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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.

Tunneling through a controllable vacuum gap by Gerd Binnig and others made barrier width experimentally adjustable. Their apparatus used a pointed electrical conductor opposite a conducting plate, with fine positioning by piezoelectricity. Rubber-tube support and magnetic levitation above a cooled bowl exhibiting superconductivity suppressed vibrations; the junction itself remained near room temperature.^[ref]

The readout was electrical resistance versus displacement at a fixed electric potential difference of 60 mV of either polarity. They observed an exponential function dependence spanning four orders of magnitude. Cleaning and better vacuum increased the inferred barrier from approximately 0.6-0.7 eV to 3.2 eV. Reproducible curves and the cleaning response argued against deforming contamination layers; an appropriate slope also helped exclude an ordinary conducting contact. The paper reports residual contamination, electronics-limited range and a jump attributed to release of thermal stress; its curve origins generally do not determine absolute gap widths.^[ref]

The ideal tunnel junction model explains the slope. At fixed small electric potential difference, fixed tip geometry and approximately unchanged density of states, take an effective barrier energy $\Phi$ above the relevant electron states and approximate the current magnitude byThe coefficient $C$ includes the applied electric potential difference, available occupied and empty quantum states, and tip geometry. A single transmission probability is not an electric current: a junction combines contributions from many electron states in both directions, with the electric potential difference producing a net imbalance. Its tilted, material-dependent barrier is only approximated by the equal-exterior rectangular potential barrier; the exact single-particle formula above is not a complete junction model.

For the chosen model value $\Phi =4$ electronvolts, $\kappa \simeq 10.25{\mathrm{nm}}^{-1}$. Increasing separation by $0.1\mathrm{nm}$ multiplies $I$ by about $0.129$, a reduction by about $7.8$. The following diagram chooses $I(0.4\mathrm{nm})=1\mathrm{nA}$ solely for illustration; it contains no measured data and is not a fit to the 1982 curves.

Distance calibration is essential. In Scanning Tunneling Microscopy: From Birth to Adolescence by Gerd Binnig and others, the authors describe an initial inferred barrier near 1 eV. Recalibration found only half the quoted piezo displacement sensitivity and raised the inferred barrier above 4 eV. Thus an exponential trace alone was insufficient evidence for tunnelling through vacuum. The model makes the sensitivity explicit: the inferred $\Phi ={\hslash }^2(d\ln I/ds{)}^2/(8m_e)$ depends quadratically on the distance scale.^[ref]

For scanning tunnelling microscopy, feedback control adjusts the vertical position to hold electric current fixed during a lateral scan; the recorded height follows separation only when the surface is electronically homogeneous. Changes in the local density of states can also change apparent height. This limitation is already explicit in the authors' account of constant-current imaging.^[ref]

A current element is the oriented infinitesimal wire contribution $Id{\mathbf{\ell }}^{\prime }$ inside the Biot-Savart law. For uniform axial current density across wire cross-sectional area $A$, $I=JA$ and $d^3{r}^{\prime }=Ad{\ell }^{\prime }$ imply $\mathbf{J}{d}^3{r}^{\prime }=Id{\mathbf{\ell }}^{\prime }$. This reduces a volume integral to a line integral in the thin-wire approximation. It is an integration contribution, not an isolated steady source. Conventional current runs opposite the drift of negatively charged electrons.

An eight-membered cyclic conjugated system with eight $\pi$ electrons. A planar bond-alternating model has the D4h molecular point group. Constraints on planarity matter when discussing its molecular geometry and electronic stability.

The cyclopentadienyl ligand is based on ${\mathrm{C}}_5{\mathrm{H}}_5^{-}$. In its common ${\eta }^5$ mode it binds through five ring carbons and donates six electrons in ionic electron counting. Ring substitution can create distinct NMR environments.

