Electric current ($I=d{Q}_{\mathrm{cross}}/dt$)

An electric current measures how quickly signed electric charge crosses a chosen fixed oriented surface $S$. Choose its unit normal $\mathbf{n}$; positive electric charge crossing along $\mathbf{n}$ counts positively, while crossing against it counts negatively. If $Q_{\mathrm{cross}}(t)$ is the accumulated signed electric charge transported through $S$, thenHere $\mathbf{J}$ is the current density; the surface integral collects its local normal contributions. Reversing the unit normal reverses $I_S$. The electric current is a signed scalar associated with the whole chosen oriented surface, whereas current density is a vector at each point. The SI unit is the ampere, one coulomb per second. ^[ref]; ^[ref].

For a constant electric current of $2\mathrm{A}$ over a time interval of $3\mathrm{s}$, the transported electric charge is $6\mathrm{C}$. This need not be electric charge accumulating in the wire: in steady transport, electric charge enters and leaves. For a closed oriented surface with outward unit normal, electric charge conservation instead gives $I_{\mathrm{out}}=-d{Q}_{\mathrm{inside}}/dt$. Keeping transported and enclosed electric charge distinct connects electrical circuit bookkeeping to the local continuity equation.

Historical terminology for electrical effects supplied by electrochemical cells. The source supplies energy through chemical changes; the resulting electric charge and electric current obey the same electrical laws as charge supplied by other means.

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 electric current that reverses direction periodically. A sinusoidal alternating current has the displayed form, where $I_0$ is its peak amplitude, $\omega$ its angular frequency, and $\phi$ its phase.

Electronic propagation through a device region with few momentum-randomizing scattering events. The mean free path should exceed the relevant trajectory length. Boundary specularity and aperture geometry can matter separately from the bulk mobility; electron phase coherence is an additional condition for interference.

A four-terminal bend resistance uses adjacent current contacts and the other adjacent pair as voltage probes. It can be negative when a ballistic injector preferentially populates the opposite voltage probe. Its sign is a nonlocal routing property, not the sign of the total electrical dissipation.

For a balanced nonnegative transmission matrix and $I_i=g_0(N_i{V}_i-{\sum }_j{T}_{ij}{V}_j)$, electrical power obeyswith diagonal reflection included consistently or removed from both sides. A negative four-terminal voltage-to-current ratio is compatible with this positive total power because the measured voltage pair is not the current source's pair.

For left-deflected injection in the left 1,top 2,right 3,bottom 4 geometry, use 1 to 2 with $M$ modes, 2 to 3 with $M$, 3 to 4 with $N$, and 4 to 1 with $M$ plus 4 to 3 with $N-M$. The zero-current probes give $V_2=V_1$ and $V_3=M{V}_1/N$, hence $R_{14,23}=(h/e^2)(1/M-1/N)$. Reversing the transmission matrix gives equal probe potentials and zero bend resistance. Fixed-lead resistance need not be even in field; reciprocity also exchanges current and voltage pairs.

Number the leads left 1,top 2,right 3,bottom 4. Let narrow leads 1,2 have $M$ spin-resolved modes and wide leads 3,4 have $N>M$. Straight-through injection sends $M$ modes between 1 and 3 and between 2 and 4; each excess $N-M$ mode connects 3 and 4. With current 1 to 4, $V_4=0$ and zero current at 2,3, $V_2=0$, $V_3=M{V}_1/N$ and $I=(e^2/h)M(N-M)V_1/N$. Thus $R_{14,23}=-h/[e^2(N-M)]$. At $N=M$ this ideal network disconnects the current source from its drain.

A diffusion current results from a spatial carrier density gradient. For semiconductor electrons and holes in one dimension, conventional-current densities are $J_n=e{D}_{n}dn/dx$ and $J_p=-e{D}_{p}dp/dx$. Their different signs reflect negative and positive carrier charge. At equilibrium each species' diffusion current is balanced by its drift current, giving zero net current while the individual transport contributions need not vanish.

A drift current is an electric current caused by the response of mobile charge carriers to an electric field. It is distinct from the diffusion contribution due to concentration gradients. At equilibrium in a p-n junction, the two contributions cancel although each can be nonzero.

