Maxwell's Equations

Theorems #

Gauss Divergence Theorem #

$$ \iiint_\Omega \nabla \cdot \vec{F}\\ dV = \iint _{d\Omega} \vec{F}\cdot dS $$

Stokes Theorem #

$$ \iint_\Sigma \nabla \times \vec{F}\\ dS = \oint_{d\Sigma} \vec{F}\cdot dl $$

Equations #

Differential form #

$$\begin{array}{r} Gauss\\ Law:\nabla\cdot\mathbf{D}= \rho \\ For \\ magnetism:\nabla\cdot\mathbf{B}=0 \\ Maxwell-Faraday:\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B} }{\partial t} \\ Ampere’s\\ Circuitual\\ Law:\nabla\times\mathbf{H}= \mathbf{J}+\frac{\partial\mathbf{D} }{\partial t} \end{array} $$

Integral form #

$$ \begin{array}{r} Gauss\\ Law:\\ \iint_{d\Omega}D\cdot dS= \rho \\ For \\ magnetism:\iint_{d\Omega}B\cdot dS= 0 \\ Maxwell-Faraday: \oint_{d\Sigma}E\cdot dl= -\frac{d}{dt}\iint_\Sigma B\cdot dS \\ Ampere’s\\ Circuitual\\ Law: \oint_{d\Sigma}H\cdot dl= \iint_\Sigma J\cdot dS+\frac{d}{dt}\iint_\Sigma D\cdot dS \end{array} $$

E: Electric field D: Electric displacement B: Magnetic field H: Magnetizing field p: charge density J: current density