Electromagnetic interaction · R.A.V.E.N. Living Encyclopedia

The electromagnetic interaction is one of the four fundamental interactions, a gauge interaction with symmetry group U(1) mediated by the photon, and its strength at low energy is set by the fine-structure constant α = 7.297 352 5643(11) × 10⁻³ (Mohr et al. 2025). It is carried by the electromagnetic field, whose observable content is the field tensor F_μν formed from the electric and magnetic fields. The classical limit of that field obeys the Maxwell equations, the quantum limit is quantum electrodynamics with the photon as the field quantum, and the scalar and vector potentials from which the tensor derives are defined only up to a gauge transformation, so that the field and not the potential is the measured quantity. Electric charges and currents source the field, and the field acts on every particle that carries electric charge or a magnetic moment; the neutrinos carry neither and do not couple to it at tree level (Navas et al. 2024). Tests of the force law give an inverse-square exponent 2 + q with q = (2.7 ± 3.1) × 10⁻¹⁶ (Williams et al. 1971), a photon charge below 1 × 10⁻⁴⁶ e (Altschul 2007), and a coupling that runs to α⁻¹(m_Z) = 127.930(8) at the Z-boson mass (Navas et al. 2024). The photon mass is below 1 × 10⁻¹⁸ eV (Ryutov 2007), so the range of the interaction exceeds ℏ/m_γc = 2 × 10¹¹ m and is consistent with an unbounded range (computed from Ryutov 2007). The interaction binds electrons to nuclei and atoms into molecules, carries the radiation by which astronomical objects are observed, and governs the magnetised plasmas of stellar coronae and pulsar magnetospheres (Goldreich & Julian 1969). The inverse-square law of the electrostatic force was established by Coulomb (1785), the description of electricity, magnetism and light by one field theory by Maxwell (1865), and the free propagation of electromagnetic waves by Hertz (1888).

02 · Equations

Governing equations

The electromagnetic interaction is described by the Maxwell field coupled to charges and currents, and at the quantum level by quantum electrodynamics (QED). Relations are written in SI units with vacuum permittivity ε₀ and permeability μ₀; numerical values are from the CODATA 2022 adjustment (Mohr et al. 2025) and the Particle Data Group review (Navas et al. 2024).

Master · Maxwell Lagrangian

Electromagnetic Lagrangian density

$$\mathcal L=-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}-J^\mu A_\mu,\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,\qquad \partial_\mu F^{\mu\nu}=\mu_0 J^\nu$$

Variation with respect to the four-potential A_μ gives the inhomogeneous Maxwell equations, and the homogeneous pair follows from the definition of F_μν (Maxwell 1865). In quantum electrodynamics the current of the electron field is J^μ = −e ψ̄γ^μψ, and the relations below follow from this Lagrangian as classical limits or as expansions in α (Navas et al. 2024).

01 · Force

Coulomb's law

$$\begin{gathered}\mathbf F_{12}=\frac{q_1q_2}{4\pi\varepsilon_0 r^2}\,\hat{\mathbf r}_{12},\\ \frac{1}{4\pi\varepsilon_0}=8.987\,551\,79\times10^{9}\ \mathrm{N\,m^2\,C^{-2}}\end{gathered}$$

Force between two point charges at rest, repulsive for like signs and attractive for unlike signs; the constant is computed from the CODATA value of ε₀ (Mohr et al. 2025). The exponent is measured as 2 + q with q = (2.7 ± 3.1) × 10⁻¹⁶ (Williams et al. 1971).

02 · Field

Electric field and Gauss's law

$$\begin{gathered}\mathbf E=\frac{q}{4\pi\varepsilon_0 r^2}\,\hat{\mathbf r},\qquad \mathbf F=q_{\rm t}\,\mathbf E,\\ \oint_S\mathbf E\cdot d\mathbf A=\frac{Q_{\rm enc}}{\varepsilon_0}\end{gathered}$$

The field is the force per unit test charge and is defined at every point whether or not a test charge is present; Gauss's law is the integral form of ∇·E = ρ/ε₀ (Maxwell 1865). The permittivity is ε₀ = 8.854 187 8188(14) × 10⁻¹² F m⁻¹ (Mohr et al. 2025).

