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The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

In the magnetic Aharonov-Bohm effect, a coherent electron wave takes two routes around inaccessible magnetic flux. Recombination converts their relative wave phase into displaced interference fringes. In 1959, Yakir Aharonov and David Bohm proposed a shielded solenoid geometry and identified the obstruction to removing the magnetic vector potential throughout the accessible region.^[ref]

Use International System of Units, electric charge $q$, and magnetic minimal coupling $H=(-i\hslash \nabla -q\mathbf{A}{)}^2/(2m)$. Let ${\gamma }_1,{\gamma }_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows ${\gamma }_1$ and then ${\gamma }_2$ backwards. For a static magnetic vector potential, the magnetic contribution to the relative wave phase isHere ${\Phi }_C={\int }_{\Sigma }\mathbf{B}\cdot d\mathbf{S}$ is the oriented magnetic flux through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an electron, $q=-e$, where $e>0$ is the elementary charge; reversing $C$ reverses the sign. The observable is the gauge-invariant holonomy $e^{i\delta }$, rather than a unique local value of the magnetic vector potential.

The accessible region is a multiply connected region: $C$ cannot contract to a point without entering the excluded flux region. Stokes theorem does not set the line integral to zero, because a spanning surface must cross that excluded region, where the magnetic field need not vanish. There is real magnetic flux in the apparatus; only the accessible paths are idealized as free of the magnetic field. The exterior of a finite straight solenoid also permits routes around its ends, so its exact topology and leakage differ from those of an infinite solenoid or an excluded solid torus.

An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose cylindrical coordinates $(r,\theta ,z)$ andIts curl vanishes there, but a loop with winding number $n$ has ${\oint }_C\mathbf{A}\cdot d\mathbf{l}=n\Phi$. Locally $\mathbf{A}=\nabla (\Phi \theta /(2\pi ))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi /\hslash$, that candidate gauge removal changes the wave function's single-valued boundary condition. On a simply connected region, a smooth static magnetic vector potential with zero curl instead has a globally defined scalar primitive and is a pure gauge potential. Thus local zero magnetic field and globally removable magnetic vector potential are different conditions.

For an original interference example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in off-axis holography. ThenThe interference fringe spacing is $d=2\pi /k$. An electron enclosing $\Phi =h/(4e)$ has $\delta =-\pi /2$, shifting maxima by $d/4$ for this orientation. Increasing the magnetic flux by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the electron Aharonov-Bohm effect has magnetic flux quantum $h/e$. This uses $h=2\pi \hslash$, the Planck constant. It does not require the source's magnetic flux to vary continuously during a single exposure.

For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max }=9/4$, $I_{\min }=1/4$, and the interferometric visibility is $4/5$. A fixed Aharonov-Bohm phase translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed magnetic flux. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.

The appeal of the Aharonov-Bohm effect is that a local Lorentz force calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of holonomy and supplies the encircling phase used in Chern--Simons flux attachment. This alone does not establish intrinsic topological order. The static comparison also matters: changing magnetic flux can induce an electric field, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial line integral.

In Mass spectra and isotopes by Francis William Aston, printed pp. 10–12, Fig. 1, Francis William Aston describes successive electric and magnetic deflections in one plane, with opposite senses. This geometry focuses a range of ion speeds; it does not require the straight-through cancellation of a velocity selector.

The reason compensation is possible follows from the Lorentz force. In a short region of length ${\ell }_E$, a small electric deflection angle has magnitude ${\theta }_E\simeq qE{\ell }_E/(m{v}^2)$: the transverse acceleration acts for a time ${\ell }_E/v$, and the deflection angle is the acquired transverse velocity divided by $v$. A short magnetic region of length ${\ell }_B$ gives ${\theta }_B\simeq qB{\ell }_B/(mv)$, since the magnetic force already contains one factor of speed. Slower ions therefore receive a disproportionately larger electric deflection.

An original local approximation makes the focusing idea explicit. At fixed mass-to-charge ratio, replace each region by a small angular kick and include the drift distances in two positive constants $a$ and $b$. With $t=v/v_0$, write the detector coordinate asChoosing the geometry and fields so that $b=2a$ eliminates the first-order speed sensitivity. Indeed,A fractional speed error $\epsilon =t-1$ then contributes only at order ${\epsilon }^2$ in this approximation. This is a local thin-kick explanation, not Aston's exact pole geometry, focal-plane equation, or a prediction of its measured mass resolution. Finite slit widths, initial directions, and field boundaries still matter. Unlike a velocity selector, focusing can collect nearby speeds into a narrow image instead of discarding all but a narrow accepted range.

