What to read about magnetic fields and magnetic interactions?

Question

3D

What it's about
I want to simulate the interaction of two or more ferrite permanent magnets, varying their size, shape and location in the mechanism. There is already an engine in mind in which you can do this. It's up to the math.
What I want
We need books and websites that explain in detail and / or extremely accessible the physics of MT and everything connected with them, in order to understand for ourselves which formulas to take into account and which characteristics to change.

Answer 1

Once upon a time, I was solving a similar problem: I wanted to plot the dependence of the force of attraction on the distance between two magnets, I modeled the magnets simply as rings with current.
1. To calculate the force, I used the well-known formula, the force is equal to the potential energy gradient. Here I made a slight simplification, considered only the force applied to the center of the magnets - and did not consider that the magnets may try to turn each other. But this is also easy to calculate.
2. The energy of the magnetic field, in turn, is the spatial integral of the product B*H = mu*B*B over dV. We limit the space to some reasonable limits, I used ~ 200-1000 ring diameters.
3. Magnetic induction is additive, that is, B at the "point" dV is the sum of B1 and B2 - the fields of one and the other ring. The rings are the same, so it is enough to pre-calculate the field around one ring, and then add the already calculated fields to take the integral.
4. The magnetic field of the ring is axisymmetric, so it is sufficient to calculate the field only on one quarter of the plane built on the radius of the ring and its axis.
5. To calculate this field - the Biot-Savart-Laplace formula, and the circular integral along the contour of the ring. Here you probably don’t optimize in any way, except to count only half of the ring.
Thus, in essence, the problem is reduced to: the Biot-Savart-Laplace circular integral over half of the ring, calculated at each point of the quarter-plane, and integrating this field over the volume with the necessary displacements and rotations.
There are problems in three places:
1. The field near the ring. There is a good method in physics textbooks (unfortunately, I don’t remember where) - to consider a ring that is not infinitely thin, then this problem is easily solved.
2. Grid granulation for faster and more accurate calculation. The standard way - near the ring we beat the plane into many segments, and further from the ring - more and more segments.
3. Mismatch of the grids of the two rings, if the rings are rotated relative to each other - I would try Bresenham's algorithm to find the nearest segment or even interpolation. I repeat, I did not implement the turns.
This was done about five years ago on a laptop with a Core2Duo 8500, on Delphi using SSE2, in 5 threads. Depending on the granulation, the calculation took up to a day. As a result, an inverse-cubic dependence was obtained, as expected.
If you want to determine the behavior of the system - direct the system to a minimum of energy by varying the positions / rotations of the magnets, I would act by Newton's method. Fortunately, the recalculation of the system with new positions does not require recalculation of the magnetic field of the ring, but only requires the calculation of the spatial integral.
What would I do now:
1. transferred vector calculations (circular integral) to CUDA, space integral - probably, too.
2. I checked whether the approximation of the magnet by a ring with current is correct - I would simply place many small rings inside the volume of the magnet (and of course, in one direction, like domains), and compare the field of such an imitation with the field of one ring.
3. made full-fledged graphs of energy gradients for all degrees of freedom. I think there would be a lot of interesting things.

Answer 2

The Feyman Lectures are worth reading. Electricity and magnetism are very clearly stated there, with all the mathematics.