Maxwells Equations

Maxwell's equations relate the fields (E and H) and their sources (p and J) to each other. The electric field strength E and the magnetic flux density B may be considered the basic quantities, because they allow calculation of a force F, applied to a charge, q, moving at a velocity, v, in an electromagnetic field; this is obtained using Lorentz's force law:

The electric flux density D and the magnetic field strength H take into account the presence of materials. The electric and magnetic properties of media bind the field strengths and flux densities; the constitutive relations are

where e is the permittivity [F/m = As/Vm] and / is the permeability [H/ m = Vs/Am] of the medium.

Maxwell's equations in differential form are

IV V X H = J + Ampere's law and Maxwell's addition (2.7)

As also mentioned in the above equations, a lot of the knowledge of electromagnetic theory was already developed before Maxwell by Gauss, Faraday, Ampère, and others. Maxwell's contribution was to put the existing knowledge together and to add the hypothetical displacement current, which then led to Hertz and Marconi's discoveries and to modern radio engineering.

How did Maxwell discover the displacement current? We may speculate and simplify this process of invention as follows (see [1], Chapter 18): Maxwell studied the known laws and expressed them as differential equations for each vector component, because the nabla notation (curl and divergence of a vector quantity) was not yet known. Nevertheless, we use the nabla notation here. He found that while Gauss' and Faraday's laws are true in general, there is a problem in Ampèere's law:

If one takes the divergence of this equation, the left-hand side is zero, because the divergence of a curl is always zero. However, if the divergence of J is zero, then the total flux of current through any closed surface is zero. Maxwell


correctly understood the law of charge conservation: The flux of current through a closed surface must be equal to the change of charge inside the surface (in the volume), that is,

In order to avoid the controversy, Maxwell added the displacement current term, dDidt, to the right-hand side of Ampere's law in a general case and got

With this addition the principle of charge conservation holds, because by using Gauss' law we obtain

The differential equations, (2.4) through (2.7), describe the fields locally or at a given point. In other words, they allow us to obtain the change of field versus space or time. Maxwell's equations in integral form describe how the field integrals over a closed surface S (§s) or along a closed loop T ) depend on the sources and changes of the fields versus time. Maxwell's equations in integral form are:

where dS is an element vector perpendicular to surface S having a magnitude equal to the surface element area, dl is a length element parallel to the loop, and dVis a volume element. The volume, V, is enclosed by the closed surface S. Equations (2.11) and (2.12) are obtained by applying Gauss' theorem, according to which for any vector quantity A it holds

and (2.13) and (2.14) are obtained by applying Stokes' theorem

Maxwell's equations would be symmetric in relation to electric and magnetic quantities if magnetic charge density pM [Wb/m3 = Vs/m3] and magnetic current density M [V/m2 ] were also introduced into them. However, there is no experimental evidence of their existence.

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