A direct current is an electric current that flows in one direction; its magnitude may vary with time. A constant or steady electric current additionally has a value independent of time. Thus a direct current need not be constant. The steady direct current model of a voltage divider assumes a constant input electric potential difference and fixed electrical resistances.

An ammeter measures electric current through a branch of an electrical circuit. Connect it in series; its own finite electrical resistance changes the measured branch's electric current unless that effect is negligible or corrected.^[ref]

A galvanometer detects electric current through a mechanical response, such as the deflection of a pointer.^[ref]

A sensitive galvanometer makes a small electric current visible; a calibrated deflection also estimates its magnitude. A center-zero scale distinguishes both signs of electric current. The moving-coil galvanometer rotates a coil in an imposed magnetic field, whereas a moving-magnet galvanometer rotates a magnetic needle in the coil's magnetic field. The spring-balanced model below describes the moving-coil galvanometer, not every instrument called a galvanometer.

A current-measuring coil and compass arranged so that the coil's central magnetic field is perpendicular to the horizontal terrestrial field. With unchanged coil geometry and terrestrial field, the electric current ratio is the ratio of the tangents of the needle deflections. A small needle and controlled alignment make the central-field approximation useful.

For fixed mean turn length and fixed conductor volume, a coil of $N$ turns has conductor length proportional to $N$ and cross-sectional area proportional to $1/N$. Its electrical resistance is therefore $R_{\mathrm{coil}}=a{N}^2$. With source electromotive force $\mathcal{E}$ and internal resistance $R_s$, suppose the small-deflection driving torque is proportional to $NI=N\mathcal{E}/(R_s+a{N}^2)$. Its derivative has the sign of $R_s-a{N}^2$, so the maximum occurs at $R_{\mathrm{coil}}=R_s$. Geometry, damping and finite response time can alter the practical optimization; this is a conditional winding model, not a universal rule for all galvanometers.

A galvanometer using oppositely directed, nearly equal magnetic dipole moments to reduce the net restoring torque of the approximately uniform terrestrial field. It still needs a weak restoring mechanism for a reproducible deflection; it does not shield the needles from that field.

The historical Leopoldo Nobili instrument combines a many-turn coil with an astatic needle pair. In a common geometry one needle responds strongly to the coil while its oppositely magnetized companion reduces the terrestrial restoring torque.^[ref]

A small-angle galvanometer model combines its moment of inertia $J_g$, damping coefficient $b_g$ and restoring coefficient ${\kappa }_g$ with a torque per electric current $K_g$. The resulting damped harmonic oscillator responds to the current pulse through the displayed equation. Its angular deflection $\theta$ can continue after the driving electric current has ceased. Inferring the current's detailed time dependence requires the instrument response, rather than equating needle angle with instantaneous current. This is a generic readout model, not a calibration of Faraday's historical instrument.

A moving-coil galvanometer has a winding that rotates about an axis, a restoring spring, and a pointer or optical readout. In the radial-field construction, a cylindrical iron core and shaped magnetic poles keep the magnetic field approximately in the coil's plane as it turns. The resulting moving-coil galvanometer torque is nearly independent of the deflection angle; the spring then converts electric current into a proportional settled deflection. ^[ref]

For $\ddot{\theta }+2{\omega }_0\dot{\theta }+{\omega }_0^2\theta =F$ switched on from rest, $\theta (t)=(F/{\omega }_0^2)[1-(1+{\omega }_{0}t)e^{-{\omega }_{0}t}]$. critical damping is the fastest nonoscillatory step response at fixed natural frequency within the ordinary second-order model.

With $e^{i\omega t}$ phasors, critical damping gives $\Theta (\omega )=F_0/({\omega }_0+i\omega {)}^2$, amplitude $F_0/({\omega }_0^2+{\omega }^2)$ and phase $-2\arctan (\omega /{\omega }_0)$. The response strongly suppresses alternating currents well above the mechanical frequency.

