Introduction

In his famous article A Dynamical Theory of Electromagnetic Field of 1864 James Clerk Maxwell wrote down differential equations that describe the laws of electromagnetism in full generality. The four equations of Maxwell,describe the dynamics of the five vector fields E, D, B, Hand J. Here E(x, t) is the electric field, D(x, t) the electric displacement, B the magnetic induction or magnetic flux density, H(x, t) is the magnetic field and, finally, J (x, t) is the electric current density. Since modern vector calculus was unknown to Maxwell, he formulated these equations as twenty scalar equations. The present form of these equations originates from Oliver Heaviside from the 1880's.

Equation (0-1) is Gauss’ law and it says that infinitesimally the total flux of the electric displacement is equal to the density of free charges. The scalar field p here is the free charge density. Equation (0-2) is the magnetic analogue of Gauss’ law saying that there are no free magnetic charges. Equation (0-3), called Faraday's law, explains how a changing magnetic flux creates an electric current in a conductive loop, a law that is based on a series of experiments that Faraday performed during 1831 and 1832.