De-puzzle it by thinking of magnetism as nature
attempting to cancel out rotating electric charge by bending the
paths of all other moving charge.
That's it.
This simple way of thinking about magnetism worked well for me when
I was taking physics courses, but that was before there was much
talk of magnetic monopoles or quantum magnetism. Nevertheless,
for introductory physics courses, it can provide some valuable
insight.
As a thought experiment, take a point of view far away from a
problem involving magnetism and expect the solution to minimize the
net rotation of electric charge. Not only is this helpful as a
"reality check" for your solution to 3D vector equations such as
F=q(vxB), but it can help you setup your attack on the problem.
I do not intend to duplicate the efforts of physics textbooks, but
it may improve your interpretation of their mathematical frameworks
to think of it as a description of HOW nature tries to cancel out
rotating charge by changing the direction of any other moving
charges.
Note that I don't have a solution for WHY nature seems to try to
cancel out rotating charge by bending the paths of any moving charge
in the vicinity and I take solace from this video of the great
educator Richard Feynman's handling of the subject (especially after
minute 6) :
This page still needs work, but here are some miscellaneous comments
that might be worked in eventually:
An important concept is that superposition applies. In a
complicated system, there will be interactions among various
applied forces (electrical attraction/repulsion, mechanical,
etc.) as time progresses, but at any given instant they can be
solved seprately and added together.Think of the Right-Hand-Rule
as a tool for book-keeping.
If a charge isn't already moving, then it won't participate
in the act of rotation cancellation.
A framework could be devised around a Left-Hand-Rule.
Magnetism can be puzzling, but you can gain intuition about
it by thinking that nature tends to try to cancel things.
In the case of magnetism, it's rotating electric charge that is
being cancelled. Just as electric charges try to cancel
each other out as seen from a distance by attracting their
opposites to move closer, spin as seen from a distance also
tends to be cancelled.
A charge moving in a straight line can be thought of as
rotating with an infinite radius.
Charges that are moving in the vicinity will try to change
direction such that, from a distance, the net rotation of moving
charges will be lower.
Permanent magnets or outside sources of magnetic fields can
be thought of as the rotating charges in an electromagnet that
would create the same field. The force on any other moving
charge will change direction so as to cancel the rotation as
seen from a distance. This is just a way of thinking about
magnetism that may provide some intuition. If nothing else
it can help when doing a "reality check" on a problem solution.
The permeability constant of the materials involved may seem
to prevent prefect cancellation of spinning charge, but at the
atomic scale cancellation is going on inside those
materials. The permeability constant is a characterization
of that cancellation.