On the determination of the internal magnetic structure by magnetic force microscope

In this contribution it is proven mathematically that it is in principle impossible to determine the magnetic charge distribution inside a magnetic material by a method which measures the stray field outside the sample, such as magnetic force microscopy (MFM). A general source of stray field, E_σ, is defined and it is shown that different solutions can be found for E_σ that result in the same stray field. It is also shown how both a perpendicular and a longitudinal medium can be described with the same E_σ. Using the equations for stray field, resulting from Eσ, it is also proven that performing the same MFM measurement at different scanning heights does not provide any new information on the stray field for sample; from a measurement at one (constant) height, the stray field at all other heights can be calculated. Moreover, the component of the field parallel to the same plane can be obtained from a measurement of the field component perpendicular to the sample plane.