A massless-cone description permits a nonzero cyclotron effective mass. For a circular constant-energy contour of area $S(E)=\pi E^2/(\hslash v_F{)}^2$, the semiclassical definition givesThe vanishing mass parameter in a Dirac Hamiltonian and the finite cyclotron response away from neutrality are different statements. This makes the temperature dependence of Shubnikov-de Haas oscillations a useful test of the dispersion rather than a measurement of an electron's vacuum rest mass.

The radius of a charged particle orbit perpendicular to a uniform magnetic field. In a parabolic band $r_c=m^{\ast }{v}_{\perp }/(|q|B)$, or $r_c=\hslash k_F/(e|B|)$ for a Fermi-level electron. Charge sign sets the curvature direction, while the radius uses magnitudes.

A Daniell cell uses the zinc/${\mathrm{Zn}}^{2+}$ and ${\mathrm{Cu}}^{2+}$/copper pairs. In the separated-compartment teaching version, a zinc electrode contacts an aqueous chemical solution containing ${\mathrm{Zn}}^{2+}$, and a copper electrode contacts one containing ${\mathrm{Cu}}^{2+}$. A salt bridge supplies an ionic connection; a wire and load supply an electronic connection. During discharge,The zinc electrode loses material and copper is deposited at the cathode. Each half-reaction conserves chemical atom counts and electric charge: the first has net charge zero on each side, the second $+2e-2e=0$ on the left and zero on the right. The net reaction has $+2e$ on both sides, with no electrons left in its chemical equation. The Zn/Cu teaching cell and these half-reactions are described in Florida State University, Redox Reactions in Voltaic Cells: Construction of a Potential Series, lab background.

Creating one mole of ${\mathrm{Zn}}^{2+}$ adds $2F$ of positive electric charge to the anode compartment's ionic inventory; consuming one mole of ${\mathrm{Cu}}^{2+}$ removes $2F$ of positive electric charge from the cathode compartment. Here $F$ is the Faraday constant. In an ideal bookkeeping example using a potassium nitrate salt bridge, two moles of nitrate anions entering the anode compartment compensate the first change, and two moles of potassium cations entering the cathode compartment compensate the second. Other ions in the cell can share the actual transport. The two compensating inventory changes do not mean that $4F$ flowed through the load: the same series electric current transports $2F$ through each cross-section of the circuit per mole of net reaction. See salt bridge.

Under standard-state conditions near $298\mathrm{K}$, tabulated reduction potentials give a cell electromotive force of approximately $1.10\mathrm{V}$. This is a reversible reference value, not a promised loaded reading; actual composition matters through thermodynamic activity. The tabulated copper and zinc values and their subtraction are given in Purdue University, Electrochemical Reactions: standard-state reduction potentials. For the net chemical equation as written, the molar Gibbs free energy change obeysThus $\mathcal{E}=1.10\mathrm{V}$ corresponds to about $-212\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ of net reaction. At fixed temperature and pressure, $-{\Delta }_{r}G$ is the maximum reversible electrical work per mole for this cell reaction. It is not automatically all the heat released by a finite-current cell. See 17.4 Potential, Free Energy, and Equilibrium (§17.4: cell potential and free energy).

The Davisson–Germer experiment turned elastic scattering from nickel into evidence for a matter wave. The search became a deliberate wave test in 1926, with diffraction beams found early in 1927. The earlier crystal investigation had begun in 1925; the de Broglie wavelength proposal was already present in the 1924 thesis.^[ref]

A heated tungsten ribbon supplied electrons; gun apertures selected a narrow beam. Two $1\mathrm{mm}$ apertures were $8\mathrm{mm}$ apart; the gun ended $7\mathrm{mm}$ from a nickel crystal cut parallel to its $\{111\}$ planes. An adjustable electric potential difference set the speed. The beam struck normally inside an evacuated, field-free enclosure with target and outer electrodes at one potential. A double-box collector, $11\mathrm{mm}$ from the target with a $1\mathrm{mm}$ entrance, selected directions. A retarding electric potential difference rejected slower electrons; the inner collector fed a sensitive galvanometer. Moving the collector scanned colatitude; rotating the crystal scanned azimuth.^[ref]