03 · Potential

Scalar and vector potentials

$$\begin{gathered}\mathbf E=-\nabla V-\frac{\partial\mathbf A}{\partial t},\qquad \mathbf B=\nabla\times\mathbf A,\\ V(r)=\frac{q}{4\pi\varepsilon_0 r},\qquad U=q_{\rm t}V\end{gathered}$$

The potentials determine the fields, and the transformation V → V − ∂χ/∂t, A → A + ∇χ leaves E and B unchanged, so the potentials are fixed only up to a gauge function and are not observables. The potential of a point charge falls as 1/r, and q_tV is the potential energy of a test charge in it (Maxwell 1865).

04 · Equation of motion

Lorentz force

$$\begin{gathered}\mathbf F=q\left(\mathbf E+\mathbf v\times\mathbf B\right),\\ \frac{d\mathbf p}{dt}=q\left(\mathbf E+\mathbf v\times\mathbf B\right),\qquad \mathbf p=\gamma m\mathbf v\end{gathered}$$

Equation of motion of a charge in given fields, valid at all speeds when the relativistic momentum is used (Lorentz 1895). The magnetic term does no work; in a uniform field B the motion is a helix of angular frequency qB/γm about the field direction.

05 · Field equations

Maxwell equations

$$\begin{gathered}\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0},\qquad \nabla\cdot\mathbf B=0,\\ \nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\qquad \nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}\end{gathered}$$

The four field equations in SI form (Maxwell 1865). In vacuum they combine into a wave equation for E and B with speed c = (ε₀μ₀)^−½ = 299 792 458 m s⁻¹, exact by the 2019 definition of the metre (Mohr et al. 2025).

06 · Energy and flux

Field energy and the Poynting vector

$$\begin{gathered}u=\tfrac12\left(\varepsilon_0E^2+\frac{B^2}{\mu_0}\right),\qquad \mathbf S=\frac{1}{\mu_0}\,\mathbf E\times\mathbf B,\\ Z_0=\sqrt{\mu_0/\varepsilon_0}=376.730\,313\,412(59)\ \Omega\end{gathered}$$

Energy density of the field and the vector giving its energy flux (Poynting 1884). In a plane wave E = cB and the time-averaged flux is E₀²/2Z₀; the impedance of free space Z₀ is from Mohr et al. (2025).

07 · Radiation

Larmor formula

$$\begin{gathered}P=\frac{q^2a^2}{6\pi\varepsilon_0c^3},\\ \frac{dP}{d\Omega}=\frac{q^2a^2}{16\pi^2\varepsilon_0c^3}\,\sin^2\theta\end{gathered}$$

Power radiated by an accelerated charge at speeds well below c, with the angular distribution of an oscillating dipole vanishing along the acceleration (Larmor 1897). The relativistic form multiplies P by γ⁶ for acceleration parallel to the velocity and by γ⁴ for perpendicular acceleration, the basis of synchrotron radiation.

08 · Coupling

Fine-structure constant and its running

$$\begin{gathered}\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}=\frac{1}{137.035\,999\,177(21)},\\ \alpha^{-1}(m_Z)=127.930(8)\end{gathered}$$

The dimensionless coupling of the interaction; value from Mohr et al. (2025), determined by atom recoil (Morel et al. 2020) and by the electron magnetic moment (Fan et al. 2023). Vacuum polarisation makes the effective coupling increase with momentum transfer, and at the Z-boson mass α⁻¹ = 127.930(8) in the modified minimal-subtraction scheme (Navas et al. 2024).

03 · Numbers

Measured properties

Recommended values from the CODATA 2022 adjustment (Mohr et al. 2025), the exact 2019 SI definitions of the elementary charge and the metre, and the Particle Data Group photon listing (Navas et al. 2024). Parenthesised digits give the standard uncertainty in the final digits shown.

