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I was wondering is there any formula to calculate the magnetic force exerted on a metallic block in a magnetic field? Let's say there is a coil of wire with length=L, area=A, and turns=N. A metallic cylinder with permeability=μ (it is ferromagnetic but doesn't have its own magnetic dipole) is partially inside the coil. If the current=I is passed through the coil, what will be the force that pulls the metal block inside the coil? Is there a formula for it? Thanks

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Yes, there are equations for the amount of force experienced by a ferromagnetic material in an external field - e.g. see Magnetic Force on a Ferromagenetic Material

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