A galvanometer radial field is approximately radial about the rotation axis within each narrow working gap. Its direction follows the moving sides of the coil over the useful sweep. Here the magnetic field points from the left pole toward the core and from the core toward the right pole; it is not outward everywhere. This geometry keeps the magnetic force on a current-carrying wire tangential, with nearly constant lever arm. A spatially uniform magnetic field in a fixed direction would instead give an angle-dependent torque as the coil turns.

For a rectangular winding with $N$ turns, active side length $\ell$, side separation $w$, and per-turn area $A=w\ell$, the magnetic force on a current-carrying wire gives equal and opposite side forces of magnitude $N|I|\ell B$. Their force couple has lever arm $w$. Choosing the positive rotation to match positive electric current in the illustration gives the signed torque and spring balanceHere $B$ is the working-gap magnetic flux density, $k>0$ is the torsional spring constant in $\mathrm{N}m/rad$, and $\theta$ is the deflection angle in radians from the no-current zero. The balance holds after motion has settled, within the radial-field and linear elasticity ranges. Do not identify $\theta$ with the angle between a magnetic dipole moment and a fixed uniform magnetic field: that different geometry has torque magnitude $NIAB|\sin \alpha |$. ^[ref]

For an original illustrative design, choose $N=120$, $A=3.00\times 10^{-4}{\mathrm{m}}^2$, $B=0.250\mathrm{T}$, and $k=3.00\times 10^{-6}\mathrm{N}m/rad$. Then $NAB=9.00\times 10^{-3}\mathrm{N}m/A$. At $I=+40.0\mu \mathrm{A}$, the magnetic torque is $+3.60\times 10^{-7}\mathrm{N}\mathrm{m}$ and $\theta =+0.120\mathrm{rad}\simeq +6.88^{\circ }$. Reversing the electric current reverses both the torque and the deflection, giving $-6.88^{\circ }$ without changing the magnetic field.

The galvanometer current sensitivity is the change in settled deflection angle per change in electric current. In the linear model its derivative is constant, $S_I=NAB/k$. The illustrative design has $S_I=3000rad/A$, or $0.1719^{\circ }$ per $\mu \mathrm{A}$. A chosen full-scale angle ${\theta }_{\max }=0.180\mathrm{rad}$ corresponds to $|I|=60.0\mu \mathrm{A}$; a straight-line extrapolation beyond that range does not establish real instrument behavior.

Finite sensitivity and finite readable angle set different limits. Under the stipulated $0.10^{\circ }$ tolerance, a reading indistinguishable from zero means $|I|\lesssim 0.582\mu \mathrm{A}$ only if the zero is stable and the model applies. More turns can increase galvanometer current sensitivity while also increasing winding electrical resistance; the response to an applied electric potential difference depends on both. Friction, zero drift, departures from the radial magnetic field, and spring nonlinearity can spoil the calibration. Damping sets how long one must wait for a settled reading. A rapidly reversing electric current can produce little mean pointer deflection even when its instantaneous magnitude is large.

NCERT's teaching experiment uses a moving-coil galvanometer, a 0–6 V supply, a 0–10 kΩ series electrical resistance box $R$, a 0–200 Ω parallel electrical resistance box $S$, two switches and a voltmeter. With the parallel path open, close the supply switch and record the pointer deflection. Keep $R$ fixed, close the parallel path, and adjust $S$ until the deflection halves. The guide requests repeated settings, clean tight contacts and constant supply electromotive force; it introduces high $R$ before energizing to protect the winding. ^[ref]

Write the winding electrical resistance as $R_g$, the supply electromotive force as $E$, and assume negligible supply internal electrical resistance. Ohm's law gives the two winding electric currentsIf proportional deflection gives $I_2=I_1/2$, solving yields $R_g=RS/(R-S)$, requiring $R>S$. The shortcut $R_g\simeq S$ requires $R\gg R_g$; closing the parallel path generally changes the total supply electric current. For chosen values $R=9900\Omega$ and $S=99.0\Omega$, the exact result is $R_g=100\Omega$, rather than $99\Omega$. The manual gives this exact relation and asks how finite supply internal electrical resistance affects it. ^[ref]

As an additional calibration control, record the supply-off zero, apply several small positive and negative electric currents, and recheck zero after each sequence. With $R_g$ known, calculate each electric current from the measured supply electric potential difference and the total series electrical resistance; plot settled deflection against that electric current. Equal-magnitude reversed readings should lie equally far from a stable zero. A fitted relation ${\theta }_{\mathrm{read}}={\theta }_0+S_{I}I$ separates zero offset from galvanometer current sensitivity. The manual's “figure of merit” is electric current per scale division, not the mechanical torsional spring constant $k$ used above. These reversal and zero checks extend the documented procedure; they are proposed controls, not reported measurements.