α = 7.297 352 5643 × 10⁻³

Coupling · 1.5 × 10⁻¹⁰ relative · Mohr et al. 2025

1.602 176 634 × 10⁻¹⁹ C

Elementary charge · exact · SI 2019

299 792 458 m s⁻¹

Speed of the free field c · exact · SI 2019

m_γ < 1 × 10⁻¹⁸ eV

Photon mass · upper limit · Ryutov 2007

Quantity	Value	Status	Meaning & convention
Fine-structure constant α	7.297 352 5643(11) × 10⁻³	measured · 1.5 × 10⁻¹⁰	Dimensionless coupling e²/4πε₀ℏc; the inverse is 137.035 999 177(21) (Mohr et al. 2025).
Elementary charge e	1.602 176 634 × 10⁻¹⁹ C	exact · SI 2019	Unit of the charge that sources the field; fixed by definition in the 2019 SI (Mohr et al. 2025).
Vacuum permittivity ε₀	8.854 187 8188(14) × 10⁻¹² F m⁻¹	measured · 1.6 × 10⁻¹⁰	Constant of the electric field equations; ε₀ = 1/μ₀c², and its uncertainty is that of α (Mohr et al. 2025).
Vacuum permeability μ₀	1.256 637 061 27(20) × 10⁻⁶ N A⁻²	measured · 1.6 × 10⁻¹⁰	Constant of the magnetic field equations; μ₀ = 4παℏ/e²c, and it is no longer exact in the 2019 SI (Mohr et al. 2025).
Speed of light c	299 792 458 m s⁻¹	exact · SI 2019	Propagation speed of the free field, c = (ε₀μ₀)^−½; exact by definition of the metre (Mohr et al. 2025).
Coulomb constant 1/4πε₀	8.987 551 79 × 10⁹ N m² C⁻²	derived · 1/4πε₀	Constant of Coulomb's law and of the point-charge potential (computed from Mohr et al. 2025).
Impedance of free space Z₀	376.730 313 412(59) Ω	derived · (μ₀/ε₀)^½	Ratio of E to H in a plane wave in vacuum (Mohr et al. 2025).
Mediator	photon γ · J^P = 1⁻ · Q = 0	exact · representation	Massless vector boson with two helicity states ±1; the quantum of the field (Navas et al. 2024).
Photon mass	< 1 × 10⁻¹⁸ eV	limit · Ryutov 2007	From the magnetohydrodynamics of the solar wind out to the orbit of Pluto; the analysis quotes no confidence level (Ryutov 2007; Navas et al. 2024).
Photon charge	< 1 × 10⁻⁴⁶ e	limit · Altschul 2007	From the absence of an Aharonov–Bohm phase in very-long-baseline interferometry (Altschul 2007; Navas et al. 2024).
Inverse-square exponent	2 + q, q = (2.7 ± 3.1) × 10⁻¹⁶	measured · Williams et al. 1971	Deviation of the electrostatic force law from 1/r² in a laboratory test with concentric shells (Williams et al. 1971).
Range	> 2 × 10¹¹ m	derived · ℏ/m_γc	Reduced Compton wavelength at the photon mass limit; consistent with an unbounded range (computed from Ryutov 2007).
Running coupling α⁻¹(m_Z)	127.930(8)	measured · MS-bar	Effective coupling at the Z-boson mass; the increase from the low-energy value is due to vacuum polarisation (Navas et al. 2024).

Sources · NIST CODATA 2022 (physics.nist.gov/constants) · PDG 2024 photon listing S000 (pdglive.lbl.gov) · Williams, Faller & Hill PRL 26, 721 (1971) · Ryutov Plasma Phys. Control. Fusion 49, B429 (2007) · Altschul PRL 98, 261801 (2007)

04 · Situations

Representative configurations

Six configurations computed from CODATA 2022 constants and the cited measurements: positronium as a Coulomb bound state, an electron in crossed electric and magnetic fields, Rutherford scattering, pair production in a nuclear field, Coulomb's torsion balance of 1785, and the magnetosphere of the Crab pulsar. Each scene states any scale factor applied to the rendering.

