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I understand mathematically why it is the case that the H must be in the opposite direction to the magnetization of the bar magnet, but I'm not quite sure what this means intuitively. This is due to $\nabla.\bf{H} = - \nabla.\bf{M}$, so $- \nabla.\bf{M}$ acts like a source of "fictional magnetic charge". But if our H-field is in the opposite direction to M does this then try to reduce the overall magnetization of our bar magnet and realign the magnetic dipoles inside in the direction of H? If so, then wouldn't it cause the magnetic dipoles to oscillate left and right all the time? Why do we have bar magnets at all in this case?

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Sorry for my poor english. My native language is french.

We reason in steady state and therefore we look for an operating point (no oscillations a priori!)

In general, the problem is complicated because the demagnetizing field is not uniform.

However, we can illustrate what happens with the simple case of a sphere for which, if the magnetization is uniform, the demagnetizing field is also uniform : ${\vec{H}}_{dm}=-\frac{1}{3}\vec{M}$

Let's suppose that the sphere is made of a linear magnetic material of magnetic susceptibility $\chi_m$.This sphere is placed in the field ${\vec{H}}_a$ of a long solenoid which applies a uniform exitation. The magnetization is written : $\vec{M}=\chi_m\vec{H}=\chi_m({\vec{H}}_a+{\vec{H}}_{dm})=\chi_m({\vec{H}}_a-\frac{1}{3}\vec{M})$ This gives :

$\vec{M}=\frac{\chi_m}{1+\chi_m/3}{\vec{H}}_a$

This magnetization is weaker than the magnetization $\vec{M}=\chi_m{\vec{H}}_a$ acquired by a material whose demagnetizing field is negligible (like a long cylinder).

In the case of a magnet, again if we can assume uniform fields, and a relation ${\vec{H}}_{dm}=-a\vec{M}$, the operating point is obtained by looking for the intersection of the demagnetization curve $M(H)$ of the material and the line $M=-\frac{1}{a}H$.

The greater the demagnetizing field (the greater $a$), the weaker the magnetization of the magnet.

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There is a "structural" difference between the field in the very middle of a "static dipole" of two "magnetic charges" and that of a current loop for their fields point oppositely to each other even though their respective far fields look the same both in the sense of direction and in shape.

This structural difference is at the heart of the very different macroscopic behavior between a bar magnet and a solenoid's field induced by a current and the true difference between the $H$ and $B$ fields.

Indeed, in the middle of a bar magnet the field is nearly zero but that does not mean that the domains are fluctuating or even effected for the molecular field that is frozen within a mesoscopic magnetic domain is orders of magnitude larger than anything we can generate form the outside. Since the direction of the polarization of these domains in a ferromagnet are essentially static and only their sizes by moving the domain walls change the external field can effect the total sum of the magnetization of the otherwise randomly oriented domains but not their individual orientations.

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