In a Wheatstone bridge, the galvanometer connects the two comparison junctions. Its purpose is to locate a null measurement, rather than to measure the branch electric current through the unknown resistor. Using Wheatstone bridge detector loading and Thévenin's theorem, a settled moving-coil galvanometer with winding electrical resistance $R_g$ obeysHere $V_{\mathrm{oc}}$ is the bridge's open-circuit voltage and $R_{\mathrm{th}}$ its Thévenin resistance. For a functioning passive detector with finite positive electrical resistances, exact $I_g=0$ implies $V_{\mathrm{oc}}=0$, preserving Wheatstone bridge balance. Away from balance, the detector loads the junctions; its electrical resistance cannot be ignored.

For the illustrative $R_g=100\Omega$, $S_I=3000rad/A$ and a chosen $R_{\mathrm{th}}=1900\Omega$, a $1.00\mathrm{mV}$ bridge open-circuit voltage produces $0.500\mu \mathrm{A}$ and $0.0859^{\circ }$. That lies inside the stipulated $0.10^{\circ }$ zero tolerance. The corresponding resolvable open-circuit voltage scale is about $1.16\mathrm{mV}$, not arbitrarily small. Actual limits also depend on the zero stability and calibration.

Use the detector checks in the slide-wire resistance bridge experiment: perturb the comparison to either side of the proposed balance and require opposite signs, while confirming the supply is on. A broken connection, a jammed pointer or an inactive supply can also give a zero reading. Conversely, a healthy bridge detector reads zero while the bridge arms still carry electric current. A zero deflection alone proves neither a broken conducting path nor infinite sensitivity.

Conventional current uses the direction in which positive electric charge flows. A moving negative electric charge contributes electric current in the opposite direction to its velocity. Thus electrons drifting left produce conventional current to the right; the convention describes signed electric charge transport, without requiring positive charge carriers. ^[ref].

The ampere is the SI unit of electric current. Since 20 May 2019 its definition fixes the elementary charge exactly at $e=1.602176634\times 10^{-19}\mathrm{C}$, with $\mathrm{C}=\mathrm{A}\mathrm{s}$. This gives electrical measurements a reference based on a fixed elementary charge. ^[ref]; ^[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.

The wire cross-sectional area is the area of a slice perpendicular to the wire's axis. For a round wire of radius $a$, it is $A=\pi a^2$. It enters both the uniform-flow formula $I=jA$ for current density and the formula $R=L/(\sigma A)$ for the electrical resistance of a uniform Ohmic conductor.

A charge carrier transports electric charge by moving: electrons are one example, and ions are another. For species $s$ with number density $n_s$, electric charge $q_s$ and mean velocity ${\mathbf{v}}_s$, the transport contributions add as $\mathbf{J}={\sum }_s{n}_s{q}_s{\mathbf{v}}_s$. A stationary charged background contributes to charge density but not to this current density. ^[ref].

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].

The mean velocity of the electron population associated with a net electric current, distinct from random thermal or Fermi motion. Its sign is opposite to conventional current for negative charge carriers. Its usually small magnitude is not the propagation speed of an electromagnetic signal in a transmission line.

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.

A charge carrier population per unit volume or, for a sheet, per unit area. The dimensional convention and whether different carrier species are counted separately must be specified.

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 ${\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$.

A transport quantity obtained by combining longitudinal electrical conductivity and a Hall coefficient. It equals single-carrier drift mobility only under the corresponding one-carrier and scattering assumptions. This does not independently determine effective mass and scattering time.

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.

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.

