All the two important problem that the EM try to describe and explain are propogation and radiation of EM wave, on the basics of Maxwell Equations. So we have to talk about the Maxwell Equations ,the greatest achievement in my opinion, first , then the propogation problem and then, the radiation problem.
- 1 the Maxwell Equations
In most conditions,except in static electronic field or static magnetic field, the two are never exits indepently. The Maxwell Equations are explicitly summarized the relationship between E and H, and between the source of them , a time-varied electric current or a time-varied charge.
The first two equations are Gauss‘s law in E field and H field. That is, Equation  is true at any point in space. That is, if there exists electric charge somewhere, then the divergence of D ( electronic displacement vector) at that point is nonzero, otherwise it is equal to zero.Gauss‘ Law states that electric charge acts as sources or sinks for Electric Fields.You see that both of these equations specify the divergence of the field in question. For the top equation, we know that Gauss‘ Law for Electric Fields states that the divergence of the Electric Flux Density D is equal to the volume electric charge density. But the second equation, Gauss‘ Magnetism law states that the divergence of the Magnetic Flux Density (B) is zero.
Why? Why isn‘t the divergence of B equal to the magnetic charge density?
Well - it is. But it just so happens that no one has ever found magnetic charge - not in a laboratory or on the street or on the subway. And therefore, until this hypothetical magnetic charge is found, we set the right side of Gauss‘ Law for Magnetic Fields to zero.
Faraday‘s law shows that a changing magnetic field within a loop gives rise to an induced current, which is due to a force or voltage within that circuit. We can then say the following about Farday‘s Law:
Ampere‘s Law with the contribution of Maxwell nailed down the basis for Electromagnetics as we currently understand it. And so we know that a time varyingD gives rise to an H field, but from Farday‘s Law we know that a varying H field gives rise to an E field.... and so on and so forth and the electromagnetic waves propagate.