Electric charge

Electric charge is an additive, signed property of matter that determines its coupling to the electromagnetic field. Write a particle's electric charge as $q$, or a collection's total electric charge as $Q$; the SI unit is the coulomb, $\mathrm{C}$. A stationary test particle in an electric field experiences electric force $\mathbf{F}=q\mathbf{E}$, while motion adds the magnetic part of the Lorentz force. Electric charge both produces an electric field and determines the response to one. For two stationary point particles in vacuum, Coulomb's law makes like signs repel and opposite signs attract. The choice of which sign to call positive is a convention, used consistently: protons carry $+e$ and electrons carry $-e$, where $e$ is the elementary charge. ^[ref].

Additivity means adding signed electric charges, rather than their magnitudes:for ordinary matter containing $N_p$ protons, $N_e$ electrons, and electrically neutral neutrons. A toy collection with seven protons and ten electrons has $Q=-3e$, however much neutral material accompanies it. Charge neutrality means $Q=0$; it does not mean an absence of charged constituents. Nor does it guarantee an absence of electric force: the spatial arrangement of positive and negative electric charges matters, as illustrated by an electric dipole. This makes a tiny imbalance in an otherwise nearly neutral object physically important. ^[ref].

Charge conservation states that a collection closed to transport of electric charge keeps its total signed $Q$. If $n$ electrons move from object A to object B without other exchange, then $\Delta Q_A=+ne$ and $\Delta Q_B=-ne$; the sum is unchanged. The amount of positive electric charge alone need not be conserved: creation of an electron and a Positron adds opposite electric charges with zero sum. These bookkeeping rules let us distinguish separation or transport of electric charge from creation of net electric charge. ^[ref].

A simple motivating experiment uses two charge-producing paddles, a conducting pail on an insulating support, a grounded outer shield and a high-input-impedance electrometer. Zero the pail, then isolate it. Rub the initially neutral paddles together, keeping their charged surfaces away from other objects. Lower each deeply into the pail without touching: the electrometer gives opposite signs. Inserting both together tests cancellation. For a separate contact test, insert one charged paddle without touching and withdraw it: the reading returns to zero. Repeat after zeroing, this time touching the inner wall before withdrawal: a residual reading shows transfer of electric charge. Zero the pail and reinsert the paddle to test how much electric charge remains on it. PASCO describes these procedures in Basic Electrostatics System manual, demonstrations 1A and 1B, printed pp. 13–16.

The crucial control is withdrawal without contact: a response during insertion can come from electrostatic induction, which redistributes the pail's existing electric charge. In the fixed-geometry approximation, the electrometer measures electric potential difference proportional to enclosed electric charge, with conversion set by effective capacitance; a voltage reading is not already a measurement in coulombs. Opposite readings and their cancellation test additivity and charge conservation within resolution. Leakage or accidental contact spoils that interpretation, and contact need not transfer all the paddle's electric charge. This macroscopic experiment cannot resolve elementary charge steps: that requires the much smaller particles of the oil drop experiment.

Electric charge concentrated on a surface. Its local amount per unit area is the surface charge density $\sigma$; in vacuum the jump of the outward normal electric field is $\sigma /{ϵ}_0$ by Gauss's law.

The spatial arrangement of electric charge, described by a charge density, a surface charge density, a linear charge density, or discrete point charges as appropriate.

Electron repulsion is the repulsive electrostatic interaction between electrons. In a metal complex it affects pairing costs and many-electron transition energies, so a measured absorption maximum is not always identical to a one-electron orbital splitting.

In macroscopic electromagnetism, free charge is the part of the charge density represented explicitly as ${\rho }_f$, separately from bound charge associated with the electric polarization of matter. The electric displacement field satisfies $\nabla \cdot \mathbf{D}={\rho }_f$. The word free does not imply zero force or zero energy: it distinguishes this source from the polarization response in the chosen material description.

Charge invariance means that the total electric charge of a specified collection of particles is unchanged by a Lorentz transformation. It differs from conservation of electric charge, which compares the same isolated system at different times. For a convective charge distribution with rest density ${\rho }_0$, the four-current is $J^{\mu }={\rho }_0{U}^{\mu }$, where $U^{\mu }$ is its four-velocity; its measured charge density increases under a boost while the simultaneous spatial volume contracts, leaving total charge unchanged.

