Magnetic moment of bar magnet, estimate

Oct 2016
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The treatment for the magnetic moment of a bar magnet has been exhibited in most general physics and EM texts. How can the magnetic moment of a bar magnet be reasonably estimated from the total magnetic flux passing through the magnetized material in the plane that is perpendicular to the magnet's axis and cuts the magnet at its centroid (i.e. the cross-section of maximum flux) without regard to the other dimensions of the magnet? In other words, can the magnetic moment be estimated from the closed line integral around the bar magnet that has maximum magnetic flux? The maximum flux case will occur at the midpoint of the bar magnet (assuming symmetry along its length) with the closed path of integration being on or slightly below the surface of the magnet in the plane which contains the midpoint and is orthogonal to the magentic axis. This assumes that demagnetization effects are small or well known.

So one method is to replace the bar magnet with a solenoid that duplicates the bar magnet's magnetic field. In this case there are n turns of I amperes which can be reduced to one turn with nI amperes. The moment of this single turn would simply be nIA where A is the area of the solenoid surviving single current loop. Seems like this is the same problem as the bar magnet. The surface integral of the B field in the magnet (at the mid point) represents the moment. Is this thinking on the right path?
 
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