Maxwell’s Equations

Abstract

Maxwell’s equations are based on the following empirical facts:

• (1) The electric charges are the sources and sinks of the vector field of the dielectric displacement density D. Hence, for the flux of the dielectric displacement through a surface enclosing the charge we have

  $$ \frac{1}{{4\pi }}\oint_{area} D \cdot nda = Q = \int_v {\rho dv} $$

  Here, n is the outward pointing unit normal vector. This relation can be derived from Coulomb’s force law.

• (2) Faraday’s induction law:

  $$ V = \oint E \cdot {\text{dr}} = - \frac{1}{c}\frac{{\partial \phi }}{{\partial t}}{\text{ }}with{\text{ }}\phi = \int_{area} {B \cdot {\text{nda}}} $$

• (3) The fact that there are no isolated monopoles implies

  $$\oint_{area} B \cdot {\text{nda}} = 0$$

• (4) Oersted’s or Ampère’s law:

  $$ \oint {H \cdot dr = \frac{{4\pi }}{c}} I = \frac{{4\pi }}{c}\int j \cdot nda $$