Electric charge distributed through a region, whose own electric field can alter the motion of further charged particles.

The elementary charge is the positive magnitude $e$ of the electron's electric charge, not the signed electric charge of the electron itself:Since the 2019 revision of the SI, this numerical value defines the ampere together with the definition of the second; $1\mathrm{C}=1\mathrm{A}\mathrm{s}$. Thus today's numerical exactness is a choice of units, whereas measured electric charges still have experimental uncertainty. An oil drop experiment can test charge quantization and the apparatus calibration; it does not supply a new uncertain defining value of $e$ in today's SI. ^[ref].

For ordinary isolated chemical atoms, ions and charged objects whose net electric charge is counted through protons and electrons, charge quantization gives $Q=ne$ with signed integer $n$. A neutral chemical atom that loses two electrons becomes an ion of electric charge $+2e$. A toy object with $10^9$ excess electrons has $Q=-0.1602176634\mathrm{nC}$. Conversely, the scale $1\mathrm{nC}/e\simeq 6.24\times 10^9$ explains why laboratory electric charge often looks continuous: billions of microscopic increments fit into a small macroscopic amount. Averaged charge density is a continuum description, not a claim that each particle has an arbitrarily divisible electric charge.

The ordinary-matter qualifier matters. In the Standard Model, quarks have electric charges $+2e/3$ or $-e/3$, and antiquarks have the opposite signs. Confinement means these are constituents of hadrons, not isolated fractionally charged particles collected by an oil drop experiment. The usual valence quark sums give $2(2e/3)-e/3=e$ for a proton and $2e/3-2(e/3)=0$ for a neutron. Thus elementary charge should not be defined as the smallest nonzero electric charge of every constituent in particle theory. ^[ref].

The beautiful experimental idea behind the oil drop experiment is to determine a microscopic electric charge by watching a visible carrier move. In his 1913 paper, Millikan timed the descent and electrically driven ascent of individual oil drops, repeatedly changing a drop's electric charge through capture of air ions. His apparatus used parallel plates separated by $16\mathrm{mm}$, a roughly $5300\mathrm{V}$ source, and an optical scale to measure travel distances. He reported multiple relationships among a given drop's inferred electric charges. He also found that uncorrected Stokes's law gave a size-dependent inferred elementary unit, and introduced a pressure-dependent slip correction. These were measured relationships with calibration and model uncertainties, not exact readings of today's defined $e$. ^[ref].

A modern instructional implementation sprays low-volatility oil of known density through a plate aperture, illuminates the drops, and follows one through a calibrated optical scale. Measure the plate spacing $d$, electric potential difference $V$, air temperature and pressure. Time a fixed distance $L$ with the electric field off to get a terminal falling speed $v_f=L/t_f$. Then choose the field polarity to drive the same drop upward and measure $v_r=L/t_r$. Repeat timings at a fixed electric charge, and compare before and after an ion-capture event. The same drop is especially useful because its radius and mass need not be inferred anew for each charge state. ^[ref].

Here is the leading continuum model, retaining buoyancy. Assume a drop shaped as a sphere of radius $a$, oil and air densities ${\rho }_o$ and ${\rho }_a$, air dynamic viscosity $\eta$, and gravitational acceleration $g$. Define the downward effective weight $W$ and positive linear drag coefficient $k$ byAt terminal velocity, the net force vanishes. Falling with no electric field, the upward drag force balances weight minus buoyancy. Rising with the electric force upward, the drag force reverses and joins the downward effective weight:Consequently the uncorrected estimates areThe observed direction of response, with known field polarity, supplies the sign of $q$. A suspended drop gives the alternative balance $|q|E=W$, but it still requires a determination of $W$: balancing a drop alone does not determine its electric charge.

