What is the force on an iron object due to a magnetic field?

Question

Say there is a solenoid that creates a constant magnetic field in the space around it. An iron ball bearing is placed at one end of the solenoid where the field lines spread out. What would be the formula for the force on the ball bearing? I imagine it has something to do with the iron being assumed to be a collection of dipoles?

Answer 1

You would calculate an expression for the magnetic potential energy of a particular configuration and then the gradient of this would give the magnitude and direction of the force.

For a small sphere of radius $r$ in an approximately uniform field ${\bf B}$ the induced magnetic dipole moment is (in cgs units) $$ {\bf p} = \left(\frac{\mu -1}{\mu+2}\right) r^3 {\bf B}\ .$$

Since iron has $\mu \gg 1$, then this reduce to ${\bf p} = r^3 {\bf B}$.

The potential energy is $$ U = -{\bf p} \cdot {\bf B} = -r^3 B^2$$ and the force on the dipole is the gradient of this expression and thus depends on how ${\bf B}$ (which is assumed uniform on the scale of the sphere in this treatment) varies with position.

For example if you assumed the field had a dipole form as it emerged from the end of the cylindrical solenoid, then the force on the sphere would vary as $\sim z^{-7}$.