In magnetism, what is the difference between the B and H fields?

[cabraham]

Formally it's pretty simple: The fundamental microscopic description is quantum electrodynamics for many-body systems, macroscopic is a description derived from that via the one or other type of coarse graining, i.e., the derivation of tranport equations for the matter coupled to mean fields (Vlasov-Boltzmann-Uehling-Uhlenbeck -> Vlasov-Boltzmann) and some simplifications. The usual "macroscopic classical electrodynamics" we learn in introductory E+M is then linear response theory, where "matter" is reduced to the electromagnetic four-current and consititutive relations (response functions like the dielectric function, conductivity, ...).

To really establish these connections is, of course, rather complicated.

My point exactly. To establish B or H as a basis, with the other being derived, certainly is complicated. You'll get no argument from me at all. But why sweat it?

A person who goes through an entire career thinking of B as "B" (the basis), but regarding H as "B/μ" (derived) should not encounter any problems as long as they do their math correctly. Every equation from Maxwell to Biot-Savart to Lorentz, etc., can be expressed in terms of either, B or H. It's arbitrary so why waste energy arguing?

But my caveat above must be remembered. For non-ferrous or soft ferrous material operating in non-saturated mode, it is safe to say that B = μH, treating μ as a constant, so that B and H have a linear relation. But with hard ferrous material operating into saturated mode, the linear equation is not accurate as μ varies with flux level. We then must use graphical methods to compute energy, force, induced emf/mmf, etc.

We seem to have a consensus on that point. Other thoughts are welcome. BR.

Claude

 

[mikeph]

I just think of mu as dB/dH in nonlinear materials. Surely that works in analytic evaluations of energy etc.?

 

[cabraham]

I just think of mu as dB/dH in nonlinear materials. Surely that works in analytic evaluations of energy etc.?

I would say that dB/dH works for computing inductance when an inductor has a dc bias, and we use the small signal permeability for inductance computation. Then dB/dH is the slope of the B-H curve at the operating point (flux).

But for energy, I don't think that dB/dH does a lot of good. If a magnetic core is driven into saturation, the area inside the B-H loop represents ir-recoverable energy (loss), while the area between the curve and vertical (B) axis represents recoverable stored energy, which gets returned on alternating ac cycles.

Also, for core losses, like hysteresis and eddy currents, empirical measurements are plotted on a graph, which are what nerds like me use when designing xfmrs, or motors/generators, etc.

The dB/dH quantity has useful value, but that alone does not fully characterize a magnetic core specimen when subjected to large flux swings into the saturated zone of operation. A combination of dB/dH, area outside curve, area inside curve, and empirical graphs, provide us with detailed insight into magnetic material behavior and optimization.

Claude

 

[DrDu]

Maybe the article by L. L. Hirst,The microscopic magnetization: concept and application, Reviews of Modern Physics, Vol. 69, No. 2, April 1997
may be useful to all interested in this question.

 

[tiny-tim]

… L. L. Hirst,The microscopic magnetization: concept and application, Reviews of Modern Physics, Vol. 69, No. 2, April 1997 …

ftp://ftp.phy.pku.edu.cn/pub/Books/%CE%EF%C0%ED/%CE%EF%C0%ED%D1%A7%CA%B7/Review_of_Modern_Physics/microscopic%20magnetization.pdf

 

[Philip Wood]

DrDu and tiny-tim: Many thanks. That should keep me quiet for a while!

 

[mikeph]

DrDu and tiny-tim: Many thanks. That should keep me quiet for a while!

Seconded


I'm always looking for clear, reliable, and descriptive works on these topics.

 

[rebel rider]

hi friends,
B is a vector quantity which is magnetic flux density
and
H is magnetic field intensity and the relation between them is B=mu*H

comparing with electric field...
B is analogous to D(electric flux density)...vector quantity
H is analogous to E(electric field intensity)...vector quantity
mu is analogous to epsilon(permitivity)
and the relation B=mu*H is analogous to D=epsilon*E

 

[kaustubhan]

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 ?

 

[kaustubhan]

correction, I mean B = {mu} {H+M}

 

[DrDu]

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 ?

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.