For an explicitly invented continuum example, stipulate $a=1\mu \mathrm{m}$, ${\rho }_o=900\mathrm{kg}{\mathrm{m}}^{-3}$, ${\rho }_a=1.2\mathrm{kg}{\mathrm{m}}^{-3}$, $\eta =1.8\times 10^{-5}\mathrm{Pa}\mathrm{s}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, $q=-5e$ and a downward $E=10^5\mathrm{V}{\mathrm{m}}^{-1}$. The balances predict $W\simeq 3.69\times 10^{-14}\mathrm{N}$, $v_f\simeq 0.109\mathrm{mm}{\mathrm{s}}^{-1}$ and $v_r\simeq 0.127\mathrm{mm}{\mathrm{s}}^{-1}$. Substituting those speeds recovers $|q|=5e$ within rounding. This checks the algebra; it is not a precision prediction for real micrometre drops, which need a slip correction.

The inference has two distinct parts. Compare inferred electric charges or charge changes with a common unit, seeking $q_i\simeq n_{i}e$ within uncertainty, rather than demanding exact ratios of rounded measurements. Then establish the absolute scale through electric field, density, dynamic viscosity and distance/time calibration. If the observed steps are all even multiples, their common spacing alone cannot exclude an unobserved smaller unit. Air motion, evaporation and mistimed crossings can alter inferred electric charge without an ion-capture event. Small Reynolds number supports Stokes's law, but does not by itself justify its no-slip continuum coefficient: a significant ratio of air mean free path to drop radius requires a slip correction. Millikan's comparison of different drop sizes and pressures addressed that systematic limitation; the modern PASCO manual also includes the correction. Agreement supports charge quantization for the observed carriers within experimental resolution, rather than proving a universal statement about every possible particle.

Multiplying $e$ by the Avogadro constant gives the Faraday constant, $F=N_{A}e$, the electric charge per mole of singly charged carriers. This connects counting individual electric charges with bulk electrochemistry. ^[ref].

The oil drop experiment infers an oil drop's electric charge from its motion under gravity, buoyancy, drag and a calibrated electric field. Repeated changes of a single drop's electric charge reveal discrete steps; see the model, experimental procedure and limitations in elementary charge.

Robert Andrews Millikan's 1911 protocol alternated electrically driven ascent with timed descent after short-circuiting the plates; a changed ascent speed signalled ion capture.^[ref]

His 1913 refinements checked voltage before and after runs, calibrated optical distance and the clock, compared independent air-dynamic viscosity measurements, and varied pressure to test slip.^[ref] His 1924 lecture describes suppressing convection, switching the electric field, neutralizing drops, and reproducing smallest observed electrical speed increments.^[ref]

Harvey Fletcher's posthumous account recalls building the first oil apparatus and measuring with Robert Andrews Millikan; priority for suggesting oil remains uncertain.^[ref]

To interpret the readout, keep one drop and one electric field polarity. Let $u_i$ be its signed upward terminal velocity in charge state $i$, and let $v_f>0$ be its field-off falling speed. Use the linear drag coefficient $k$ already defined in elementary charge, with any required gas slip correction to Stokes drag included. Provided the drop's radius, surrounding air and electric field remain fixed, its electrical speed increment isThese relations apply while the electric force points upward in every state. A drop may still descend, with $u_i<0$, when that force is too weak to support it. A held drop has $u_i=0$ and therefore $S_i=v_f$, rather than zero electrical response. A neutral drop has $S_i=0$: its signed terminal velocity agrees with field-off descent. Thus neither raw upward speed nor the ability to hold a drop is itself a count of electrons.

The ratios cancel a fixed $k/E$, even if its value is poorly known. This is why relative charge quantization and absolute measurement calibration are different questions. A changed field-off $v_f$ warns that evaporation, air motion or another change has spoiled comparison of charge states. A stable $v_f$ is useful evidence but cannot exclude every compensating change. Repeat field-off checks around a proposed jump, allow the terminal velocity under linear drag to settle after switching, and propagate measurement uncertainty in both the field-on and field-off readings. Reusing a single estimate of $v_f$ gives the different $S_i$ a shared covariance.

An integer pattern must survive the stated measurement uncertainty. For an original simulated example, choose a reference electric charge $Q_{\ast }>0$, define $y_i=|q_i|/Q_{\ast }$, and stipulate state meansHere each ${\sigma }_i$ is an independent standard uncertainty of a state mean, before shared calibration effects. The generator constructs repeated noisy observations with these means; the numbers are deliberately chosen, not historical data. There is no conversion of $Q_{\ast }$ to coulombs and no measured value of the elementary charge in this example.

