# Electrical resistivity and conductivity **Electrical resistivity**, written ρ, is the property of a material that measures how strongly it opposes the flow of [[Electric_current|electric current]]; **electrical conductivity**, σ = 1/ρ, measures how readily it lets current through. They are intrinsic to the material, where the resistance of a wire depends also on its length and cross-section: `R = ρ·L/A`.[^openstax-93] No other material property spans so wide a range: [[Copper|copper]] has ρ = 1.68×10⁻⁸ Ω·m, pure [[Silicon|silicon]] about 2×10³ Ω·m, [[Glass|glass]] 10¹⁰ to 10¹⁴ Ω·m and polytetrafluoroethylene (PTFE) beyond 10¹⁶ Ω·m, some twenty-five orders of magnitude between the best ordinary conductor and the best insulator.[^openstax-93][^spec-m18] In the microsim below the reader slides the temperature from 4 K to 1,000 K and turns an impurity dial while a box of conduction [[Electron|electrons]] drifts in a field. The readout computes the Drude conductivity `σ = n·e²·τ/m`, the number of carriers times the square of their [[Electric_charge|charge]] times the mean time τ between collisions over the electron mass; the resistivity it implies, `ρ(T) = ρ₀·[1 + α·(T − T₀)]`, with Matthiessen's impurity floor `ρ = ρ_lattice(T) + ρ_imp`; and where the material lands on the ladder from copper through silicon and glass to PTFE. The copper preset (1.68 μΩ·cm, α = 0.0039/K) is the sim's home position.[^spec-m18] The equation answers one question: how many carriers a material has and how far each gets before it is scattered. On the [[Materials_science]] flagship this article serves the *Electrical conduction* section of Part III — Fundamentals › Properties, with the shared transport sim (C35); its thermal sibling is [[Thermal_conductivity_and_resistivity]], and the copper preset is the variant placed on [[Copper]] for the [[Chemistry]] flagship's "Cu → conductivity" spine. ## Definition Resistivity is defined to depend on the material alone; the definition runs from a uniform wire to the tensor a crystal needs. ### Ideal case For a uniform bar of length L and cross-section A carrying a current I under a [[Voltage|voltage]] V, [[Ohm's_law|Ohm's law]] gives R = V/I, and the resistance scales with the geometry as `R = ρ·L/A`.[^openstax-93] Doubling the length doubles the resistance, doubling the area halves it, and what is left after the geometry is divided out is the resistivity, in ohm-metres. A kilometre of 2.5 mm² copper wire has R = 1.68×10⁻⁸ × 1,000/(2.5×10⁻⁶) = 6.7 Ω.[^derived-er] The conductivity is the reciprocal, σ = 1/ρ, in siemens per metre; copper's is 6.0×10⁷ S/m.[^derived-er] ### General scalar quantities Inside the bar the current is spread over the area as a current density J = I/A and the voltage along the length as an [[Electric_field|electric field]] E = V/L. Ohm's law then takes its local form, `J = σ·E` or `E = ρ·J`, which holds point by point and needs no particular shape.[^openstax-93] This is the form the microsim uses: a field is applied across the electron box, the electrons drift, and the resulting current density divided by the field is the conductivity on the readout.[^spec-m18] ### Tensor resistivity In a crystal without cubic symmetry the current need not be parallel to the field, and σ becomes a symmetric tensor, J_i = Σ σ_ij·E_j, with three principal conductivities; [[Graphite|graphite]], conducting far better within its layers than across them, is the extreme case. A [[Magnetic_field|magnetic field]] adds antisymmetric off-diagonal terms, the Hall effect. Polycrystalline metals average over grain orientations and behave as scalars. ## Causes of conductivity A material conducts if it has charge carriers and the carriers are mobile. ### Band theory simplified In a solid the electron [[Energy_level|energy levels]] of the atoms broaden into bands, and whether the solid conducts depends on how the bands are filled.