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In Gaussian units, ${\bf B=H}+4\pi{\bf M}$. If it is long and thin, $\bf H$ is negligible, and ${\bf M =B}/4\pi$ for most of the length. The magnetic moment is $\bf M$ times the volume. When $\bf B$ is removed, $\bf M$ will reduce to smaller value, depending on the material.

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They do, for a short time. However, a spin which is precessing has higher magnetic potential energy than a spin which is completely aligned with the field, so the precessing state is unstable. The mechanism by which spins "decay" from a precessing state to a state in which they are aligned with the external field is provided by the spin-phonon ...

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In a classical experiment Rasetti proved that $\mathbf{B}$ was the effective field within a magnetized matter. Here is the abstract: The deflection of mesons in a magnetized ferromagnetic medium was investigated. A beam of mesons was made to pass through 9 cm of iron, and the resulting distribution of the beam was observed. Two arrangements were employed. ...

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The reason is that $\vec{B}$ is the fundamental field. Traditionally (and in Maxwell’s traditional notaion), this was not really appreciated, and was $\vec{H}$ treated as if it were the fundamental one; however, this was incorrect. The reason for the mistake was that $\vec{H}$ was easier to measure, as its sources was the free (i.e., experimenter-controlled)...

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If the object precesses or not, or does something inbetween, namely nutate, depends on the ratio of Larmor frequency to the rotation frequency of the object. If you start from a situation where the magnet is tilted with respect to the field and then you let go, you obtain perfect precession only, if the rotation frequency is infinite. One may think of ...

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The magnetization in a bar magnet is due to the alignment of the spin magnetic moments of the electrons, not to currents. The precession would depend on the cross product $\mu\times{\bf B}$. Since $\mu$ is aligned along $\bf B$, there is no torque and no precession.

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