Here's a simple "barnyard engineering" explanation.
First of all, the proper term is "quasi-static electric field." "Quasi-" meaning "sort of" or "like." So before we look at the quasi-static field, we will look at the static field, which is easy.
If you have two electric charges of different polarity (+ and -) separated by a distance, there is an electric field between them. You can picture the two sources of charge as the terminals of a battery. The field between the terminals looks exactly like the field between the poles of a bar magnet, which everybody sees in school at some time. Of course, the field between the battery terminals is an electric field, and the field between the magnet poles is a magnetic field, but otherwise they look identical.
Now in both of these cases the source of the fields is unchanging - battery potential or the strength of the bar magnet. Hence these are both static fields. Picture the battery so oriented that the battery "+" terminal is vertically above the battery "-" terminal, and similarly for the magnet, the north pole is up and the south pole is down. Now we start to slowly rotate the battery so that after a 180 degree rotation, the "-" terminal is on top and the "+" terminal is below it. The orientation of the field is opposite what it was at the beginning. And after another 180 degree rotation, it is back to where we started. Ditto, once again, for the bar magnet.
At any point in time, the field lines' orientation relative to the battery terminals (or magnet poles) always looks the same, but since the source is rotating, the fields rotate with them. Now if this was all that happened, there wouldn't be much point in renaming these rotating fields "quasi-static" instead of just plain static, but something else is going on as well. In the static case, the field lines simply get weaker as you move farther away from the source. Think of those iron filings lining up on the paper above the bar magnet, very precisely close in, but more and more ragged looking as you move away. When you rotate the source, there is a tiny bit of lag between the time you move the source and when the effect of moving the source causes the field to rotate. This effect is technically termed the "retarded potential." An imperfect analogy is the cracking of a whip. If you wave the whip slowly through the air, not much happens. But if you move it fast enough, the whip will "crack." This happens when you move your end of the whip so fast that it is moving opposite to the direction that the whip tip is traveling. Note that a "wave" travels down the whip, because of the delay between the movement at your end and the tip. The force acting on the tip originated in time after the movement of your hand - hence the term "retarded." You will never notice this effect when rotating an electric or magnetic field source by hand, nor even at several thousand rpm, which is what an electric motor or generator is doing with a magnet. But the effect is there, and if you rotate the charges or poles fast enough, what happens is that some of those lines of force break away from the terminals or poles and travel away from the source. That is termed radiation, and it is how an antenna works. And it is what you hear when the whip cracks. And of course, it is what you hear when you tune your radio across the AM or FM broadcast bands.
Note that radiation requires a source of changing polarity, either electric or magnetic. Further, recall that the field close in to the source looks like the static field would if the charges or poles held steady. The field near the changing source is termed quasi-static to differentiate it from a purely static field, and also to differentiate it from that portion of the field which breaks away and radiates.
You can get much more complex than this, but in a nutshell, this explains and differentiates the quasi-static field from the other types of electromagnetic fields.
I spent four years at a technical college, and one year in grad school, and never got an explanation like this. I got math - lots of it, but no clear picture of what was going on. The type of explanation above comes from books of an earlier era - WWII and before, when people were more interested in undertanding at a basic level, and left the math to the Ph.D.s. Hope it works for you - it was a revelation to me!
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