Is there a formula, which describes the force on diamagnetic and paramagnetic matter in a magnetic field?

Question

To prove that a element is diamagnetic or paramagnetic, you can hold it into a inhomogeneous magnetic field and look in wich direction the force acts. (see description here) I am interested in a formula, which describes the force on the matter, similar to this one ($F_z = m_z *\frac{\partial B_z(z0)}{\partial z}$). The force should not just depend on $\partial B$ but also on the matter, which is used.

Thanks in advance

EDIT: would the formula look like this? $F = \chi * m * \frac{\partial B_z(z0)}{\partial z} \text{ where } \chi \text{ is the magnetic susceptibility.}$ I would need this formula for a simulation of a spheric. So I would just assume that the spheric is just a point (not in terms of the mass, but in terms of the force acting on it and in terms of the magnetic field), so I can ignore the geometry. Do I need any other factors or constants?

Answer 1

The formula contains the contribution of the matter. If you are looking for the force on a piece of matter in magnetic field, the magnetic moment of the piece of matter depends on the susceptibility of the material and the size of the sample. But a formula for the actual force is not trivial. It will depend, in general, on the actual geometry of the sample and the field, as the magnetic moment of the piece depends on the field too. So the force will be proportional to a term like $$ B \frac{\partial B}{\partial z} $$

The proportionality constant will contain the magnetic susceptibility and some geometric factor.