# Force exerted on an uncharged particle in a uniform magnetic field?

I'm struggling to understand how to calculate the force exerted on an uncharged particle in a magnetic field. Consider the following setup: a ball of ferromagnetic iron weighing 1kg is placed in a uniform magnetic field of 1 Tesla. What is the force exerted on the iron ball?

Most of the information I've found so far is limited to charged particles in magnetic fields, which results in the textbook helical motion. However, uncharged particles (such as an atom of iron) also clearly show an attraction toward magnets. What equation can I use to relate the strength of the magnetic field to the force exerted on an uncharged particle suspended in it?

A similar situation for electric forces would use $$F=qE$$, but there's no charge on an atom of iron so I'm not sure what to use.

• You use the dipole moment. There are charges in the Iron atom and there is a coupling of those charges with the field. The net dipole moment can be modeled classically or using quantum shell wave functions for the atom. The cross(dipole, field) gives you the torque. – ggcg Dec 14 '18 at 23:09
• Your second paragraph presents a different problem. A magnet has a nonuniform field around it, so the field gradient exerts a force on an atom with a magnetic moment. – A. Newell Dec 14 '18 at 23:10

The item will distort the magnetic field (because the field lines would "prefer" to travel through the iron than free space). It will experience forces/torques to enable more field lines to pass through the object.

In the case of a (initially) uniform magnetic field and a symmetrical sphere, there will be zero force because there is no difference in any position or orientation.

Whereas a normal magnet creates a very non-uniform field. The increasing strength of the field near the poles creates the lower potential (force toward) at those locations.

Typically a uniform magnetic field will NOT exert a net force on a magnetic dipole. It will cause it to rotate. A field gradient is needed to create a net force on the com of the magnetic material. Of course that analysis requires (1) the external field to be uniform (which it is based on your description) and (2) the dipole to have a fixed value. When you place unpolarized material in a field it will alter the material and that may create an opportunity for a force to be set up. This type of problem is addressed in most undergrad physics books to some degree of approximation. More elaborate treatments can be found in upper level EM physics texts.

• Hm. So an atom of iron wouldn’t move in a magnetic field? And an iron nail moves toward a magnet only because the magnetic field isn’t uniform? – Dubukay Dec 14 '18 at 23:03
• Have you done the calculations, or the experiments? – ggcg Dec 14 '18 at 23:06
• Think of the Stern-Gerlach experiment. – ggcg Dec 14 '18 at 23:12
• You should look for the Kelvin force... You need a magnetic field gradient to exert a force on – nodarkside Dec 14 '18 at 23:28
• That seems to be the consensus – ggcg Dec 14 '18 at 23:30