Electromagnetic waves consist of electric (E) and magnetic (B) fields propagating through space. These fields are orthogonal (at right angles to each other), in phase (reach same peak at same time), and fluctuate perpendicular to the direction of motion.
There you see an EM wave propagating outwardly from a metal rod (antenna) given a high frequency signal. The electric field and current oscillate vertically within the antenna, radiating off a vertically polarized electric field. Because fluctuating electric fields induce fluctuating magnetic fields at right angles and vice versa, electromagnetic waves consist of both coupled together.
An easier way to understand such waves is to visualize them in terms of the vector potential rather than magnetic or electric field. The vector potential is a more fundamental field, analogous to the momentum carried by flowing water. If a thick rope is dragged through water, some of the water surrounding it will be dragged along. Likewise with a wire or antenna through which current flows. The current (I) drags some “ether” along with it, and that flow is the vector potential (A).
Whenever this flow accelerates or decelerates, that brings about an electric field in the direction of flow. Whenever the flow contains some vorticity, that creates a magnetic field along the vortex’s axis of rotation, an axis that is necessarily perpendicular to the direction of flow. This is what the math says, but diagrams say as much.
Because the flow is stronger near the wire, an inequality exists between near and far fields, and this makes for some vorticity.
A magnetic field therefore follows along the axis of this vorticity, wrapping around the wire like so:
Rather than drawing the 3D view all the time, we can diagram only the side-view and use a circle with a dot to mean “coming out of screen” and circle with an X to mean “going into screen:
The antenna shown earlier is just a vertical wire with an oscillating rather than steady current. So let’s look at the vector potential field around the antenna:
In this diagram, only a slice of the right side of the field is shown. Here you see the vector potentials varying over distance. If this were animated you would see each arrow oscillate vertically, and the train of these would move out and away from the antenna. The electric field is also oriented vertically since it arises from changes in the vector potential, but with a 90 degree phase lag.
As mentioned earlier, a current-carrying wire is surrounded by a circular magnetic field due to differences between adjacent parts of the vector potential field creating vorticity. Same holds true for the antenna:
Only in this case the rotation of the vortices varies over distance and time, meaning the magnetic field also fluctuates, as expected:
So now you should have a better understanding of how electromagnetic waves are generated, and how the electric and magnetic components are simply different aspects of a single and more fundamental field, the vector potential. In fact, the vector potential is more “real” than either the electric or magnetic fields which are just our measurable interpretations of different distortions in the vector potential.
This brings up an interesting question. Modern physics dispels the idea of “ether” because it claims the concept is unnecessary to explain electromagnetism. Its reasoning is that an electromagnetic wave can easily travel through the vacuum since the fluctuating electric and magnetic fields both generate each other, that a photon is a self-contained entity requiring no medium in which to travel. But if the vector potential is what actually fluctuates then there is no other field component and the rationalization fails, meaning there must be a medium to support the wave. Some might respond that the vector potential is just a mathematical convenience, an arbitrary concept with no tangible existence of its own, but that could not be further from the truth.