Let the specimen rotate uniformly with angular velocity $\mathbf{\Omega }=\Omega \hat{\mathbf{z}}$. In its rotating reference frame, the internal Hamiltonian containsHere $J_z$ is the internal angular momentum operator about the rotation axis, and ${\mathcal{H}}_0$ contains the internal interactions and any imposed magnetic field expressed in that frame. It does not include the separately treated rigid body's rotational kinetic energy. For a sector with magnetic moment $m_z=\gamma J_z$, the rotational term has exactly the form of a magnetic dipole potential energy:This Barnett equivalent magnetic field has tesla units, with the angular convention for the signed gyromagnetic ratio: $\Omega$ is in radians per second, not revolutions per second. Its sign matters. For an electronic contribution with $\gamma <0$, positive $\Omega$ favors positive $J_z$ and negative $m_z$. A sector with $\gamma >0$ has the opposite magnetic sign. Equal $\Omega$ need not correspond to equal Barnett fields for sectors having different gyromagnetic ratios.

A short derivation fixes the sign independently of any material model. Put $U(t)=\exp (-\mathrm{i}\Omega t{J}_z/\hslash )$ and transform a laboratory quantum state by $|{\psi }_{\mathrm{rot}}⟩=U^{†}|{\psi }_{\mathrm{lab}}⟩$. The Schrödinger equation givesThe transformed laboratory term is denoted by ${\mathcal{H}}_0$; a laboratory perturbation not stationary in the rotating frame can make it time dependent. Establishing a steady thermal response also requires exchange of internal angular momentum with the rotating body, for example through spin-lattice relaxation. A mere relabeling of coordinates cannot polarize isolated, unrelaxed spins.

The Barnett field is a useful description of spin-rotation coupling, not a claim that empty laboratory space acquires a uniform magnetic flux density $\Omega /\gamma$. In particular, inserting it into the Lorentz force on an unrelated test electric charge is unjustified. A nearby magnetometer detects the specimen's actual stray magnetic field and backgrounds, rather than directly detecting this internal Hamiltonian parameter.

Stereoscopy supplies depth because different cameras see different projections of the same track. A functional reconstruction first matches the same interaction vertex and track segments across views. A calibrated lens and window model maps each measured image coordinate back to a ray inside the liquid. Refraction at the windows must be included. Intersecting the rays, or finding their closest agreement when measurements have errors, estimates three-dimensional coordinates. Several points then determine a trajectory, rather than only a projected arc.

An original ideal-camera model makes the geometry explicit. Take two parallel pinhole cameras with centers $(0,0,0)$ and $(b,0,0)$, focal length $f$, and optical axes along $z$. Use virtual image planes in front of the pinholes so that image coordinates have the same orientation as the scene. A point $(X,Y,Z)$ then givesSubtracting the horizontal image coordinates gives disparity $d=u_L-u_R=fb/Z$, henceFor stipulated $b=0.20\mathrm{m}$, $f=0.050\mathrm{m}$, $u_L=0.010\mathrm{m}$, $u_R=0.005\mathrm{m}$ and $v_L=0.002\mathrm{m}$, the reconstructed point is $(0.40,0.08,2.0)\mathrm{m}$. These are ideal optical inputs, not dimensions of Alvarez's apparatus. This model omits refraction and lens distortion; actual calibration replaces its simple ray geometry. Since $|dZ/dd|=fb/d^2$, the same disparity error gives worse depth precision for distant points.

The Lorentz force then supplies a momentum scale. With magnetic field $B$ uniform and negligible stopping power over a fitted segment, bubble-chamber track readout gives $p_{\perp }=|q|Br$. The local three-dimensional helix tangent supplies the missing direction: if $\alpha$ is its angle to the field, $p=p_{\perp }/\sin \alpha$. In practical units,Here $e$ is the elementary charge and $c$ the speed of light. A short, almost straight arc poorly constrains its radius of curvature; a visually sharp track can therefore have poor momentum precision.