A current dipole idealizes a localized impressed electric current density by the displayed Dirac delta function distribution. Its moment $\mathbf{Q}$ has units of current times length. In a homogeneous volume conductor of conductivity $\sigma$, its primary electric potential is $\Psi (\mathbf{r};{\mathbf{r}}_0)=\mathbf{Q}\cdot (\mathbf{r}-{\mathbf{r}}_0)/(4\pi \sigma |\mathbf{r}-{\mathbf{r}}_0{|}^3)$.

A radial current dipole has moment parallel to its displacement from the sphere centre. The Sarvas formula gives identically zero exterior magnetic field for such a source in a spherical volume conductor model, although its electric potential need not vanish. This is a nonuniqueness mechanism for the inverse problem in magnetoencephalography.

The four-current combines charge density and current density into the four-vector $J^{\mu }=(c\rho ,\mathbf{J})$. With $x^{\mu }=(ct,\mathbf{x})$, its four-divergence vanishes by the charge continuity equation: ${\partial }_{\mu }{J}^{\mu }={\partial }_t\rho +\nabla \cdot \mathbf{J}=0$. It is the source in the covariant Maxwell equations.

A surface current density is electric current per unit transverse length confined to a surface. Integrating a thin layer's current density through its thickness gives $\mathbf{K}$.

A current sheet idealizes electric current as supported on a surface. It produces a jump in the tangential magnetic field.

Two broad perfect conductors of width $w$, separated by gap $d\ll w$, carry opposite steady electric currents $I$ and connect through a terminating resistor $R$. Away from edges, Ampère's circuital law and the superposition principle give gap magnetic field magnitude ${\mu }_{0}I/w$. Each conductor is an equipotential, and Ohm's law sets their electric potential difference to $IR$. Planar symmetry and charge-free Gauss's law then give uniform gap electric field magnitude $IR/d$. Injection parallel to the upper current makes the electric and magnetic Lorentz forces opposite. Their cancellation selects speed $Rw/({\mu }_{0}d)$ independent of nonzero electric charge and of operating current; nonrelativistic kinetic energy is $m[Rw/({\mu }_{0}d){]}^2/2$.

The Lorentz force per unit area on an ideal current sheet is computed using the mean of its limiting magnetic fields. The sheet's own local magnetic field is equal and opposite on its two sides and cancels in this mean. Equivalently, integrate the Lorentz force density through a thin regularized layer: its self-field contribution averages to zero, leaving the external field acting on the surface current density.

For a surface current density $\mathbf{K}$ on an interface with unit normal $\mathbf{n}$, Ampère's law givesin Gaussian units, or ${\mu }_0\mathbf{K}$ in SI units.

For a thin layer with essentially uniform tangential electric field, integrating electrical conductivity through its thickness gives the sheet conductivity tensor ${\mathbf{\sigma }}_s$. The two-dimensional form of Ohm's law is $\mathbf{K}={\mathbf{\sigma }}_s{\mathbf{E}}_{\parallel }$, with surface current density $\mathbf{K}$. Its SI unit is $\mathrm{S}=A/V$: amperes per metre divided by volts per metre. When response is the same in every in-plane direction, the sheet conductivity reduces to a scalar $\Sigma$; for uniform bulk electrical conductivity $\sigma$ and thickness $t$, $\Sigma =\sigma t$.

A line current idealizes an electric current as flowing along a curve; its current-density volume integral reduces to $Id\ell$ along that curve.

Electromotive force is the driving work per unit electric charge around an electrical circuit, measured in volts, despite its name containing "force". For a stationary loop its electromagnetic contribution is $\mathcal{E}={\oint }_C\mathbf{E}\cdot d\mathbf{l}$; for a moving electrical conductor, include the motional electromotive force. A nonzero closed-loop line integral of the electric field cannot be represented by a single-valued electric potential alone. An electromotive force specifies the drive; the resulting electric current also depends on the electrical resistance and self-inductance. ^[ref].

Motional electromotive force is the contribution ${\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{l}$ from an electrical conductor moving with local velocity $\mathbf{v}$ through a magnetic field. The cross product comes from the magnetic part of the Lorentz force. For a straight rod of length $\ell$, perpendicular to both its velocity and a uniform magnetic field, its magnitude is $B\ell v$. The sign depends on the direction of integration along the rod. ^[ref].

