Electromagnetic Field Dynamics

Variables [> s.a. electromagnetism]
* 3+1 version: The 3-vectors (Eⁱ, Bⁱ), the 3-vectors (Eⁱ, A_i), or the potentials (, A_i), with

E = – ,   B = × A .

* Covariant version: The 4-vector A_a = (, A_i), or the Faraday field strength tensor F_ab = _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 = – _ab^cd u^b F_cd , the magnetic field.
@ 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.

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

* 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 = –4 J ^b ,

and in terms of the electric and magnetic fields they are

_a(B^au^b – B^bu^a) + _a(^abcd u_cE_d) = 0 ,   _a(E^au^b – E^bu^a) – _a(^abcd u_cB_d) = 4 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]; Singh & Dadhich IJMPA(01)gq/99 [derivations]; Bracken IJTP(05)mp/06 [and Lagrangian, Hamiltonian, Poisson brackets]; Heras AJP(07)jul, EJP(09) [from continuity equation]; Pierce a0807 [from covariance requirements]; Fleisch 08 [+ CUP resource site]; Huang PS(09) [new approach to relativistic transformations].
> Related topics: see history of physics.

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