For a candidate spacing $a>0$ in units of $Q_{\ast }$ and proposed integers $n_i$, use weighted least squares:The last two expressions follow by differentiating the quadratic in $a$ and propagating the independent errors. They are conditional on fixed, correct integer assignments and the stated error model. They do not include uncertainty from choosing those assignments. For $\mathbf{n}=(2,3,5,4)$, they give $\hat{a}=1$, $u(\hat{a})\simeq 0.01089$, and ${\chi }^2=0.46875$. Such a small residual is compatible with this chosen error scale; it is not evidence that the errors vanish. A chi-squared distribution interpretation additionally needs the appropriate normal distribution error model and allowance for the fitted parameter; searching many assignments changes that interpretation.

The obstruction is identifiability: replacing $(a,n_i)$ by $(a/m,m{n}_i)$ for any positive integer $m$ leaves every predicted reading unchanged. In particular, an $e$-spaced pattern also fits an $e/2$ lattice using even indices. Even the coarsest well-fitting spacing need not be the physical unit: a dataset containing only even charge states can have twice that unit. Worse, if arbitrarily fine spacings and unrestricted integers are allowed, rounding each $y_i/a$ givesConsequently a finite fit cannot prove a minimal transferable electric charge. Reproducible smallest observed jumps supply additional evidence; interpreting them as single-carrier transfers requires independent physical support. One visibly abrupt transition could contain several unresolved transfers. In the simulation the successive differences are $0.93$, $2.06$, and $-1.03$, each with standard uncertainty $\sqrt{2}(0.08)\simeq 0.1131$. These support changes near $a,2a,-a$ conditional on that unit; they equally support $2,4,-2$ units of $a/2$. Neighboring differences share a state reading and are therefore correlated, even though the state means were stipulated independent.

A shared scale error can preserve every apparent integer relation. To first order, an independent fractional scale standard uncertainty $s$ adds $s^2{a}^2{n}_i{n}_j$ to the covariance matrix of the $y_i$. For the simulation, stipulating $s=0.02$ changes the conditional uncertainty of $\hat{a}$ to approximately $\sqrt{0.01089^2+0.02^2}=0.02277$, despite the same residual pattern. Repeating timings cannot average away this common scale contribution. In the uncorrected continuum model in elementary charge, holding the other measured inputs fixed gives $|q|\propto {\eta }^{3/2}$, so a small fractional error in air dynamic viscosity causes $3/2$ times that fractional error in inferred electric charge. The gas slip correction to Stokes drag requires its own propagation. Pattern agreement, drop-stability controls and absolute measurement calibration together make the inference persuasive within resolution.

A charged particle carries nonzero electric charge.

An electrometer measures small electric potential differences or electric charges while drawing very little electric current. The conducting-pail experiment in electric charge uses a voltage response to infer electric charge with a fixed effective capacitance.

An electroscope indicates electric charge through the deflection of conducting leaves or fibers.

Electrostatic induction is redistribution of electric charge in response to an external charged object. An isolated electrical conductor can acquire positive and negative surface regions while its total electric charge stays fixed; see the contact control in electric charge.

Conductors with net charges of the same sign can attract because their internal charge distributions respond to one another. A point-charge calculation misses this electrostatic induction contribution. Exact conducting-sphere results are given in Electrostatics of Two Charged Conducting Spheres by John Lekner.^[ref]

A small conducting sphere of radius $a$ has electric polarizability $\alpha =4\pi {ϵ}_0{a}^3=a^3/k$. In the field $E\simeq k{Q}_1/R^2$ of a much more strongly charged distant source, its leading induced contribution is $-\alpha E^2/2$, in addition to $k{Q}_1{Q}_2/R$ for its fixed net charge. Taking $-dU/dR$ gives $F_R\simeq k{Q}_1{Q}_2/R^2-2k{Q}_1^2{a}^3/R^5$. The approximation requires $a/R\ll 1$ and control of the source's finite-size corrections; the threshold is not an exact finite-sphere result.

