# 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:

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