Stereoscopy enables three-dimensional track reconstruction.^[ref] Calibrated camera positions and optical geometry are needed before a photographed arc becomes a physical radius of curvature. Neutral particles do not directly leave the same continuous ionization trail as charged particles; an interaction or decay can instead reveal them through charged secondary tracks. A photon's inferred path between interactions is not a photographed luminous line.

In a uniform magnetic field, with negligible stopping power and elastic scattering over the fitted segment, the transverse projection is a circle. The Lorentz force turns transverse momentum at angular rate $v_{\perp }/r$, givingHere $r$ is measured in the plane perpendicular to the magnetic field. The relation remains valid with relativistic physical momentum; it does not require $p=mv$. Thus curvature determines magnetic rigidity, not mass or kinetic energy alone. For chosen $|q|=e$, $B=1.5\mathrm{T}$ and $r=0.40\mathrm{m}$, $p_{\perp }\simeq 0.180\mathrm{GeV}/c$. This is an invented example, not a fit to a historical photograph. Reconstructing a three-dimensional helix also determines the component parallel to the magnetic field; the transverse radius of curvature alone does not. Determining the sign of electric charge additionally requires the field direction and the direction of travel, as in the charge-sign ambiguity of a curved track.

Suppose $V$ is a Killing vector field, $F$ is an electromagnetic field tensor satisfying $dF=0$ and ${\mathcal{L}}_{V}F=0$, and a particle of nonzero mass $m$ obeys the Lorentz force equation ${\nabla }_{u}u=(q/m)F^a{}_b{u}^b$. Cartan's magic formula gives $d({\iota }_{V}F)=0$, so the local potential of a closed differential one-form proof gives $d\Phi ={\iota }_{V}F$ locally. The Killing equation eliminates $u^a{u}^b{\nabla }_a{V}_b$, and the force term gives $d(V\cdot u)/d\tau =(q/m)d\Phi /d\tau$. Therefore $mV\cdot u-q\Phi$ is locally conserved. The sign assumes contraction in the first argument of $F$. A global potential additionally requires the closed one-form to have zero periods; invariance alone is insufficient without the homogeneous Maxwell equations.

For identical particles of rest mass $m$ and electric charge $q$, the four-current is $j^{\mu }=(q/m)\rho u^{\mu }$. The matter stress-energy tensor obeys ${\nabla }_{\mu }{T}_{\mathrm{m}}^{\mu \nu }=F^{\nu \alpha }{j}_{\alpha }$, with the opposite force density in the electromagnetic stress-energy tensor. Antisymmetry of the electromagnetic field tensor makes its contraction with two four-velocities vanish, so mass continuity still holds. Removing that term leaves the displayed covariant Lorentz force law. The matter tensor alone is not divergence-free when an electromagnetic force acts.

A photograph records a path but supplies no arrow of time. The Lorentz force depends on $q\mathbf{v}\times \mathbf{B}$: reversing both electric charge and direction of travel leaves the spatial bending compatible with the same path. Thus a particular arc can represent a Positron travelling one way or an electron travelling the other. Knowing the magnetic field's polarity is necessary but insufficient. Gravity does not justify assuming that every cosmic ray track travels downward.

The lead plate adds a physical ordering. For a single particle crossing a passive absorber without an accelerating interaction, the outgoing momentum is smaller, so the outgoing gyroradius is smaller in the same magnetic field. The less curved segment comes first. Once this direction is assigned, the cross product in the Lorentz force determines the sign of electric charge. For the convention below, $\mathbf{B}=B\hat{\mathbf{z}}$ points out of the page; at upward velocity $\mathbf{v}=v\hat{\mathbf{y}}$, $\mathbf{v}\times \mathbf{B}=vB\hat{\mathbf{x}}$. Bending to the right therefore means positive electric charge.

Magnetic forces do no work: the change of kinetic energy occurs through interaction with the material, not through the bending magnetic field. The ideal construction suppresses multiple scattering, kinetic energy fluctuations, and loss of kinetic energy in the gas. In an actual analysis one compares arcs away from the plate and allows a change of tangent from multiple scattering; one cannot interpret every kink as a change of electric charge. Short visible arcs, uncertain projection and small bending also limit a determination of magnetic rigidity.