For a computed example, take $\ell =0.50\mathrm{m}$, velocity $\mathbf{v}=2.0\hat{\mathbf{x}}m/s$ to the right, and magnetic field $\mathbf{B}=0.40\hat{\mathbf{z}}\mathrm{T}$ out of the page. The rod lies along $y$. ThenPositive electric charge is driven downward. For an isolated rod after charge separation has equilibrated, $\mathbf{E}=-\mathbf{v}\times \mathbf{B}$ inside it, and the electric potential difference is $V_{\mathrm{bottom}}-V_{\mathrm{top}}=+0.40\mathrm{V}$; the bottom is positive. This open-circuit statement does not imply a sustained electric current.

Close the rod's path with stationary U-shaped rails and an electrical resistance $R>0$ at the left. With out-of-page unit normal, the positive loop direction is counterclockwise, hence upward along the rod. If the rod's horizontal position is $x(t)$, the enclosed area is $A=\ell x$ and the external magnetic flux is ${\Phi }_{\mathrm{ext}}=B\ell x$. ThusHere the resistive approximation neglects self-inductance. The negative electric current is clockwise, downward along the rod, consistent with the charge-separation calculation and Lenz's law.

The closed-loop result illustrates why the moving-circuit form of Faraday's law of induction includes motional electromotive force: the magnetic field is constant in time, so the electric field circulation alone would miss this drive. Its mechanical consequence is electromagnetic braking of a sliding rod.

A vertical field linking a conducting loop on an incline contributes flux $Bl\cos \theta s$. For zero resistance and zero initial current, flux conservation gives $LI=Bl\cos \theta s$. The uphill magnetic force component then makes the downhill coordinate oscillate about $s_e=mgL\sin \theta /(B^2{l}^2{\cos }^2\theta )$. Release from $s=0$ first stops at $2s_e$, not at the force-free point $s_e$. The energy is stored in the inductor, not dissipated.

For positive loop resistance, coupled mechanical and circuit equations relax toward constant sliding speed on sufficiently long rails. The steady current balances the downhill gravitational force, and motional electromotive force balances the resistive voltage. The undamped zero-resistance limit has no relaxation, so its late-time limit is singular.

For laboratory velocity $v\hat{r}+w\hat{\theta }$ and field $B\hat{z}$, $F_{B,r}=qBw$ and $F_{B,\theta }=-qBv$. Their powers are opposite. The field-dependent guide-reaction power is $-qBvw$; the total ideal radial reaction for bead mass $m_b$ and constant expansion speed additionally contributes the ordinary centripetal term $-m_{b}v{w}^2/R$.

A conducting rod sliding on rails separated by $\ell$ in a transverse magnetic field has induced current magnitude $B\ell v/R$ when self-inductance is neglected and resistance is $R$. The Lorentz force opposes motion with magnitude $B^2{\ell }^{2}v/R$, so free motion decays exponentially on the displayed time constant. Joule heating integrates to the rod's lost kinetic energy. Current direction follows the Lenz's law.

For a smoothly moving circuit and a divergence-free magnetic field, the magnetic flux derivative is $d\Phi /dt={\int }_S{\partial }_t\mathbf{B}\cdot d\mathbf{S}-{\oint }_C(\mathbf{v}\times \mathbf{B})\cdot d\mathbf{r}$. The minus sign follows by orienting the swept side surface compatibly with the two spanning surfaces. Combining this transport identity with Faraday's law yields the moving-loop electromotive-force law.

For an ohmic element, electrical resistance is $R=V/I$ and conductance is its reciprocal $G=I/V$. They describe the same voltage-current response using reciprocal units. The relation $V=IR$ is Ohm's law.

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 resistance measures the electric potential difference needed to drive a given electric current through a component. For an Ohmic conductor under fixed material conditions, $R=V/I$ is constant and Ohm's law gives $V=IR$. The SI unit is the ohm, $1\Omega =1V/A$. At a fixed positive electric potential difference, increasing electrical resistance reduces electric current. Electrical conductance $G=1/R$ expresses the same response reciprocally. ^[ref].