[^openstax-v3-95] A band that is completely full carries no current, because every state that could move an electron forward is already occupied; only a partly filled band conducts, and a metal has one. An insulator has a full valence band separated by a large [[Band_gap|band gap]] from an empty conduction band, so that thermal energy cannot lift electrons across; a [[Semiconductor|semiconductor]] is an insulator whose gap is small enough that some electrons do cross at room temperature, and whose conductivity can be set at will by doping. The gaps of the [[Electronic_band_structure|band-structure]] sibling sim are the numbers that matter: about 1.1 [[Electronvolt|eV]] for silicon and 5.5 eV for [[Diamond|diamond]], against a thermal energy kT of 0.026 eV at room temperature, k being the [[Boltzmann_constant|Boltzmann constant]].[^spec-m18][^openstax-v3-95] The ratio of gap to kT, in the exponent of the carrier density below, is what the twenty-five decades of the ladder measure. ### In metals A metal has of order one free electron per atom, so many carriers that their number never limits the current; what limits it is scattering. In the [[Drude_model|Drude model]] of 1900 the electrons form a gas that accelerates in the field between collisions and loses its drift at each one, giving `σ = n·e²·τ/m` with τ the mean free time.[^drude1900][^likharev-ch6] Copper makes the numbers concrete. With about 8.5×10²⁸ conduction electrons per cubic metre (one per atom) and the measured σ = 6.0×10⁷ S/m, the Drude time is τ = σ·m/(n·e²) = 2.5×10⁻¹⁴ s, and the [[Electron_mobility|mobility]] μ = e·τ/m = 4.4×10⁻³ m²/(V·s).[^derived-er][^cu-density] The drift is slow: a current of 1 A in a 1 mm² wire is J = 10⁶ A/m², and the drift velocity v_d = J/(n·e) is 0.07 mm/s, although the electrons themselves move at the Fermi velocity, of order 10⁶ m/s, and travel some 40 nm between collisions.[^derived-er][^openstax-92] The signal travels at nearly the speed of light because the field, not the electrons, carries it. In the sim the impurity dial adds scattering centres to the box and shortens τ; the temperature slider shortens it further as the lattice vibrates harder.[^spec-m18] *Try:* hold T at 300 K and turn the impurity dial up: the electrons' random flights shorten and σ falls; then drop T toward 4 K with impurities present and watch ρ flatten onto the residual floor instead of vanishing. ### In semiconductors and insulators In a semiconductor the number of carriers, not their mobility, is the variable. The intrinsic density follows `n_i = (N_c·N_v)^{1/2}·exp(−E_g/(2·k·T))`, about 10¹⁰ per cm³ for silicon at 300 K, ten trillion times fewer than in copper, and the conductivity is `σ = e·(n·μ_n + p·μ_p)` with both electrons and holes contributing.[^spec-m38][^openstax-v3-96] [[Doping_(semiconductor)|Doping]] with one donor atom in a million raises n by six orders of magnitude and turns the same crystal from a poor insulator into a fair conductor; a [[Transistor|transistor]] is that change under electrical control. An [[Insulator_(electricity)|insulator]] is the same physics with a gap too wide to cross: the exponential makes diamond, glass and PTFE effectively carrier-free, and their residual conductivity comes from impurities, surface moisture and ionic motion rather than from the band. ### In ionic liquids/electrolytes In an [[Electrolyte|electrolyte]] the carriers are [[Ion|ions]], each dragging its solvent shell through the liquid, and the conductivity is the sum over ion species of concentration times charge times mobility. It rises with concentration until the ions begin to hinder one another, rises with temperature as the liquid thins, and is orders of magnitude below a metal's, because ions are heavy and slow where electrons are light and fast. [[Molar_conductivity|Molar conductivity]], the conductivity per unit concentration, is the chemist's measure of how completely a solute dissociates; molten salts and the electrolytes of [[Electric_battery|batteries]] follow the same rules with no solvent at all.