire of diameter $3\mathrm{mm}$, and take the current density as uniform. One conduction electron per atom givesUsing $I=neA|v_d|$, the drift velocity corresponding to a current magnitude of $13\mathrm{A}$ isIt points opposite to conventional electric current. A simple distance scale over the $1/50=0.020\mathrm{s}$ cycle is $\boxed{|v_d|/f=2.71\mu \mathrm{m}}$, but this treats the magnitude as constant. For sinusoidal mains, alternating-current drift excursion gives zero net displacement after a full cycle.

If $13\mathrm{A}$ is the usual root mean square current, $v_{\mathrm{pk}}=\sqrt{2}|v_d|$ and the displacement amplitude and total absolute drift distance per cycle areIf $13\mathrm{A}$ instead denotes peak current, these become $0.431\mu \mathrm{m}$ and $1.72\mu \mathrm{m}$. The paper does not state peak versus RMS; the characteristic drift is the boxed $1.35\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-1}$ on whichever current convention is chosen. It also does not explicitly say diameter; taking $3\mathrm{mm}$ as the radius would reduce all these velocities and drift distances by four. None of these distances is the microscopic path travelled at roughly the Fermi velocity between scattering events; that path cannot be obtained from the supplied current alone.

Treat sodium as monovalent, with one mobile electron per atom. Its mass density is ${\rho }_m=0.97\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}=970\mathrm{kg}{\mathrm{m}}^{-3}$, and its molar mass is $M=0.023\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$. Using the Avogadro constant,The electron mobility $\mu =e\tau /m_e$ relates the magnitude of drift velocity to electric field. The Drude model then givesUsing $m_e=9.109\times 10^{-31}\mathrm{kg}$,The inferred time is the momentum relaxation time. Identifying it with the mean time between collisions is the intended simple isotropic Drude model assumption; small-angle scattering need not relax directed momentum in a single event.

The Hall effect is a transverse electric response to a magnetic force on moving charge carriers. Take conventional current density $\mathbf{J}=J_x\hat{\mathbf{x}}$ and magnetic field $\mathbf{B}=B_z\hat{\mathbf{z}}$ in a conducting slab. For one carrier type of number density $n$ and signed electric charge $q$, the drift velocity is ${\mathbf{v}}_d=\mathbf{J}/(nq)$. Carriers initially deflect sideways until a Hall electric field balances the magnetic Lorentz force:The Hall coefficient is thereforeFor electrons it is $-1/(ne)$, whereas a simple hole-dominated conductor gives $+1/(pe)$. The sign reveals the effective carrier sign; the magnitude estimates their density in the one-carrier approximation.

For width $w$ along $y$ and thickness $t$ along $z$, $I=J_{x}wt$. Define the signed Hall voltage by $V_H=E_{y}w$, namely the electric potential at the lower-$y$ edge minus that at the higher-$y$ edge. ThenThis explicit contact convention fixes the voltage sign. A Hall bar with longitudinal and transverse contacts measures both electrical resistivity and the Hall response. Reverse the field and take the part of the transverse voltage odd in $B_z$ to suppress longitudinal-voltage pickup from misaligned contacts; reversing current provides another useful check.

A Hall measurement is useful for carrier identification, semiconductor doping measurements and magnetic-field sensing. Combining a one-carrier $R_H$ with electrical conductivity gives a mobility magnitude $\mu =\sigma |R_H|$. However, mixed electron and hole conduction is not described by a single density: in weak fields,The response weights mobilities differently from ordinary conductivity. Complicated energy bands, anisotropic Fermi surfaces and multiple bands can likewise change the simple interpretation in metals; a positive Hall coefficient does not mean that the material contains positively charged fundamental electrons.

In a metal, a partially filled energy band or overlapping energy bands supplies electrons with nearby unoccupied states. At thermal equilibrium, oppositely directed group velocities are equally populated and the net electric current vanishes. An applied electric field distorts the occupation distribution, producing a small mean drift velocity. A filled energy band alone has no such low-field intraband response because its velocity contributions cancel over the entire Brillouin zone.