For a straight uniform wire at rest, with constant electrical conductivity $\sigma >0$, length $L$ and wire cross-sectional area $A$, the uniform Ohmic conductor model givesThe electrical resistivity ${\rho }_{\mathrm{el}}$ characterizes the material; electrical resistance also includes geometry. Doubling length doubles $R$, while doubling wire cross-sectional area halves $R$, with material and temperature fixed. The uniform-field assumption includes insulated sides and equipotential end faces, as explained in uniform current in a straight homogeneous wire. ^[ref].

As a defined wire example, take $L=1\mathrm{m}$, $A=1\mathrm{m}{\mathrm{m}}^2$ and $\sigma =5\times 10^5(\Omega \mathrm{m}{)}^{-1}$. Then $R=2\Omega$; applying electric potential difference $5\mathrm{V}$ drives electric current $2.5\mathrm{A}$. With the same length and material, a second wire of wire cross-sectional area $2\mathrm{m}{\mathrm{m}}^2$ has $R=1\Omega$ and carries $5\mathrm{A}$ at that same electric potential difference. Fixing electric potential difference here differs from fixing electric current in the current density comparison: stating what is held fixed is essential.

The fractional change in electrical resistance per unit temperature change, with geometry and other conditions specified. For a small interval around $T_0$, $R(T)\simeq R(T_0)[1+{\alpha }_R(T-T_0)]$. Low $|{\alpha }_R|$ is useful in stable resistors.

The continuum spreading resistance for current entering a circular aperture of radius $a$ from one uniform semi-infinite reservoir of electrical resistivity $\rho$ is $\rho /(4a)$. Two symmetric reservoirs contribute $\rho /(2a)$ in series with a cylindrical pore resistance $\rho \ell /(\pi a^2)$. Unobstructed access is not the same as zero spreading resistance; molecular-scale pores also require non-continuum transport physics.

The scale $R_K=h/e^2\simeq 25.8\mathrm{k\Omega }$. Spin degeneracy perfectly transmitting channels instead have conductance $2e^2/h$. Suppressing charge fluctuations in a simple tunnel-island picture requires nanowire junction resistances large relative to $h/e^2$.

Electrical resistance introduced by a connection between conducting regions. Interface chemistry, contact geometry and microscopic barriers can dominate over the bulk resistance of the connected materials. Four-terminal measurements help separate contact drops from the resistance of a test section.

The resistance per square of a uniform thin conducting sheet. For bulk resistivity $\rho$ and film thickness $t$, $R_{\square }=\rho /t$ and a rectangular stripe has $R=R_{\square }L/W$. The number of squares is dimensionless and independent of an overall change of lateral scale. Networks and grids use an effective sheet resistance that may depend on finite-size connectivity and contacts.

A change of electrical resistance associated with an applied magnetic field, with material, temperature, measuring electric current and preparation specified. The response can depend on magnetic history as well as the instantaneous field. A magnetoresistance measurement therefore compares defined states, rather than treating every material as a single-valued function of field magnitude.

A large field-induced resistivity reduction in correlated magnetic materials. Double exchange links carrier mobility to spin alignment, while lattice coupling, localization and competing phases can magnify the response.

A junction with magnetic electrodes separated by a thin insulating barrier. Spin-dependent tunnelling can make resistance depend strongly on the relative electrode magnetizations.

tunnelling magnetoresistance is the dependence of a magnetic tunnel junction's electrical resistance on the relative magnetizations of its electrodes. A common convention is $(R_{AP}-R_P)/R_P$, where the subscripts denote antiparallel and parallel alignment. Its magnitude and sign depend on effective spin-polarized transmission through the particular interfaces and barrier.

Assume spin-conserving, low-voltage tunnelling and a spin-independent matrix element. Then parallel conductance is proportional to $D_{1\uparrow }{D}_{2\uparrow }+D_{1\downarrow }{D}_{2\downarrow }$, while antiparallel conductance has the crossed products. Define $P_i=(D_{i\uparrow }-D_{i\downarrow })/(D_{i\uparrow }+D_{i\downarrow })$. The two conductances are $G_0(1\pm P_1{P}_2)$; taking their reciprocals and using the resistance-ratio convention proves the displayed formula. Orbital filtering and interface-dependent matrix elements can invalidate a bulk-density-of-states estimate.

