Why Electromagnetic Fields "Rotate"? I've been watching several videos on DC electric motors and they all mention how current through a wire creates a perpendicular rotating electromagnetic field around the wire in a direction that can be found using the hand rule.
If electromagnetism is nothing more than the attraction of electrons due to relativity as they move, why is this field rotating, what causes it to rotate?
 A: You're probably talking about the physics summarized in this photo:

The arrow next to the letter I is the direction that current is moving, which refers to the actual direction that positive charges would move. I stress actual there because the  arrows representing the magnetic field B do not represent a movement of any kind. The magnetic field is not rotating in the sense you may be imagining; it's not some substance that's moving around the wire in a swirling pattern.
Historically, magnetic field vectors were chosen to point "around" the wire because that is how tiny compasses would align if near the magnet:

That's really all the circular lines mean: The direction at any point along the circle tells us how a compass will point (in the absence of Earth's magnetic field). Experimentally, I would say this is "why" magnetic field lines wrap around current-carrying wires.
It turns out (turn, get it?!) that this direction also helps determine the direction in which a moving charged particle would experience a force due to the wire. (If you're interested, the force is found using a mathematical operation known as the cross product.) 
The typical view that student in introductory physics classes have of the magnetic field is that there's nothing physically there pointing in that direction, nor is there anything actually rotating. The direction (and strength) just help us make sense of other observations.
