Eikonal Blog

2010.02.05

Equations of electro+magneto-dynamics in 3+1 dimensions

Filed under: physics — Tags: , — sandokan65 @ 22:49

Maxwell equations:

  • \nabla \vec{D} = \rho_e,
  • \nabla \vec{B} = \rho_m,
  • \partial_t  \vec{B} + \nabla \times \vec{E} = - \vec{j}_m,
  • \partial_t  \vec{D} - \nabla \times \vec{H} = - \vec{j}_e.

Integral form of Maxwell equations:

  • \oint_S \vec{D}\cdot d\vec{S} = Q_e :\equiv \int_V \rho_e dV,
  • \oint_S \vec{B}\cdot d\vec{S} = Q_m :\equiv \int_V \rho_m dV,
  • \oint_l  \vec{E} \cdot d\vec{l} = - \int_S ( \vec{j}_m + \partial_t  \vec{B})
  • \oint_l  \vec{H} \cdot d\vec{l} = + \int_S ( \vec{j}_e + \partial_t  \vec{D})

Moments of the field (aka Energy-Momentum(-Pressure) complex of the electromagnetic field):

  • Flow of energy (Poynting vector): \vec{\Gamma}:\equiv\vec{E}\times\vec{H},
  • Linear momentum of field: \vec{G}:\equiv \vec{B} \times\vec{D},
  • Stress-pressure tensor: {\cal T}:\equiv \frac12 \vec{E}\otimes\vec{D} + \frac12 \vec{H}\otimes\vec{B} - \frac14 {\bf 1} \frac12 (\vec{E}\cdot\vec{D} + \vec{H}\cdot\vec{B}).

Material equations: in the case of the weak fields
the polarizable and magnetizable environment yields the linear responses:

  • \vec{D} = \epsilon \vec{E} + \vec{P} = \hat{\epsilon} \vec{E},
  • \vec{B} = \mu \vec{H} + \mu \vec{M} = \hat{\mu} \vec{H},

where the linear coefficients are the permittivity (\epsilon) and the the permeability (\mu), and the polarization and the magnetization of the substance are measured by vector fields \vec{P} and \vec{M}.

When \rho_m\ne 0 or \vec{\jmath}_m\ne 0 one can not use the signgle complex of potentials (\Phi, \vec{A}) (as used in ordinary electrodynamics), but should add one more complex of potentials (\Psi, \vec{F}). Then:

  • \vec{E} = - \partial_t \vec{A} -\nabla \Phi -c^2 \nabla \times \vec{F},
  • \vec{D} = - \partial_t \vec{L} -\nabla \Psi - \nabla \times \vec{U},
  • \vec{H} = - \partial_t \vec{U} -\nabla \phi + c^2 \nabla \times \vec{L},
  • \vec{B} = - \partial_t \vec{F} -\nabla \psi + \nabla \times \vec{A}.

The source constrainst:

  • \partial_t \rho_e + \nabla \cdot \vec{\jmath}_e = 0,
  • \partial_t \rho_m + \nabla \cdot \vec{\jmath}_m = 0.
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