The charge-to-mass ratio is a body's electric charge $q$ divided by its mass $m$. Its SI unit is the coulomb per kilogram, $\mathrm{C}\mathrm{k}{\mathrm{g}}^{-1}$. In a prescribed electric field and magnetic field, nonrelativistic acceleration depends on $q/m$ through the Lorentz force. Two bodies with the same initial position, velocity, and charge-to-mass ratio therefore have the same trajectory in this ideal model. For an electron, write $q=-e$ with $e>0$ the elementary charge: $q/m=-e/m$, whereas the positive number $e/m$ is the magnitude of the charge-to-mass ratio.

For constant masses, a time-independent potential energy $V$ and forces ${\mathbf{F}}_i=-{\nabla }_{i}V+{\mathbf{R}}_i$ with zero total extra power ${\sum }_i{\mathbf{R}}_i\cdot {\mathbf{v}}_i=0$, Newton's second law gives $dT/dt=-dV/dt$. Thus mechanical energy is conserved. Smooth stationary constraint reactions and a purely magnetic Lorentz force supply zero power. This is more restrictive than conservation of energy in general, which may include thermal or other internal forms.

Take an ion of mass $m$ and electric charge $q\ne 0$, with velocity perpendicular to a uniform magnetic field of magnitude $B>0$. In the classical mechanics limit, the Lorentz force provides the centripetal acceleration:Here $r$ is the local radius of curvature and ${\omega }_c$ is the cyclotron frequency, expressed as an angular frequency. A half-turn takesIncreasing speed increases the path length in exactly the same proportion. Drive the gap at angular frequency ${\omega }_{\mathrm{RF}}={\omega }_c$, or ordinary frequency $f_{\mathrm{RF}}=|q|B/(2\pi m)$. In each half-turn the gap electric field reverses once, just as the particle reverses its direction of passage through the gap. The particle therefore receives two accelerating kicks per complete turn, using one oscillation per complete turn. This is the fundamental resonance; transit time through the gap and departures from a uniform magnetic field have been neglected.

With a fixed axial seed field and prescribed localized differential rotation, the winding source is time independent. Early induction gives $B_{\varphi }\sim B_0\Omega t$, while the steady balance has $B_{\varphi }\sim B_0\Omega {\ell }^2/\eta$. Their comparison gives the magnetic diffusion time, not merely the rotation period. It diverges as conductivity becomes infinite. In an unbounded domain, this is a local equilibration estimate on scale $\ell$, not finite-time exact convergence everywhere. Lorentz force feedback may invalidate the prescribed-flow approximation sooner.

A motivating laboratory realization is The Hall Effect and Conductivity in Germanium by Rex Tayloe. The handout specifies a germanium sample holder with temperature and magnetic field control, a low-voltage supply, digital meters, and a current-reversing switch. Measure the sample dimensions, map the contacts, and establish the directions of electric current, magnetic field, and transverse voltage. At fixed temperature and zero magnetic field, sweep positive and negative electric currents and record longitudinal electric potential differences. A straight voltage-versus-electric current relation tests Ohm's law; sample geometry converts its slope to electrical resistivity. For the Hall effect, minimize the zero-field transverse offset and measure transverse voltage at all four combinations of $\pm I$ and $\pm B$, repeating for several electric currents. The readouts test longitudinal proportionality, transverse polarity, and transverse proportionality to electric current. These are protocol instructions, not reported measurement outcomes.^[ref]

For interpreting such measurements, an ideal uniform bar of wire cross-sectional area $A$ with longitudinal sensing contacts separated by $L$ gives $\sigma =LI/(A{V}_L)$. Four-terminal sensing suppresses voltage drops in current contacts when the voltage meter draws negligible electric current; it does not fix uncertain dimensions or a nonuniform current density. Reverse electric current and use $[V_L(+I)-V_L(-I)]/2$ to reject a constant voltage offset. Keep temperature stable, allow the current density to settle, and reduce electric current to test for Joule heating. Those are controls for interpreting the material coefficient, rather than reasons to assume every measured curve is linear.