The readout was collected electric current, not a photograph of electron paths. Voltage scans held the gun-plate current constant because incident-beam current could not be held constant; the overall current trend therefore needed caution. A strong feature developed at $54\mathrm{V}$ near $50^{\circ }$ colatitude. Surface heating distinguished crystal beams from adsorbed-gas beams; finite apertures and background shifted or broadened apparent peaks.^[ref]

At the bright beam's colatitude, an azimuth scan revealed three companion directions, matching crystal symmetry. The test compared wavelengths inferred from diffraction angles and known crystal spacings with wavelengths inferred from accelerating voltage and momentum; the 1937 lecture displayed this comparison against $U^{-1/2}$. Agreement across beam directions supplied more evidence than one angular peak.^[ref]

The inference is about coherent quantum phases in scattering probabilities. It does not establish a wavy particle trajectory, and a collector-current experiment does not resolve individual detection events. Predicting the full intensity additionally requires the material's scattering amplitude, its surface condition, and instrumental angular acceptance; the de Broglie wavelength alone predicts none of those strengths. The following finite-row electron diffraction model isolates phase addition without attempting to reproduce the nickel measurements.

For mobile thermal electrons with a fixed neutralizing background, the Debye length is the characteristic distance over which plasma screening suppresses an electrostatic perturbation. Linearizing the Boltzmann density response and using Poisson's equation gives the displayed scale. Other mobile species contribute additively to ${\lambda }_D^{-2}$ through $n_s{q}_s^2/({ϵ}_0{k}_B{T}_s)$.

Scattering that resolves the constituents of a hadron at large spacelike momentum transfer, with an inclusive sum over final hadron states.

In electron scattering from a proton, write the reaction as $e^{-}p\to e^{-}X$: the outgoing electron is measured, while $X$ denotes all allowed final hadron states. Large momentum transfer supplies a short-distance probe; a large final invariant mass separates the deep continuum from elastic scattering and individual particle resonances. The useful surprise is that an extended proton can have a response characteristic of much smaller constituents. Deep inelastic scattering turns measurements outside the proton into constraints on its internal parton distribution functions.

The SLAC-MIT deep inelastic electron scattering experiment connects this definition to an actual measurement. Its evidence came from distributions of outgoing electron energies and scattering angles, rather than a photograph of a quark. Bjorken scaling concerns deep-inelastic structure functions after known scattering factors have been accounted for; it does not say that the raw differential scattering cross-section is constant.

Deferred charge in CCD readout reaches later output samples than its originating packet. George Elwood Smith describes surface states at the silicon–silicon dioxide interface capturing electrons and releasing them later, smearing the image. A buried channel keeps the stored packet away from that interface.^[ref]

An original comparison separates two interpretations of the same main-packet deficit. If a single transfer defers fraction $ϵ$ by exactly one sample and loses nothing, an isolated input packet $N$ produces samples ${\eta }_{\mathrm{tr}}N$ followed by $ϵN$: summing both recovers $N$. A truly lossy transfer yields only ${\eta }_{\mathrm{tr}}N$, with no later recovery. For $N=10000$ and $ϵ=0.01$, both give $9900$ in the intended sample, but only the deferred case gives a $100$-electron trail. Such a trail redistributes position information even when total electric charge is conserved. Repeated delays require a time-response model; $N_{\mathrm{in}}{\eta }_{\mathrm{tr}}^n$ alone cannot predict trail shape. Electron-hole recombination, escape to another contact, or release beyond the recorded window can instead remove signal from the measured sum.