[^openstax-chem16] ### Superconductivity Below a critical temperature certain materials lose their resistivity entirely: a current set up in a superconducting ring persists without a source. Heike Kamerlingh Onnes found the effect in [[Mercury_(element)|mercury]] at 4.2 K in 1911, shortly after he had learned to liquefy [[Helium|helium]]; niobium superconducts below 9.3 K, the wires of [[Superconducting_magnet|superconducting magnets]] below about 10 to 18 K, and the cuprate YBa₂Cu₃O₇ below about 92 K.[^openstax-v3-98] The [[BCS_theory|BCS theory]] explains the conventional cases as electrons pairing through the lattice into a state that scattering cannot break up one electron at a time; zero resistivity comes with the expulsion of magnetic flux, the [[Meissner_effect|Meissner effect]].[^openstax-v3-98] A superconductor is a separate state, not a very good metal, and the sim's ladder stops at copper. ### Plasma A [[Plasma_(physics)|plasma]] is a gas hot enough that its atoms are ionized, and its free electrons conduct much as a metal's do, with collisions against ions setting τ. Because the Coulomb cross-section falls as the electrons speed up, the conductivity of a fully ionized plasma rises as T^{3/2} and is nearly independent of density; the plasma of a [[Tokamak|tokamak]] conducts better than copper. ## Resistivity and conductivity of various materials The sim's ladder and the Portal Book's table agree on the ordering: the pure metals within a factor of two of one another, the alloys and resistance wires ten to a hundred times worse, the semiconductors in between, and the insulators fifteen or more decades further up.[^openstax-93][^spec-m18] ### Table Values near 20 °C from the Portal Book's Table 9.1, with the temperature coefficient α where the book gives it (page to pin); the sim's four ladder rungs are marked.[^openstax-93][^spec-m18] | Material | ρ (Ω·m) | α (per K) | Note | |---|---|---|---| | Silver | 1.59×10⁻⁸ | 0.0038 | best metallic conductor | | Copper | 1.68×10⁻⁸ | 0.0039 | sim preset, ladder rung | | Aluminium | 2.65×10⁻⁸ | 0.0039 | overhead lines | | Tungsten | 5.60×10⁻⁸ | 0.0045 | lamp filaments | | Nichrome | 1.00×10⁻⁶ | 0.0004 | heating elements | | Silicon (pure) | 2.3×10³ | −0.075 | ladder rung | | Glass | 10⁹–10¹⁴ | — | ladder rung | | PTFE (Teflon) | >10¹³ | — | ladder rung; handbook values reach 10¹⁶–10¹⁸ | The negative α of silicon is the semiconductor signature: warming it creates carriers faster than it slows them. ## Temperature dependence The sign of the change of ρ with temperature says what kind of conductor a material is. ### Linear approximation Over a modest range the change is linear, `ρ(T) = ρ₀·[1 + α·(T − T₀)]`, with ρ₀ the resistivity at the reference temperature T₀ (usually 20 °C) and α the temperature coefficient of resistivity.[^openstax-93] For copper, α = 0.0039/K means that a coil at 100 °C has 1.31 times its room-temperature resistance, which is how a copper winding reports its own temperature and how a [[Platinum|platinum]] resistance thermometer works.[^derived-er] Extrapolated to 77 K the same line predicts ρ = 2.6×10⁻⁹ Ω·m, a sixth of the room-temperature value; the true value depends on purity, because the line has no floor.[^derived-er] The sim draws the linear law as a dashed line over its full Matthiessen curve.[^spec-m18] ### Metals In a pure metal the lattice vibrations, the [[Phonon|phonons]] of the [[Debye_model|Debye model]], scatter the electrons, and ρ_lattice rises linearly with T above the Debye temperature and far faster, as T⁵, below it, where fewer and softer phonons are available. Impurities and [[Crystallographic_defect|defects]] scatter independently of temperature, and Matthiessen's rule of 1864 states that the two contributions add: `ρ = ρ_lattice(T) + ρ_imp`.[^matthiessen1864][^likharev-ch6] Cooled toward absolute zero, a metal's resistivity does not vanish but settles onto the residual value ρ_imp, and the ratio of room-temperature to residual resistivity is the standard measure of a metal's purity. This floor is the impurity dial of the sim: with it at zero the curve heads for the origin, with any impurity at all it flattens.