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, Ai), or the potentials (φ, Ai), with

E = –∇φ ,   B = ∇ × A .

* Covariant version: The 4-vector Aa = (φ, Ai), or the Faraday field strength tensor Fab = εabc Bc + 2 E[a vb] , or

\[ F_{ab} = \left(\matrix{0 & E_{_1} & E_{_2} & E_{_3} \cr -E_{_1} & 0 & -B_{_3} & B_{_2} \cr
-E_{_2} & B_{_3} & 0 & -B_{_1} \cr -E_{_3} & -B_{_2} & B_{_1} & 0}\right)\;, \]

in terms of the spatial tensors Ea = ub Fba, the electric field, and Ba = –\(1\over2\)εabcd ub Fcd , 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]; > 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 Aa) and d*F = *J, or

[a Fbc] = 0 ,   ∇a Fab = –4π J b ,

and in terms of the electric and magnetic fields they are

a(BaubBbua) + ∇a(εabcd ucEd) = 0 ,   ∇a(EaubEbua) – ∇a(εabcd ucBd) = 4π J b ,

where ua is the timelike unit vector field used to define the space + time decomposition that gives the fields Ea and Ba.
@ 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 a1505 [as a Bäcklund transformation].
@ 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.


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