Electromagnetic Field Dynamics

Variables > s.a. electromagnetism / electricity; magnetism; electromagnetism in media; tensor decomposition.
* 3+1 version: The electric and magnetic field 3-vectors (Eⁱ, Bⁱ), the 3-vectors (Eⁱ, A_i), or the potentials (φ, A_i), with

E = −∇φ ,   B = ∇ × A .

* Covariant version: The 4-vector A_a = (φ, A_i), or the Faraday field strength tensor F_ab = ε_ab^c B_c + 2 E_[a v_b] , or

in terms of the spatial tensors E_a = u^b F_ba, the electric field, and B_a = −\(1\over2\)ε_ab^cd u^b F_cd , the magnetic field.
@ General references: Armand-Ugón et al PRD(94)ht/93 [action in terms of loop variables]; Heras & Báez EJP(09) [covariant, for general units]; Rácz a1811 [Gauss law constraint as a first-order hyperbolic pde]; > s.a. gauge theories.
@ Relativistic transformation rules: Huang PS(09) [new approach]; Gomori & Szabó a0912, PE-a1109.

Maxwell's Equations > s.a. Coulomb's Law; Faraday's Law; Gauss' Law; electricity; magnetism [Ampère-Maxwell law].
* The Maxwell equations: If we include hypothetical magnetic charges and currents, they are

\[ \def\dd{{\rm d}} \matrix{\underline{\rm Differential\ version\ (cgs,\ in\ a\ medium)}
& \underline{{\rm Integral\ version\ (SI},\ \rho_{\rm m} = {\bf J}_{\rm m} = 0)}\hfil \cr
\nabla\cdot{\bf B} = 4\pi\,\rho_{\rm m}\hfil & \oint_S {\bf B}\cdot\dd{\bf a} = 0\hfil \cr
\nabla\times{\bf E} + {1\over c}\,{\partial{\bf B}\over\partial t} = -{4\pi\over c}\,{\bf J}_{\rm m}\hfil
& \oint_C {\bf E}\cdot\dd{\bf s} = -{\dd\over\dd t}\int_S {\bf B}\cdot\dd{\bf a}\hfil \cr
\nabla\cdot{\bf D} = 4\pi\,\rho_{\rm e}\hfil & \oint_S {\bf E}\cdot\dd{\bf a} = {Q\over\epsilon_0}\hfil \cr
\nabla\times{\bf H} - {1\over c}\,{\partial{\bf D}\over\partial t} = {4\pi\over c}\,{\bf J}_{\rm e}\hfil
& \oint_C {\bf B}\cdot\dd{\bf s} = \mu_{_0}I + \mu_{_0}\epsilon_{_0}\,{\dd\over\dd t} \int_S {\bf E}\cdot\dd{\bf a}}\]

* Covariant versions: In terms of the Faraday tensor they can be written as dF = 0 (local existence condition for A_a) and d*F = *J, or

∇_[a F_bc] = 0 ,   ∇_a F^ab = −4π J ^b ,

and in terms of the electric and magnetic fields they are

∇_a (B^au^b − B^bu^a) + ∇_a (ε^abcd u_cE_d) = 0 ,   ∇_a (E^au^b − E^bu^a) − ∇_a(ε^abcd u_cB_d) = 4π J ^b ,

where u^a is the timelike unit vector field used to define the space + time decomposition that gives the fields E^a and B^a.
@ References: Soodak & Tiersten AJP(94)oct [interpretation, sources]; Barut FP(94) [from Coulomb-Clausius potential]; in Sonego & Abramowicz JMP(98) [in terms of electric and magnetic fields]; Bracken IJTP(05)mp/06 [and Lagrangian, Hamiltonian, Poisson brackets]; Fleisch 08 [+ CUP resource site]; Kulyabov & Korolkova a1211 [in arbitrary coordinate systems]; Papachristou AEM(15)-a1505 [as a Bäcklund transformation]; Sebens FP(19)-a1902 [and photon wave function].
@ Derivations: Singh & Dadhich IJMPA(01)gq/99; Heras AJP(07)jul, EJP(09) [from continuity equation]; Pierce a0807 [from covariance requirements]; Diener et al AJP(13)feb [heuristic]; Kosyakov EJP(14) [and spacetime geometry].
@ Related topics: Heras AJP(11)apr [interpretation of the displacement current]; > s.a. history of physics.
> Online resources: see HyperPhysics page; Wikipedia page; World of Physics page.

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 9 sep 2019