- #106
Philip Wood
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DrDu and tiny-tim: Many thanks. That should keep me quiet for a while!
Philip Wood said:DrDu and tiny-tim: Many thanks. That should keep me quiet for a while!
I think this was "decided" very early in the history of magnetism when people had still no idea of the microscopic basis of magnetism. People were observing the magnetic flux density B via the Lorentz force on one side and the magnetic field as the force with which two fields were interacting. Units were defined by the experimental setup and also chosen in analogy with electric fields.kaustubhan said:The question remains, and I am looking for an intuitive answer :- Why exactly was it decided that "B" and "H" should have different units ? especially when B = {1/ mu } {H + M} . Is it because field "B" becomes more dense inside a ferromagnetic material placed inside an external magnetic field ?
It depends on the system of units you're using. In the SI ("MKS") system, B and H have different units. In the Gaussian system, they have the same units.kaustubhan said:Why exactly was it decided that "B" and "H" should have different units ?
Charles Link said:In an in-depth analysis of it, ## H ##, unlike ## B ##, turns out to not represent a magnetic field, but rather it is a mathematical construction which results from ensuring that the equation ## B=\mu_o H+M ## holds when ## H ## consists of contributions from currents in conductors as well as from magnetic poles. In any case, it is an extremely useful mathematical construction for which ampere's law for magnetic materials ## NI=\oint H \cdot dl ## can be used to greatly simplify some mathematics in solving for the magnetic field ## B ##. Se e.g. https://www.physicsforums.com/threads/mmf-flux-density-across-air-gap-for-a-salient-pole.925295/
In the first case, I believe you are really measuring ## B ##, and in the second case, you are measuring ## E ##. In the first case, I also question the technique. For a superconductor, the ## B ## field inside is extinguished by surface currents, but not for ordinary conductors. For a much more recent discussion, where I think I presented a good case for my statement of ## D ## and ## H ## being mathematical constructions, see https://www.physicsforums.com/threads/understanding-gauss-law-diff-b-w-e-and-d-flux.929601/mairzydoats said:The only problem with saying H is just a mathematical construct and not a real field is that H can actually be directly measured by experiment at a particular point in space, and all without needing to know either B or M or even J locally. In this respect it is just as fundamental as E or B.
1) At the point in space in which you wish to know the magnitude and direction of vector H, place a small cylindrical perfect conductor, small enough that any currents induced in its surface won't affect the H field you want to measure more than negligibly.
2) Rotate it the cylinder around every conceivable axis until you maximize the induced surface current around the cylinder's curved surface.
3) Magnitude of field-H will be equal to this maximum induced surface current. Direction of field-H will be along the cylinder's axis and will be left-hand (and not right hand) rule with respect to the induced current.
This theoretical experiment makes use of the boundary conditions for field vectors and a perfect conductor, in particular the condition that relates the tangential component of vector H to the surface current at a boundary.
And you can theoretically measure vector-D as well, using the same small perfectly conducting cylinder. Just change step 2) from maximizing induced surface current on the curved surface to maximizing the surface charge induced on the cylinder's two flat surfaces. Induced surface charged density will be the magnitude of D, with the direction being along the cylinder's axis, going from - to +.
If experiments can be devised - even theoretical ones - to directly measure a field without needing to know any other fields, than the field is just as real/fundamental as the others.
Charles Link said:In the first case, I believe you are really measuring ## B ##,
Charles Link said:In the first case, I also question the technique. For a superconductor, the B field inside is extinguished by surface currents, but not for ordinary conductors.
Charles Link said:and in the second case, you are measuring E
mairzydoats said:This theoretical experiment makes use of the boundary conditions for field vectors and a perfect conductor, in particular the condition that relates the tangential component of vector H to the surface current at a boundary.
vanhees71 said:It's a bit tricky. The auxiliary field components ##\vec{D}## and ##\vec{H}## are always model dependent, i.e., it's more or less your choice, what you call "free charges/currents" and what "polarization/magnetization" of the matter. You can shuffle these various contributions of these forces to the physical electromagnetic field with components ##\vec{E}## and ##\vec{B}## more or less arbitrarily. The em. field is uniquely measurable by its influence on the motion of charged particles/matter.
The boundary condition which states that the discontinuity in the tangential component of the H at the border between two mediums is equal to the free surface current density J follows directly from the relationDrDu said:I think that is the point. The boundary conditions are in general quite complex and model dependent, for example if the relation between M and B is non-local.
Sure, here you made the usual assumptions, treating the conductor as a continuum and map everything to boundary conditions. From a microscopic point of view things are much different, and the OP asked even on the level of in-medium quantum electrodynamics, a topic, I'd recommend only after studying the classical theory and also the vacuum-QED case in detail first.mairzydoats said:But inside of a perfect conductor, there is no "polarization/magnetization" by definition. All charges/currents are "free charges/currents" and are at the surface.
Where does choice come in?
...And what would you "shuffle" to change the conclusion you would be unavoidably directed to about the induced surface charges and/or currents on the border of the perfect conductor, e.i., that the field strengths they are tied to are from outside the border alone?
vanhees71 said:Sure, here you made the usual assumptions, treating the conductor as a continuum and map everything to boundary conditions.
mairzydoats said:Yes, the usual assumptions in the macroscopic model.
mairzydoats said:And besides, trying to discuss vector H on the quantum level doesn't even make any sense. H is defined as:
H = B/μ° - M
And M is by definition a macroscopic value. It is the magnetic dipole moment per unit volume. Undefined on the quantum level, and hence so too is H.
mairzydoats said:The boundary condition which states that the discontinuity in the tangential component of the H at the border between two mediums is equal to the free surface current density J follows directly from the relation
∇ x H = Jfree + ∂D/∂t
mairzydoats said:And M is by definition a macroscopic value. It is the magnetic dipole moment per unit volume. Undefined on the quantum level, and hence so too is H.
As I just laid out, you can shuffle terms between M and P (at least in the time dependent case) and also the choice of J is a matter of convention. E.g. instead of J_Free you may use J_external.mairzydoats said:...which is model independent
It is not! What you define as ##\rho_{\text{free}}## and ##\vec{j}_{\text{free}}## and what as polarizations ##\vec{P}## and ##\vec{M}## is more or less arbitrary. You can easily shuffle contributions from the one to the other without changing the physical relevant fields ##\vec{E}## and ##\vec{B}##. Often, of course, there's a "natural choice", but it's still model dependent.mairzydoats said:...which is model independent
How so?DrDu said:As I just laid out, you can shuffle terms between M and P (at least in the time dependent case) and also the choice of J is a matter of convention. E.g. instead of J_Free you may use J_external.
In QM, a unique distinction between bound and free charges is problematic.
vanhees71 said:Sure, here you made the usual assumptions, treating the conductor as a continuum and map everything to boundary conditions. From a microscopic point of view things are much different, and the OP asked even on the level of in-medium quantum electrodynamics, a topic, I'd recommend only after studying the classical theory and also the vacuum-QED case in detail first.
For an excellent treatment of classical in-medium electrodynamics, see Landau&Lifshitz vol. VIII.
A classical example is the Lindhard dielectric function of the free electron gas, where a gas of electrons is described by a dielectric function, although one would be tempted to treat it as free charges.mairzydoats said:How so?
OkayDrDu said:A classical example is the Lindhard dielectric function of the free electron gas, where a gas of electrons is described by a dielectric function, although one would be tempted to treat it as free charges.
https://en.wikipedia.org/wiki/Lindhard_theory