Calculate Magnetic Pull Force for Steel Ball

[Neil_Caffrey]

Homework Statement
Hello,
I am suppose to provide an equation to calculate the force that a disk magnet pulls a steel ball with respect to the distance between those two objects.

My idea is to calculate the difference between the energy density with and without the ball and then calculate the gradient of this change.
It will look something like F=∇(W0-W),
where W0 is the energy density with the ball and W is the energy density without it. The difference between them would be the permeability of the ball which varies on its size and material.

I don't have to calculate it exactly, just present some thought process. My question is: Is the method acceptable or should i rather focus on Maxwell tensor?
Thank you in advance.

 

[tman12321]

I think you have the right idea, but I can't tell exactly what you mean. Assuming that the ball is paramagnetic steel, its magnetization would be M = χ H, where χ is the magnetic susceptibility and H is the auxiliary field. H is related to the magnetic field by H = μ B, where μ is the permeability of the material ( μ = μ0 (1+χ) ). The force on a dipole is F = grad(m dot B), where m is the magnetic dipole moment, so the force density on the ball would be f = grad(M dot B). Then ∫∫∫ f dV (integral over the volume) would give the force. So F = χ μ ∫∫∫ grad(B dot B) dV.

 

[rude man]

Magnetic pressure = d B∇B/μ₀
where d = effective thickness of ball in direction of B field. This assumes infinite μ of the ball.

This formula would be exact for a rectangular material of infinite permeabiluity but for a ball the area is the cross-section and the thickness is some kind of average I guess.