If one temperature-dependent relaxation scale controls a fixed carrier population and fixed orbital geometry, the magnetoresistance ratio can collapse onto a function of $B/\rho (0,T)$. This is Kohler's rule. For the equal-mobility compensated two-carrier magnetoresistance model, $[\rho (B)-\rho (0)]/\rho (0)=(\mu B{)}^2$ and $\mu =[2ne\rho (0){]}^{-1}$. Therefore plotting against $[B/\rho (0){]}^2$ gives a common straight line of slope $1/(4n^2{e}^2)$. A collapse tests the scaling relation; it does not by itself prove this particular microscopic carrier model. Multiple independently varying scattering times or carrier populations can spoil it.

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 dependence of electrical resistance on the angle between electric current and magnetization.

Giant magnetoresistance connects the relative magnetizations of adjacent magnetic layers to their electrical transport. In a suitable magnetic-metal/nonmagnetic-metal stack, aligning the layers can leave one spin channel with an easy conducting path through both magnetic layers; opposite alignments make each channel encounter a more resistive layer. Albert Fert's account traces the spin-dependent-conduction idea to Nevill Mott and distinguishes the later device geometries.^[ref]

The beauty is that a magnetic arrangement becomes an electrical readout. Preparing reproducible layers, establishing their magnetizations, and measuring electrical resistance are separate parts of that inference. The Fe-Cr giant magnetoresistance experiments show how those parts were joined; the two-current resistor model isolates a simple transport mechanism under stated assumptions.

When spin-flip scattering is weak, two spin populations have separately defined current responses and share a terminal electric field. Their electrical conductivities add: $\sigma ={\sigma }_{\uparrow }+{\sigma }_{\downarrow }$, so $\rho ={\rho }_{\uparrow }{\rho }_{\downarrow }/({\rho }_{\uparrow }+{\rho }_{\downarrow })$. Spin-dependent band structure and scattering set the channel parameters; being the majority population does not universally imply lower channel resistivity. The model is named for Nevill Francis Mott and is the independent-channel basis for many giant magnetoresistance models.

A channel's electrical conductivity weights its density of states by squared velocity and the transport momentum relaxation time. In an elementary spin-conserving $s$–$d$ scattering model, ${\tau }_s^{-1}\propto |V_{sd,s}{|}^2{D}_{d,s}(E_F)$, so available heavy final states can reduce the current carried by light states. Band filling, defects and interfaces affect the asymmetry; total spin population alone does not determine its sign. A density-of-states sketch without scattering matrix elements is not a universal transport prediction.

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

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

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

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

The measuring electric current flows along the layer planes. This is the geometry of the Fe-Cr giant magnetoresistance experiments. Scattering and current paths through an extended stack require a transport description; treating successive layers as ordinary series resistors would not follow from this geometry.

In a model of carriers sampling alternating magnetic and nonmagnetic layers, assume spin conservation, comparable carrier transport weights and negligible extra interface scattering. Averaging relaxation rates by the time spent in each material then gives a channel electrical resistivity $(t_1{\rho }_{\mathrm{FM},\mathrm{s}}+t_2{\rho }_{\mathrm{NM}})/(t_1+t_2)$. Here all constituent resistivities are per spin channel. A long mean free path compared with a period motivates interlayer sampling; it does not by itself prove the other assumptions. This is a homogenized current-in-plane model, rather than an independent parallel-layer limit or a current-perpendicular-to-plane resistor chain.

From the four-layer CIP sheet resistance, $R_P=ab/[2D^2(a+b)]$ and $R_{AP}=(a+b)/(8D^2)$. Subtraction gives $R_{AP}-R_P=(a-b{)}^2/[8D^2(a+b)]\ge 0$ for positive channels. The ratio in the title normalizes by the low-resistance state; normalizing by the high-resistance state instead gives $(a-b{)}^2/(a+b{)}^2$. Common spacer scattering adds equally to $a,b$, diluting the contrast. The result assumes identical magnetic layers; arbitrary materials need not have the same sign of spin asymmetry.