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If a current carrying loop has magnetic moment m, how can i determine the magnetization M, of it? Ive done some reading and have found a formula that seems to solve it, it says that M is equal to the derivative of the magnetic moment with respects to volume, however using volume makes no sense to me as the volume where all the mag field lines are enclosed is infinite. So the question becomes: what volume does this formula imply? Thanks in advance

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Magnetisation, M in a small region of volume is $\delta V$ is given by $ M=\frac{\delta m}{\delta V}$ in the limit as $\delta V $$ \rightarrow 0$,

where m is the magnetic dipole moment in that region.

The magnetic field lines from a dipole moment indeed occupy an infinite space. However, to find the Magnetisation at a given point, imagine enclosing a that point by a small box of volume $\delta V$. If the magnetic dipole moment inside that volume is $\delta m$ then the Magnetisation at that point will be given by the above expression for M. Note that in general, the magnetisation will be different at different points. In the case of a simple magnetic dipole, a graph of M vs spatial position will be a delta function, which is zero everywhere, except at the position of the magnetic dipole, such that if you integrate the function over the whole space, you will get the magnetic dipole moment.

However, the term Magnetisation is normally used when dealing with magnetism in solids. In solids, you will have several magnetic dipole moments arranged and the concept of finding the magnetisation at a given point makes more sense in this case.

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