A system is charge neutral when its total positive and negative electric charge cancel.

A sensitive charge neutrality test lets nominally neutral gas escape from an electrically isolated vessel connected to an electrometer. A net charge per escaping molecule would leave a measurable opposite charge on the vessel. Removing stray ions and charged particles is essential. Hydrogen and helium measurements constrain charge imbalance despite different particle binding and internal motions, providing evidence for velocity-independent additive charge rather than a charge proportional to relativistic particle energy. The 1960 experiment used this efflux principle. ^[ref]

Charge conservation says that the total signed electric charge of an isolated collection stays constant. More generally, a collection closed to electric charge exchange has constant total electric charge, even if energy crosses its boundary. This is the charge bookkeeping relevant to a closed system or an isolated system; isolation from energy exchange is not required. Positive and negative electric charge count with opposite signs. Creating or removing an electron and a Positron together is compatible with charge conservation, because their combined electric charge is zero. ^[ref].

Imagine a fixed box initially containing electric charge $Q(0)=10\mu \mathrm{C}$. Stipulate incoming conventional current $I_{\mathrm{in}}=3\mu \mathrm{A}$ and outgoing conventional current $I_{\mathrm{out}}=1\mu \mathrm{A}$ throughout $0\le t\le 2\mathrm{s}$, with no other boundary transport. Since one ampere is one coulomb per second, the stored electric charge grows at $2\mu C/s$:If instead the outgoing electric current also equals $3\mu \mathrm{A}$, the stored electric charge remains $10\mu \mathrm{C}$ while electric charge passes through. These are stipulated transport examples; constant terminal electric currents alone do not guarantee a time-independent interior charge density.

Locally, charge conservation is the charge continuity equation, an instance of the general continuity equation:Here $\rho (\mathbf{r},t)$ is signed volume charge density and $\mathbf{J}(\mathbf{r},t)$ is current density. A positive divergence of current density means net outward transport from a small neighborhood, so its charge density decreases. For a fixed control volume $V$ with closed surface $\partial V$ and outward unit normal $\mathbf{n}$, the equivalent integrated balance isThe surface integral is signed: incoming positive electric charge contributes negatively to outward electric current, and outgoing positive electric charge contributes positively. Integrate the charge continuity equation over fixed $V$ and apply the divergence theorem to obtain this balance. Zero net boundary transport gives constant $Q_V$; the global and local statements describe the same conservation law at different scales. ^[ref].

In a steady transport state, charge density and current density are time-independent, so the charge continuity equation reduces to $\nabla \cdot \mathbf{J}=0$. At an electrical circuit junction, choose each terminal electric current $I_k$ positive outward and assume the terminals account for all boundary transport. ThenKirchhoff's current law, ${\sum }_k{I}_k=0$, follows when accumulation of electric charge at the junction is negligible. During storage, retain the same balance with its nonzero derivative; charge conservation continues to hold. This makes charge conservation useful for checking electrical circuit models and for tracking the charging of a capacitor. ^[ref].

The beautiful consistency with Maxwell equations is that the electric field and magnetic field already enforce this transport balance. For sufficiently smooth fields in vacuum SI, take the divergence of the Ampère-Maxwell equation, use that the divergence of a curl is zero, and substitute Gauss's law, $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$. Constant vacuum permittivity ${ϵ}_0$ givesThe displacement current term supplies the part needed when electric charge accumulates. Historically, Maxwell added that term to extend the steady-current equation; see the existing charge conservation from Maxwell equations for this connection. ^[ref].

Charge density is electric charge per unit volume, area, or length according to the dimension of its support.

Linear charge density is electric charge divided by length along a charged line, or locally $dQ/d\ell$. Its SI unit is coulombs per metre. For a uniformly charged long cylindrical conductor, it equals the total surface charge density integrated around the cylinder's circumference; it need not represent a charge literally supported on a one-dimensional line.

A line charge is a charge distribution idealized as lying on a curve, described by charge per unit length. An infinite straight line with constant density $\mu$ produces the radial electric field $E_r=\mu /(2\pi {\epsilon }_{0}r)$ by Gauss's law, and a logarithmic electrostatic potential.

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.