[^spec-m18] ### Semiconductors A semiconductor's resistivity falls steeply as it warms, because the carrier density in the exponential above grows far faster than the mobility falls: between 300 K and 400 K intrinsic silicon gains carriers by a factor of about 350.[^derived-er][^spec-m38] A doped semiconductor has three regimes: at low temperature the dopants freeze out and ρ rises on cooling; over a broad middle range they are all ionized, n is fixed and ρ rises gently with T like a metal's; at high temperature intrinsic carriers overwhelm the dopants and ρ falls exponentially again.[^openstax-v3-96] ### Kondo insulators Not every rise of resistivity on cooling means a semiconductor. Jun Kondo showed in 1964 that a dilute magnetic impurity in a metal scatters conduction electrons more strongly as the temperature falls, producing a resistance minimum followed by a logarithmic rise, the Kondo effect.[^kondo1964] In a Kondo insulator the same spin-flip interaction acts at every lattice site: a narrow gap opens at low temperature, and a material that is metallic when warm becomes insulating when cold, its resistivity climbing as T falls instead of levelling onto a residual floor. Such compounds lie outside the Drude picture and the sim. ## Complex resistivity and conductivity Under an [[Alternating_current|alternating]] field the electrons cannot follow instantly, and the Drude model gives a frequency-dependent conductivity, `σ(ω) = σ₀/(1 − i·ω·τ)`, complex because the current lags the field.[^likharev-ch6] The lag is negligible until ω·τ approaches 1, which for copper's τ = 2.5×10⁻¹⁴ s is a frequency of about 6 THz, in the far infrared; below that a metal is the simple conductor of the sim, and above it the imaginary part takes over and the metal turns from an absorber of [[Electromagnetic_radiation|radiation]] into a reflector, which is why metals shine.[^derived-er] In insulators the complex conductivity describes dielectric loss, the small in-phase current that heats a capacitor's dielectric or a dinner in a microwave oven. A related high-frequency effect is the skin effect: an alternating current crowds into a surface layer of thickness `δ = (2·ρ/(ω·μ₀))^{1/2}`, about 8 mm in copper at 60 Hz and 0.07 mm at 1 MHz, so the resistance of a thick conductor rises with frequency.[^derived-er] ## Resistance versus resistivity in complicated geometries For any shape other than a uniform bar, R = ρ·L/A is replaced by an integral over the current path. A conductor whose cross-section varies slowly along its length has `R = ρ·∫dl/A(l)`; a truncated cone of length L and end radii r₁ and r₂ gives R = ρ·L/(π·r₁·r₂), which reduces to the bar when the radii are equal. A thin film of thickness t is characterized by its sheet resistance, R_s = ρ/t, in ohms per square: any square of the film, whatever its size, has that resistance edge to edge, and a strip n squares long has n·R_s. A 100 nm copper film would have R_s = 0.17 Ω per square at the bulk resistivity, and more in practice, because a film thinner than the 40 nm mean free path scatters electrons at its surfaces, which matters for every [[Integrated_circuit|integrated circuit]] interconnect.[^derived-er] The sim's readout is a bar because a bar is the one shape where resistance and resistivity differ only by L/A. ## Resistivity-density product Where a conductor must be carried, by a pylon, an aircraft or a satellite, the figure of merit is not ρ alone but the product of resistivity and [[Density|density]], ρ·d, the mass of conductor needed for a given resistance per unit length. Copper, with ρ = 1.68×10⁻⁸ Ω·m and d ≈ 8.9 g/cm³, has ρ·d ≈ 1.5×10⁻⁴ Ω·kg/m²; [[Aluminium|aluminium]], with ρ = 2.65×10⁻⁸ Ω·m and d = 2.7 g/cm³, has 7.2×10⁻⁵, about half.[^derived-er][^openstax-93][^cu-density] An aluminium line therefore weighs half as much as a copper line of the same resistance, which is why the [[Electric_power_transmission|overhead transmission lines]] of the [[Electrical_grid|grid]] are aluminium strands around a [[Steel|steel]] core, though the aluminium must be thicker and the towers must carry the extra sag. [[Silver|Silver]], the better conductor, loses on this measure because it is dense; [[Sodium|sodium]] would win if it were not soft, reactive and impossible to insulate; and [[Gold|gold]] is used for its inertness, not its conductivity. ## History The quantitative science began with Georg Ohm, whose *Die galvanische Kette, mathematisch bearbeitet* of 1827 stated that the current through a conductor is proportional to the voltage across it and inversely proportional to a property of the conductor.