Maxwell's Equations

Created by Maxwell.

Microscopic

For vacuum. $\nabla \cdot E=\frac{\rho }{{ϵ}_0}$ Gauss’ Law for Electricity $\iint \vec{E}\cdot d\vec{S}=\frac{Q_{enclosed}}{{ϵ}_0}$ $\nabla \times E=-\frac{\partial B}{\partial t}$ Maxwell-Faraday Equation $\mathcal{E}=\oint \vec{E}\cdot d\vec{l}=-\frac{d{\Phi }_B}{dt}=-\frac{d}{dt}\iint \vec{B}\cdot d\vec{S}=-\iint \frac{\partial B}{\partial t}\cdot d\vec{S}$ $\nabla \cdot B=0$ Gauss’ Law for Magnetism $\iint \vec{B}\cdot d\vec{S}=0$ $\nabla \times B={\mu }_0(J+{ϵ}_0\frac{\partial E}{\partial t})$ Ampère’s Law With Maxwell’s Addition ${\oint }_C\vec{B}\cdot d\vec{l}={\mu }_0\left(I+{ϵ}_0\frac{d{\Phi }_E}{dt}\right)={\mu }_0(\iint \vec{J}\cdot d\vec{S}+{ϵ}_0\frac{d}{dt}\iint \vec{E}\cdot d\vec{S})S$

Macroscopic

For a medium. You get the integral forms the same way. $\nabla \cdot D=\rho$ $\nabla \times E=-\frac{\partial B}{\partial t}$ $\nabla \cdot B=0$ $\nabla \times H=J+\frac{\partial D}{\partial t}$