This is a message to me as an electrical engineering student 15 years ago.
There are no magnetic fields. There are only electric fields. The magnetic field lines presented to you in all your course work are mathematical constructs that allow the convenient tabulation of electric fields around moving charges. They do not really exist.
What you see as a field force like particle deflection is actually caused by the relativistic transformation of electric force between reference frames. Read Electromagnetic Fields and Waves by Paul Lorrain and Dale Corson where there are no less than five chapters on special relativity.
Here's how it works quick and dirty with no fancy derivation.
Electrons shot out of an electron gun at low energy and slow speed relative to a stationary observer come out in a wide spray. The higher the energy and the faster the velocity relative to the observer, the tighter the beam will become. Because they have like charges, the electrons repel each other. As they fly along, their velocity relative to each other is low. So, the electrons observe themselves flying away from each other with the acceleration afforded to them by the force of electrostatic repulsion. We the observer, however, appear to the electrons to be moving very slowly just as the movement of the electrons away from each other appears slow to us. This time dilation is due to Einsteins special relativity. The faster an object moves relative to us the slower it's inner processes appear to be.
Time dilation is only part of the story. Objects also appear shorter to the stationary observer the faster they go. This all means that electrostatic force bends with all the other contractions and dilations we observe of a physical object moving in the same way. After all, it is the shape of space itself that transforms. This relativistic bending of force is what we attribute to a magnetic field.
Notice that the deflecting force on a charged particle moving in a magnetic field is dependent on the speed and direction of the particle relative to the direction of that field. If we send a conventional current down a straight stretch of wire we will observe that a proton flying parallel to the wire in the direction of current will be drawn towards the wire. A proton flying in the opposing direction will fly away from the wire. Protons flying towards the wire from a perpendicular angle turn in the direction opposite the flow of current. This is all due to the slight change in shape of the electric fields of the charge carriers as they observe each other and as they are seen by the stationary observer. I am purposely anthropomorphizing the charge carriers to maintain brevity and simplicity.
And, now it should be obvious why the mathematical construct of magnetic field lines is the preferred method of derivation of electromagnetic force. It's simpler and it works. Unfortunately, the loss of the true explanation of magnetism from engineering curriculum leaves many students like myself woefully uninspired. It cannot be overemphasized that relativistic transformation as it relates to magnetism is crucial to a proper understanding of electromagnetism and the universe as a whole. Just because a student can derive a process does not mean he or she understands how it works.
A final note to those of you who know that the drift velocity of charge carriers in a conductor is only a few meters a second, and that isn't even close to relativistic speeds. As explained by Lorraine and Corson, the magnitude of force exhibited by electric charge is immense and the observed force of deflection in a magnetic field is minute by many orders of magnitude in comparison.
And, to those of you who are simply fascinated by magnetic fields but you're not really sure why. You know there is something terribly significant about them but you don't know what it is. Very simply, every time you play with a magnet, whether you are watching metal filings dance on a piece of paper or observing the deflection of a compass needle, you are really exploring the shape of space itself.