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On a horizontal plane, there is a circular magnet and a lot of steel balls, what mathematical method can be used to express the distribution of these steel balls' positions? 


It can be many lays balls  around the magnet center, is that posible to calculate each ball location?

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The mathematical procedure that may have the best fit to your question is this: A topology optimization algorithm for magnetic structures based on a hybrid FEM–BEM method utilizing the adjoint approach.

But even this approach cannot calculate the location of each individual steel ball. The best it can do is calculate the geometry of the overall structure of the magnetic field, and even this would be limited to a small number of steel balls in the field.

In brief, I believe the problem that you posed; the way you expressed it, is unsolvable.


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