Consider an ideal degenerate electron gas at zero temperature. Counting two spin states per phase space cell gives number density $n_e=p_F^3/(3{\pi }^2{\hslash }^3)$. The isotropic momentum flux gives pressureHere $m_e$ is electron mass, $c$ is the speed of light, $\hslash =h/(2\pi )$ with Planck constant $h$, and special relativity gives $v(p)=pc/\sqrt{p^2+m_e^2{c}^2}$. The factor $1/3$ averages the directional momentum flux, and the spin factor is already included. This normalization makes $P_{\ast }$ a pressure, not an energy per electron.

For $x\ll 1$, expand the denominator: $I(x)\sim x^5/5$ and $P\simeq n_e{p}_F^2/(5m_e)$. For $x\gg 1$, the integrand approaches $t^3$: $I(x)\sim x^4/4$ and $P\simeq n_e{p}_{F}c/4$. At fixed composition, mass density is $\rho ={\mu }_e{m}_u{n}_e$, with ${\mu }_e$ the mean molecular weight per electron and $m_u$ the unified atomic mass unit. Since $n_e\propto x^3$, the logarithmic stiffness isPressure keeps rising; its fractional gain per fractional compression becomes smaller. Relativistic electrons approach a speed ceiling while their momenta keep increasing. Thus the high-density equation of state loses the extra compression leverage of a nonrelativistic white dwarf. The Chandrasekhar mass from central polytropic constants supplies the corresponding global mass calculation.

For a calculated local example with ${\mu }_e=2$, $x=1$ gives $\rho \simeq 1.95\times 10^9\mathrm{kg}{\mathrm{m}}^{-3}$, $I(1)=0.1537384$ and $P\simeq 7.38\times 10^{21}\mathrm{Pa}$. Its stiffness is $1.53314$, intermediate between the two limits. These are ideal equation of state values at one point, not a measured central mass density or a whole white dwarf model. Finite temperature and interactions are excluded.

The fraction of a specified population of chemical atoms that has become ionized. For singly ionized atoms with no external charge source, charge neutrality gives equal free-electron and ion number densities. Its equilibrium value depends on both temperature and total number density through the Saha ionization equation.

American Journal of Physics 57, 117–120. Original experimental paper reporting single-electron interference recorded with an electron biprism and a position-sensitive counting detector.

A framework describing electronic ground-state properties using the electron density. Practical approximations require an exchange-correlation functional; this does not mean that explicit electrons have been replaced by a classical atom-only force field.

Detector conversion gain is the number of collected electrons per analogue-to-digital unit. With independent Poisson distribution photoelectron counts, the shot-noise variance in count units is the mean count divided by the gain. Detected-photon and electron conventions agree only when each detected photon contributes one electron.

The Dewar-Chatt-Duncanson model describes an alkene–metal bond through donation from the filled alkene $\pi$ orbital to the metal and pi backbonding from the metal into alkene ${\pi }^{\ast }$. Backbonding weakens and lengthens the alkene double bond. A $d^0$ early-metal ion has little conventional $d$-electron backbonding.

Dielectric breakdown is failure of an insulating regime under a sufficiently strong electric field, allowing substantial electric current through the material. For example, an electron freed from a chemical atom or molecule can become a mobile charge carrier. The small-field linear electric susceptibility model does not describe this conducting state.^[ref]

A free electron is resonantly captured while exciting the ionic core, producing a doubly excited state above the parent-ion threshold. Capture becomes permanent recombination only if the state undergoes radiative stabilization rather than autoionization. For resonance $d$, entrance ion $i$, and Maxwellian electrons, ${\alpha }_d=(g_d/(2g_i))(h^2/(2\pi mkT){)}^{3/2}{e}^{-E_d/kT}{A}_{di}^a{A}_{\mathrm{stab}}^r/(A_{\mathrm{tot}}^a+A_{\mathrm{tot}}^r)$. Detailed balance determines the capture rate; a steady population balance determines the survival factor. The rate coefficient has dimensions volume per time.