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];
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 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 (Baub − Bbua) + ∇a (εabcd ucEd) = 0 , ∇a (Eaub − Ebua) − ∇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 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