Maxwell's Equations
Maxwell's Equations are a set of four vectordifferential equations that govern all of electromagnetics (except at
the quantum level, in which case we as antenna people don't care so much). They were first presented in a complete form
by James Clerk Maxwell back in the 1800s. He didn't come up with them all on his own, but did add the displacement current
term to Ampere's law which made them complete. The four equations (written only in terms of E and H, the electric field and the magnetic field), are given below.
In Gauss' law, is the volume electric charge density, J is the electric current density (in Amps/metersquared), is the permittivity and is the permeability. The good news about this is that all of electromagnetics is summed up in these 4 equations. The bad news is that no matter how good at math you are, these can only be solved with an analytical solution in extremely simple cases. Antennas don't present a very simple case, so these equations aren't used a whole lot in antenna theory (except for numerical methods, which numerically solve these approximately using a whole lot of computer power). The last two equations (Faraday's law and Ampere's law) are responsible for electromagnetic radiation. The curl operator represents the spatial variation of the fields, which are coupled to the time variation. When the Efield travels, it is altered in space, which gives rise to a timevarying magnetic field. A timevarying magnetic field then varies as a function of location (space), which gives rise to a time varying electric field. These equations wrap around each other in a sense, and give rise to a wave equation. These equations predict electromagnetic radiation as we understand it.
See also the Maxwell's Equations Tutorial.

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