For the transverse measurement define $\mathbf{J}=J_x\hat{\mathbf{x}}$, $\mathbf{B}=B_z\hat{\mathbf{z}}$, width $w$ along $y$, and thickness $d$ along $z$. Define $V_H=V(y=-w/2)-V(y=+w/2)=E_{y}w$, so $J_x=I/(wd)$ givesExtract the part odd in both electric current and magnetic field usingThis combination rejects a constant offset and longitudinal pickup even in magnetic field. It cannot reject another signal with the same two reversal symmetries. Check field polarity and magnitude independently; a sign error in the contact map changes the inferred Hall coefficient.

For one Drude model species in transverse open circuit, the steady Lorentz force balance is $0=q(\mathbf{E}+{\mathbf{v}}_d\times \mathbf{B})-m{\mathbf{v}}_d/\tau$ with $v_{d,y}=0$. Its $y$ component gives $E_y=v_{d,x}{B}_z$. ConsequentlyLongitudinal electrical conductivity and the Hall coefficient therefore separate number density from charge carrier mobility in this single-species model. Inferring a momentum relaxation time additionally requires an appropriate mass. A constant-field longitudinal measurement alone cannot supply all three.

A Hall coefficient is not universally the reciprocal of signed number density. For example, adding independent electron and electron hole contributions with densities $n,p$ and positive charge carrier mobilities ${\mu }_e,{\mu }_h$ in the weak-magnetic field limit givesThe Hall coefficient can vanish even while electrical conductivity is positive, and its sign weights charge carrier mobility as well as number density. The germanium handout explicitly asks students to track possible transverse sign changes with temperature; it does not establish a universal single-carrier interpretation. Real electronic band structure and unequal scattering also limit that interpretation.^[ref]

Dynamo quenching is the reduction or modification of magnetic-field generation by magnetic feedback on the generating flow and its correlations. The Lorentz force can change shear or turbulence, while magnetic helicity affects mean-field regeneration. A phenomenological scalar example is $\alpha (B)={\alpha }_0/(1+B^2/B_{\ast }^2)$. In the illustrative amplitude equation $\dot{B}=[c\alpha (B)-d]B$ with $c,d>0$, a growing field saturates at $B^2=B_{\ast }^2(c{\alpha }_0/d-1)$ when $c{\alpha }_0>d$. This demonstrates the negative feedback mechanism without claiming that algebraic quenching is a complete solar closure.

Earth's outer core is a liquid, electrically conducting region whose buoyant motion and rotation sustain the geodynamo. Its magnetic field both responds to the flow through induction and modifies that flow through the Lorentz force. A prescribed-velocity kinematic model describes onset rather than the entire nonlinear core dynamics.

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.

The electric force on an electric charge $q$ is ${\mathbf{F}}_E=q\mathbf{E}$, using the electric field of the other sources and excluding the charge's own self-field. Positive electric charge gives force along the electric field; negative electric charge reverses it. A field $\mathbf{E}=400N/C\hat{\mathbf{x}}$ therefore gives ${\mathbf{F}}_E=-8\times 10^{-7}\mathrm{N}\hat{\mathbf{x}}$ on $q=-2\mathrm{nC}$. This is the electric part of the Lorentz force; the magnetic contribution is $q\mathbf{v}\times \mathbf{B}$. ^[ref].

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.

The combined electric field and magnetic field that determine the Lorentz force on a charged particle.

Electromagnetism describes how electric charge and electric current interact through the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$. The Maxwell equations describe the fields and their sources; the Lorentz force describes the response of a particle of electric charge $q$ and velocity $\mathbf{v}$:The electric field can change the particle's energy. The magnetic part of the Lorentz force is perpendicular to velocity, so magnetic forces do no work; they can bend a trajectory. These are local interactions: a particle responds to the electromagnetic field at its position. ^[ref].

The remarkable unification is that the same Maxwell equations connect Coulomb's law, Faraday's law of induction, and freely propagating electromagnetic waves. A changing electric field contributes to magnetic circulation through displacement current; together with Faraday's law, this permits waves in vacuum without conducting matter. The Poynting theorem then explains how the fields store and transport energy. This connects the force on a charge to the energy carried by light.

In his 1865 paper, James Clerk Maxwell derived transverse disturbances and compared the propagation speed inferred from electrical measurements with independently measured light speeds. Part VI, especially §§95–97, argues that their agreement supports identifying light as an electromagnetic wave. This was a comparison with finite experimental accuracy, rather than a numerical identity between the historical measurements. ^[ref]; ^[ref].