Why is the induced M field proportional to H, instead of the B field?

[burgjeff]

The M field is the density of induced or permanent magnetic dipole moments. It is analogous to the P field in electrostatics. In electrostatics, the induced P field in a dielectric is proportional to the applied electric field. This is intuitive to me. Why though, in magnetostatics, is the M field proportional to the Demagnetizing field(the H field).

M=X[itex]_{m}[/itex]H=X[itex]

​

_{m}[/itex]([itex]\frac{B}{u_{0}}[/itex]+M)

I can't seem to wrap my head around this. My textbook declares it without any justification. Can anyone provide me some insight into this?

Thank you.

 

[Meir Achuz]

It is the B field in the material that produces M. When magnetization is produced by placing iron in a long solenoid or a torus, the H field in the iron is directly given by nI because H_t is continuous, so engineers like to consider M a function of H. But the iron knows that is is the B field that is producing M. When removed from the long solenoid or torus, end effects become dominant. That is why B and H are in opposite directions near the end of a magnet. At this point mu and chi become irrelevant. The H field is not 'demagnetrizing', although calling it so is a fairly common error of UG texts. It is B, not H, that affects M
You are right, but don't tell your professor, because he/she probably believes the textbook.

 

[Jano L.]

The reason behind the use of ##\mathbf H## in the definition of magnetic susceptibility is the fact that in common situations, ##\mathbf H## field is easily calculated from the electric currents in the wiring, which are easily measured by Am-meter. The field ##\mathbf B## is hard to calculate from the controlled variables such as the applied current; it has to be inferred based on the intensity of induced currents in secondary wiring.

This is similar to definition of electrical susceptibility in electrostatics: there it is defined by

$$
\mathbf P = \epsilon_0 \chi_e \mathbf E,
$$

because the ##\mathbf E## field is easily calculable from the voltage applied to capacitor, while the ##\mathbf D## field is not.

But the iron knows that it is the B field that is producing M.

In macroscopic EM theory, there is not much reason for such assertion. Both fields are only macroscopic fields, describing the physical state in a different way.

Only in microscopic theory we could attempt to identify the basic field that acts on particles and makes them magnetized. The common view is that the macroscopic field ##\mathbf B## is a kind of probabilistic description of the much more complicated microscopic magnetic field ##\mathbf b## actually exerting force on the individual charged particles. So iron responds to the presence of microscopic fields ##\mathbf e, \mathbf b## and magnetizes; but it would be a stretch to say iron feels ##\mathbf B##, because ##\mathbf B## is just a simplified concept invented by humans, incapable to account for all the myriads of nuclei and electrons in the material.

 

[burgjeff]

Thanks for the responses guys. After reading them, my understanding is that the M field is defined this way conventionally because it is more convenient. This is allowed because, by rearranging the equation M=X[itex]_{m}[/itex]H in order to solve for M, one gets:

M=[itex]\frac{X_{m}}{X_{m}+1}[/itex] [itex]\frac{B}{u_{0}}[/itex]

Thus, defining M proportional to H is equivalent to defining M proportional to B, within an arbitrary constant of proportionality.

 

[pmacias]

The relationship between the induced M field and the applied H field is a fundamental principle in magnetostatics known as the magnetic susceptibility, X_m. This relationship can be described by the equation M=X_mH, where M is the magnetic moment density, H is the applied magnetic field, and X_m is the magnetic susceptibility. This relationship is similar to the relationship between the induced P field and the applied electric field in electrostatics, as you mentioned.

The reason for this relationship can be understood by considering the microscopic behavior of magnetic materials. In a magnetic material, there are tiny atomic-level magnetic dipoles that can align themselves with an external magnetic field. When an external magnetic field is applied, these dipoles will tend to align with the field, resulting in an induced magnetic moment. This is similar to how the electric dipoles in a dielectric align themselves with an electric field.

Now, the strength of the induced magnetic moment will depend on the strength of the external magnetic field. The stronger the external field, the more the dipoles will align and the stronger the induced magnetic moment will be. This is why the M field is proportional to the H field.

On the other hand, the B field is a combination of the applied magnetic field and the M field. So, if we were to make the M field proportional to the B field, we would end up with a circular argument where the B field depends on the M field, which in turn depends on the B field.

In summary, the proportionality between the induced M field and the applied H field is a result of the microscopic behavior of magnetic materials and is a fundamental principle in magnetostatics. I hope this explanation helps to clarify the concept for you.