05 · Latest research

Recent literature

Preprints and papers retrieved at page load, ordered by submission date. arXiv: categories hep-ph, hep-ex, physics.atom-ph, physics.class-ph and astro-ph.HE, abstracts matching the fine-structure constant, the photon mass, Coulomb's law, quantum electrodynamics, the Lorentz force or the Maxwell equations. INSPIRE-HEP: the same terms in titles of high-energy-physics records. Dates are arXiv submission dates and INSPIRE record dates; no publisher issue dates are used.

Querying arXiv · INSPIRE-HEP …

Date	Source	Title and authors	Reference
Retrieving …

06 · References

References

Declared representations

Field lines and surfacesField lines and equipotential surfaces are drawn at a finite number of positions; the field is continuous and defined at every point, and the line density is a display convention for its magnitude.

Test chargeThe gold test charge in the hero samples the field at its position. Its force arrow is scaled as |E|^0.35 for display (declared), and it is drawn as if it did not disturb the source charges.

Radiation layerThe radiation layer draws the far-field pattern and the wavefronts of an oscillating dipole on the axis of the source charges, which are drawn static; the layer declares the pattern, not a motion of the charges.

Declared scalingWhere a physical length or rate is below display resolution (the gyroradius, nuclear distances, the neutron-star radius, the rotation rate), the applied scale factor is stated in the scene.

Sources

C. A. Coulomb (1785). Premier mémoire sur l'électricité et le magnétisme. Histoire de l'Académie Royale des Sciences 1785, 569.

J. C. Maxwell (1865). A dynamical theory of the electromagnetic field. Phil. Trans. R. Soc. 155, 459.

J. H. Poynting (1884). On the transfer of energy in the electromagnetic field. Phil. Trans. R. Soc. 175, 343.

H. Hertz (1888). Ueber elektrodynamische Wellen im Luftraume und deren Reflexion. Ann. Phys. 270, 609.

H. A. Lorentz (1895). Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern. E. J. Brill, Leiden.

J. Larmor (1897). On the theory of the magnetic influence on spectra; and on the radiation from moving ions. Phil. Mag. 44, 503.

E. Rutherford (1911). The scattering of α and β particles by matter and the structure of the atom. Phil. Mag. 21, 669.

H. Geiger & E. Marsden (1913). The laws of deflexion of α particles through large angles. Phil. Mag. 25, 604.

H. Bethe & W. Heitler (1934). On the stopping of fast particles and on the creation of positive electrons. Proc. R. Soc. A 146, 83.

M. Deutsch (1951). Evidence for the formation of positronium in gases. Phys. Rev. 82, 455.

P. Goldreich & W. H. Julian (1969). Pulsar electrodynamics. Astrophys. J. 157, 869.

E. R. Williams, J. E. Faller & H. A. Hill (1971). New experimental test of Coulomb's law: a laboratory upper limit on the photon rest mass. Phys. Rev. Lett. 26, 721.

A. H. Al-Ramadhan & D. W. Gidley (1994). New precision measurement of the decay rate of singlet positronium. Phys. Rev. Lett. 72, 1632.

R. S. Vallery, P. W. Zitzewitz & D. W. Gidley (2003). Resolution of the orthopositronium-lifetime puzzle. Phys. Rev. Lett. 90, 203402.

R. N. Manchester, G. B. Hobbs, A. Teoh & M. Hobbs (2005). The Australia Telescope National Facility pulsar catalogue. Astron. J. 129, 1993.

B. Altschul (2007). Bound on the photon charge from the phase coherence of extragalactic radiation. Phys. Rev. Lett. 98, 261801.

D. D. Ryutov (2007). Using plasma physics to weigh the photon. Plasma Phys. Control. Fusion 49, B429.

L. Morel et al. (2020). Determination of the fine-structure constant with an accuracy of 81 parts per trillion. Nature 588, 61.

X. Fan et al. (2023). Measurement of the electron magnetic moment. Phys. Rev. Lett. 130, 071801.

S. Navas et al., Particle Data Group (2024). Review of particle physics. Phys. Rev. D 110, 030001 · PDG photon listing.

P. J. Mohr et al. (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97, 025002 · physics.nist.gov/constants.