In the Drude model, let $n$ be the mobile electron number density, let $e>0$ be the elementary charge, and let $\tau$ be the momentum relaxation time. The mean motion obeysThe steady state is ${\mathbf{v}}_d=-e\tau \mathbf{E}/m$. The current density is opposite to the electron drift velocity:This gives microscopic Ohm's law, with electrical conductivity $\sigma$ in $\mathrm{S}{\mathrm{m}}^{-1}$ and electrical resistivity ${\rho }_{\mathrm{el}}$ in $\Omega \mathrm{m}$. For a uniform wire of length $\ell$ and cross-sectional area $A$, its electrical resistance is $R={\rho }_{\mathrm{el}}\ell /A$. The dissipated power per volume is $\mathbf{J}\cdot \mathbf{E}=\sigma E^2$, the Joule heating supplied by the external source.

The free electron model with Fermi-Dirac statistics improves the microscopic picture. The Pauli exclusion principle fills states up to the Fermi energy at low temperature. For a spherical, parabolic energy band with two spin states,Only states near the Fermi surface change their occupation in a weak-field response; the large equilibrium Fermi velocity is not the small drift velocity. The same $n{e}^2\tau /m$ formula results for this isotropic parabolic model when $\tau$ is evaluated for transport near the Fermi surface. In a crystal, the energy band group velocity, effective mass and possibly direction-dependent scattering replace the simplest bare-particle picture. The mean free path is approximately ${\ell }_{\mathrm{mfp}}=v_F\tau$.

A perfectly periodic static lattice is already included in Bloch states and does not by itself give a finite electrical resistivity. Momentum relaxation comes from defects, impurities, boundaries, and electron-phonon scattering. Independent scattering mechanisms approximately add their rates, giving Matthiessen's rule,The approximately temperature-independent ${\rho }_0$ is the residual resistivity. In ordinary simple metals, the phonon contribution is approximately proportional to temperature at high temperature; a three-dimensional acoustic-phonon transport model gives a $T^5$ contribution well below the relevant Debye temperature. Other mechanisms can dominate in particular materials or regimes, so neither power law is universal.

The classical Drude model incorrectly treats the equilibrium carrier statistics. Fermi-Dirac statistics explains the small electronic heat capacity, while electrons carrying both charge and heat give the approximate Wiedemann-Franz law ${\kappa }_e/(\sigma T)={\pi }^2{k}_B^2/(3e^2)$ for degenerate carriers with appropriate nearly elastic scattering. Both the simple electrical conductivity formula and this thermal relation depend on the assumptions about energy bands and scattering, rather than being exact for all metals.

The Drude model treats mobile electric charge carriers as independent particles with density $n$, signed electric charge $q$, isotropic mass $m$, and momentum relaxation time $\tau$. In uniform steady fields,At zero magnetic field the drift velocity is $q\tau \mathbf{E}/m$, giving electrical conductivity and electrical resistivityThe electrical conductivity is positive for either carrier sign. Scattering dissipates directed drift energy into the environment; a finite steady electric current is compatible with a short-lived drift momentum. The independent-carrier relaxation-time approximation omits detailed band anisotropy, quantum interference and microscopic collision correlations.

For electric current along $x$ and magnetic field $B\hat{\mathbf{z}}$, the magnetic force deflects carriers toward one side of the sample. electric charge accumulates until a transverse Hall field cancels that force in an open transverse circuit. With $J_y=0$, $E_y=v_{x}B$, soElectrons have negative $R_H$ in this convention, whereas positive carriers have positive $R_H$. For a strip of thickness $d$ along $z$, the magnitude of the Hall electric potential difference across its width is $|V_H|=|IB/(nqd)|$. Thus Hall measurements determine carrier sign and, within this one-carrier model, density, while combining $R_H$ with ${\sigma }_0$ determines charge carrier mobility $\mu =|q|\tau /m={\sigma }_0|R_H|$.

Keeping both in-plane electric current components, the resistivity tensor isThis fixes the off-diagonal sign convention and shows that inversion is not componentwise reciprocation. Under the Hall boundary condition $J_y=0$, the ideal one-carrier isotropic model gives $E_x/J_x={\rho }_0$, independent of magnetic field: there is no ordinary longitudinal magnetoresistance in this minimal model. Multiple carrier types or anisotropic bands can change both this conclusion and the simple $R_H=1/(nq)$ inference. Their electrical conductivity must be added before inverting to obtain the total electrical resistivity.