# Finding the force between magnetized object and conductor

Let's suppose we have an object with some constant magnetization $$\vec{M}$$ (for example, a cylinder). If we approach it to a material with infinite permeability (i.e. a ferromagnetic) we know from experience that the object will be attracted to the conductor.

My question is, how can we calculate the force of attraction between these two? We can easily calculate $$H$$ and $$B$$ from the magnrtization and the geometry of the object, and probably apply boundary conditions for these fields to further find other unknowns.

However I'm not sure how can we determine the force of the attraction between the magnetized object and the conductor, or if there's an equation or principle that I may be missing (i.e. I don't think Lorentz force can be used since I only have one magnetized object).

• A magnet will not be attracted to a metal in general. It is only attracted, if the metal itself can be magnetized. – flaudemus Feb 21 '19 at 6:58
• Thanks, indeed I forgot to point out this happens for ferromagnetic metals. So in this case, will I have an induced magnetization on the conductor? – Charlie Feb 21 '19 at 9:39
• Yes, indeed, this is what you are going to have. – flaudemus Feb 21 '19 at 13:14
• Thank you. One last question, is there an expression or a formula to find the induced magnetization of an object given an applied field? I've read Jackson and Zangwills books on the topic, but the examples provided only deal with already magnetized objects and finding their magnetic fields, and not the other way around (i.e. having a magnetic field and finding the magnetization of the object). Same would be if there's an expression for the magnetic force given the field between objects (my intuition tells me it should be proportional to the gradient of the magnetic potential $\Phi_M$). – Charlie Feb 21 '19 at 17:57
• In lowest order, the magnetization of a material is proportional to the external field, and the proportionality constant is the magnetic susceptibility. However, for your problem and the materials you have in mind, the linear approximation is not sufficient. Magnetic materials containing iron, for example, have the well-known hysteresis curves, which are curves of magnetization vs external magnetic field. These curves depend a lot on the details of your material. – flaudemus Feb 21 '19 at 21:04