 

[Charles Link]

The H was originally introduced in the magnetic pole method and has two different sources for it: 1)magnetic poles with the H obeying the inverse square law and 2)the currents in conductors such as a solenoid where the H is computed from a Biot-Savart type formula. Later it was found from magnetic surface current models of the magnetism that the H is not a second type of magnetic field, and that inside the material, the $\mu_o H$ from the poles is simply a subtractive corrective term to the $M$ from the magnetic surface currents for geometries other than a cylinder of infinite length, thereby $B=\mu_o H+M$. A student recently posted a homework problem to which I responded that illustrates these concepts: https://www.physicsforums.com/threads/magnetic-field-of-a-ferromagnetic-cylinder.863066/ In the pole model, using the equation $B=\mu_o H+M$ , the magnetization $M$ looks like it provides a local contribution to the magnetic field $B$, when in fact, surface current calculations show this $B=M$ actually originates from the surface currents and Biot-Savart. Meanwhile, the $\mu_o$ factor is just a constant. In c.g.s. units, the equation reads $B=H+4 \pi M$. The pole model and the surface current model yield identical results for the magnetic field $B$, but the underlying physics is explained only by the magnetic surface currents. The pole model with its $H$ field, which is analogous to the electric field $E$ in the corresponding electrostatic problem, can be somewhat misleading when attempting to draw conclusions about the underlying physics.

 

[jtbell]

Why exactly was it decided that "B" and "H" should have different units ?

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.

https://en.wikipedia.org/wiki/Gaussian_units

 

[vanhees71]

In sensible systems of unit all components of the electromagnetic field $(\vec{E},\vec{B})$ and the auxilliary fields $(\vec{D},\vec{H})$ of macroscopic in-medium electrodynamics have the same unit. In the SI there are conversion factors introduced for convenience to have simpler numerical values for electromagnetic phenomena in the usual range of applications in engineering. For physics the SI units are confusing, particularly if you want to describe electromagnetism as a relativistic field theory, as you should :-).

 

[Charles Link]

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/

 

[mairzydoats]

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/

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]

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.

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]

In the first case, I believe you are really measuring $B$,

Nope.

https://en.wikipedia.org/wiki/Inter...terface_conditions_for_magnetic_field_vectors

At the interface between two mediums, the free surface current J is equal to the difference between the tangential components of H, on either side of the boundary. Not B.

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.

And the technique will work precisely because of the fact that inside of a perfect conductor, all electromagnetic field vectors are null. Therefore the value of J_surface is attributable solely to the tangential component of H on the non-perfectly-conducting side of the border.

and in the second case, you are measuring E

No. Similar reasoning.

 

[vanhees71]

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.

 

[DrDu]

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.

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.

 

[mairzydoats]

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.

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?

 

[mairzydoats]

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.

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 = J_free + ∂D/∂t

 

[vanhees71]

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?

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.

 

[mairzydoats]

Sure, here you made the usual assumptions, treating the conductor as a continuum and map everything to boundary conditions.

Yes, the usual assumptions in the macroscopic model.

 

[mairzydoats]

Yes, the usual assumptions in the macroscopic model.

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]

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.

...and so they are the usual assumptions for a good reason-they are the correct assumptions for the subject matter under discussion.

 

[mairzydoats]

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 = J_free + ∂D/∂t

...which is model independent

 

[DrDu]

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.

That's not correct. M and P can be defined microscopically. Only in the simplest cases, it is the dipole moment density.
However, the definition is model dependent, namely, it depends on whether one splits the electric charges into internal and external ones or into bound and free + external. Any of these groups fulfills a continuity equation
$\nabla \cdot j =\partial \rho /\partial t$.
Introducing the four vector $j^\mu$ with $j^0=\rho$ and $j^\mu=j_i$ for $\mu \in \{1,2,3\}$ and $i \in \{x,y,z\}$, and $\partial_\mu$ as $\partial_0=\partial_t$, $\partial_\mu =-\partial_i$,
we can write this as
$\partial_\mu j^\mu=0$.
This equation will be fulfilled iff $j^\mu = \partial_\nu \Pi^{\mu\nu}$ where $\Pi$ is an antisymmetric tensor, i.e. $\Pi^{\mu\nu}=-\Pi^{\nu\mu}$.
We call it the polarisation tensor.
It may be parametrized as
$\Pi = \begin{pmatrix} 0 &P_x &P_y &P_z\\ -P_x& 0 &M_z& -M_y\\ -P_y &-M_z &0 & M_x\\ -P_z & M_y & -M_x& 0 \end{pmatrix} .$
This tensor is not unique, as the solution of $j^\mu = \partial_\nu \Pi^{\mu\nu}$ is defined only up to a solution of the homogeneous problem $\partial_\nu \Pi^{\mu\nu}=0$. I suppose this is what Hendrik meant with the possibility to shuffle terms.

 

[DrDu]

...which is model independent

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]

...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]

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.

How so?

 

[mairzydoats]

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.

When the cylinder is embedded, a secondary layer of bound densities is formed around it which oppose those of the P or M whose influence would otherwise be prevalent. That is why the method will measure H or D rather than E or B

Too see what I mean you can try a thought experiment using my cylinder method between the plates of a capacitor with a dielectic medium. Density on the cylinder should always be the density as the cap's plates even though E changes when permitivity changes

 

[DrDu]

How so?

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

 

[mairzydoats]

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

Okay