[^ohm1827] Augustus Matthiessen's measurements on [[Alloy|alloys]] in the 1860s separated the temperature-dependent and impurity parts of that property.[^matthiessen1864] Paul Drude's electron-gas model of 1900 gave the first microscopic account,[^drude1900] and the quantum theory of metals of the late 1920s repaired its failures, replacing the classical gas by a [[Fermi_gas|Fermi gas]] and explaining why a perfect lattice would not scatter at all.[^likharev-ch6] Kamerlingh Onnes's discovery of [[Superconductivity|superconductivity]] in 1911 showed that resistivity can vanish outright.[^openstax-v3-98] ### John Walsh and the conductivity of a vacuum Before Ohm, the question was what could conduct at all. John Walsh, a Fellow of the Royal Society, reported in 1773 a series of experiments on the torpedo, the electric ray, showing that its shock passed through conductors and was stopped by insulators exactly as the discharge of a Leyden jar was, which established the shock as electrical and the fish as a living battery.[^walsh1773] The pair's section is named for a further experiment of Walsh's on whether such a discharge would cross an evacuated space; no open source on this page's shelf documents it, so it is left as a claim to source rather than reproduced.[citation needed][^walsh-cn] The general answer came later: a vacuum has no carriers and does not conduct, but a discharge through a nearly evacuated tube glows, and the study of those glows led to the electron itself. ## See also - [[Drude_model]] — the electron-sea box of the sim - [[Ohm's_law]] — the definition's starting point - [[Electron_mobility]] — μ = e·τ/m - [[Copper]] — the placed preset and the Chemistry flagship's variant - [[Electrical_conductor]] - [[Insulator_(electricity)]] - [[Thermal_conductivity_and_resistivity]] — the thermal sibling and the Wiedemann–Franz link - [[Superconductivity]] ## Notes Numbers marked as computed for this article are derived from the equations on the page and the cited constants; the sim's Matthiessen curve is a model for ordinary metals and is marked ILLUSTRATIVE in its HUD outside the copper preset. ## References [^openstax-93]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 9, "Current and Resistance" (pp. 373–416), §9.3 Resistivity and Resistance: `R = ρ·L/A`, `E = ρ·J`, `ρ = ρ₀·[1 + α·(T − T₀)]`, Table 9.1 Resistivities and Temperature Coefficients (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-92]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 9, §9.2 Model of Conduction in Metals: drift velocity `I = n·q·A·v_d` (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-v3-95]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, "Condensed Matter Physics" (pp. 393–440), §9.5 Band Theory of Solids (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^openstax-v3-96]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, §9.6 Semiconductors and Doping (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^openstax-v3-98]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, §9.8 Superconductivity: Onnes 1911, the critical-temperature table, BCS pairing and the Meissner effect (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^openstax-chem16]: Flowers, Paul; Neth, Edward; Robinson, William (2019). *Chemistry: Atoms First 2e* (OpenStax). Chapter 16, "Electrochemistry" (pp. 753–790): ionic conduction in electrolytes (page to pin). https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first-2e [^likharev-ch6]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 6, "Elements of kinetics" (pp. 187–225), §6.2 The Ohm law and the Drude formula, including the AC form and the quantum (Fermi-gas) correction (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^drude1900]: Drude, P. (1900). "Zur Elektronentheorie der Metalle." *Annalen der Physik* 306 (3): 566–613. https://doi.org/10.1002/andp.19003060312 [^kondo1964]: Kondo, J. (1964). "Resistance Minimum in Dilute Magnetic Alloys." *Progress of Theoretical Physics* 32 (1): 37–49. https://doi.org/10.1143/PTP.32.37 [^matthiessen1864]: Matthiessen, A.; Vogt, C. (1864). "On the Influence of Temperature on the Electric Conducting-Power of Alloys." *Philosophical Transactions of the Royal Society of London* 154: 167–200. DOI to pin. [^walsh1773]: Walsh, J. (1773). "Of the Electric Property of the Torpedo." *Philosophical Transactions of the Royal Society of London* 63: 461–480. DOI to pin. [^walsh-cn]: Citation needed. The vacuum experiment that names the pair's section is not documented in any Portal Book or open text on this page's shelf; the record that would settle it is Walsh's correspondence and papers in the *Philosophical Transactions* of the 1770s (page to pin). [^ohm1827]: Ohm, Georg Simon (1827). *Die galvanische Kette, mathematisch bearbeitet.* Berlin: T. H. Riemann. Pre-DOI work; no DOI exists. [^cu-density]: Derived for this article: n = ρ_m·N_A/M with the handbook density and molar mass of copper (8.96 g/cm³, 63.55 g/mol) gives 8.5×10²⁸ atoms per cubic metre, one conduction electron each; density against *University Physics Volume 1* (2016), Chapter 14, Table 14.1 (page to pin), molar mass against *University Physics Volume 2*, Appendix F, pp. 727–728; aluminium density 2.70 g/cm³ from the same table (page to pin). [^derived-er]: Computed for this article from the equations on the page with the Table 9.1 values of [^openstax-93], the constants of *University Physics Volume 2* Appendix C (pp. 719–720) and n from [^cu-density]: the wire resistance, σ, τ, μ, drift velocity, mean free path (with a Fermi velocity of 1.57×10⁶ m/s), the 100 °C and 77 K linear extrapolations, the silicon carrier ratio (E_g = 1.12 eV), the Drude roll-off frequency, the skin depths, the sheet resistance and the ρ·d products. [^spec-m18]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M18: sim concept (`wt-solid.transport`), Drude `sigma = n·e^2·tau/m`, `rho(T) = rho_0·[1 + alpha·(T − T_0)]`, Matthiessen `rho = rho_lattice(T) + rho_imp`, controls T (4–1,000 K) and an impurity dial, the 25-decade ladder Cu → Si → glass → PTFE, and the Cu preset (1.68 μΩ·cm, α 0.0039/K) placed on Copper. [^spec-m38]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M38 (Doping): `n_i = sqrt(N_c·N_v)·exp(−E_g/(2·k·T))`, Si n_i ≈ 10¹⁰ cm⁻³ at 300 K, `sigma = e·(n·mu_n + p·mu_p)`, and row M39: E_g Si 1.12 eV, diamond 5.5 eV. ## Further reading - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*, Chapter 9 (current and resistance), and *Volume 3*, Chapter 9 (condensed matter). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 - Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM*, Chapter 6, for the Drude formula and its quantum correction. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics ## External links - The Wikipedia pair's *External links* section lists resistivity tables and calculators; the open texts above are the Portal Book sources of this page. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Electrical_resistivity_and_conductivity.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Electrical resistivity and conductivity* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Electrical_resistivity_and_conductivity.html" data-title="Electrical resistivity and conductivity"></div> *Built from `MICROSIM_GUIDE/specs/sims/Electrical_resistivity_and_conductivity.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity) : [Wikitube](https://en.wikitube.io/wiki/Electrical_resistivity_and_conductivity) · pinned revision [1374114780](https://en.wikipedia.org/w/index.php?oldid=1374114780) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]], [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M18 · sim pending (matter/Electrical_resistivity_and_conductivity).*