Maxwell's Equations

Maxwell’s Equations provide a general description of electromagnetic phenomena. They are named after James Clerk Maxwell, the Scottish physicist whose pioneering work unified the theories of electricity and magnetism. The theory of electromagnetism was built on the discoveries and advances of many scientists and engineers, but the pivotal contribution was that of Maxwell.

The properties of the microscopic electro-magnetic (EM) field for stationary environments are expressed with well known system of Maxwell's equations, as shown on Picture 1:

Picture 1: System of Maxwell's Equations (for stationary environments)


I. The first equation is derived in compliance of the Ampere's Law with the law of continuation.
II. The second equation is a generalized form of Faraday's Law for electromagnetic induction.
III. The third equation expresses the generalized form of the Gauss theorem.
IV. The fourth equation represents the Law of conservation of magnetic flux.


When solving concrete systems, the Maxwell's Equations system need to be upgraded or supplemented with the following equations (Picture 2):

Picture 2: Additional equations for general Maxwell's Equations System


Here, the equation (1) represents the generalized form of the Ohm's Law, while with the equations (2) are expressed the vectors of electric displacement and magnetic induction. For linear and isotropic environments, these equations (2), are expressed in the form as represented with the equations (3).

The physical quantities included in all these equations above, are:

E - Electric field;
H - Magnetic field;
D - Electric displacement;
B - Magnetic induction;
J - Electric current density;
ρ - Distribution of electric charge;
σ - Specific electric conductance;
P - Electric polarization vector;
M - Vector of magnetization;
ε - Electric constant;
μ - Magnetic constant;