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

[vanhees71]

[...]

Like I said, I've used cgs in magnetics design and consistently arrived at the correct answer. But SI for me makes more sense to a widget designer who works in industry, not a theoretical research lab. To me, I like the distinction between H & B. They are both important and neither is "derived" from the other. In Halliday-Resnick elementary physics, the authors stated that the decision to treat B as the basis, and regard H as derived from B, is purely arbitrary. I concur.

[...]

Neither is the more "fundamental" at all. Our universe consists of free space as well as molecular structures and quantized atomic energy levels. Polarization, electric or magnetic, is just as "real" as a vacuum. Besides, more than one poster has attempted to propagate the crackpot heresy that B is the counterpart of E, while H is that to D. This is nonsense.

Claude

From the point of view of classical electrodynamics (which itself is an approximation of QED) the fundamental field is the one and only electromagnetic field, whose components with respect to an arbitrary inertial reference frame (for simplicity let's consider special-relativistic spacetime, neglecting gravity) we use to call \vec{E} and \vec{B}. These are no "counterparts" but just components of the electromagnetic field, which in the manifestly covariant description are given as the antisymmetric 2nd-rank Faraday tensor field in Minkowski space, F_{\mu \nu}. This is not crackpotery but well-established since Minkowski great work on these issues in 1908.

When it comes to the electromagnetism at presence of matter, you can describe this with pretty good accuracy with linear-response theory, and the usual (relativistic!) constitutive relations. It is very clear, and has been already clarified by Minkowski in 1908, that \vec{E} and \vec{H} are the components of the corresponding antisymmetric tensor D_{\mu \nu}. Of course you have to distinguish between these two tensor fields and both are important in macroscopic electromagnetics, but it's not very clear to me, why these quantities should have different units. In the SI even the components belonging to the same tensor field have different units. This is pretty confusing rather than illuminating. It's of course not wrong, because you simply include the appropriate \mu_0 and \epsilon_0-conversion factors of the SI, but it's unnecessarily complicated for the theoretical treatment, which best is done in the manifestly covariant way, which you can easily "translate" into the 1+3-dimensional formalism by splitting into temporal and spatial components whenever necessary for applications.

You find a very good description of all this already in pretty classical textbooks like vol. III of the Sommerfeld Lectures (which, by the way also use the SI!) or Abraham/Becker/Sauter. A very nice more uptodate treatment is also found in vol. VIII of Landau and Lifshitz. A more formal transport-theoretical treatment is given in the book by de Groot and Suttorp, Foundations of Electrodynamics.

Of course, sometimes you have memory effects and spatial correlations in some materials like ferromagnets (hysteresis) etc. That's no contradiction to what I've said above.

 

[cabraham]

Claude, in macroscopic theory I agree with you, all EBDH are equally important. However nowadays we know about atoms and molecules and have the possibility to understand the macroscopic theory on the basis of microscopic theory. In microscopic theory, 4 independent field quantities make little sense. Usually we think that there is only one EM field with two components: electric and magnetic, as vanhees71 said. These may be denoted as $\mathbf e, \mathbf b$ (microscopic) and their meaning (definition) is that they give us the force on a point-like test particle

$$
\mathbf F = q\mathbf e + q\frac{\mathbf v}{c} \times \mathbf b.
$$

So in this picture, the fields e,b are more basic, since they directly give force. The macroscopic fields E,B can then be sought as a kind of average of these microscopic fields. The fields D,H are then necessarily only auxiliary quantities that play little role in the logic of microscopic theory; there is little reason to consider microscopic fields $\mathbf d,\mathbf h$.

I already acknowledged that relativistic transformations per Lorentz, Lorentz force, etc., are expressed in canonical form via E & B, since B is independent of medium for this narrow condition. Computing force acting on a charge by a mag field w/o considering what generates said mag field is done best by using B as the basis as it is medium-independent.

But take another example where we generate a mag field by setting up a current in a wire loop. The loop is circular w/ radius R, the current is I, what is B/H at the center?

Per Biot-Savart: B = μI/2R, or H = I/2R.

If we want the mag field generated by a current loop, the canonical form would be the equation with H, not B, as it is medium independent. Do you see what I mean about this question being arbitrary? A particle physicist bangs particles together in a super-conducting super-collider, cyclotron, etc. The force on a free charge in the presence of a mag field is best expressed using B.

But a widget inventing nerd like myself, deals with motors, generators, transformers, relays, solenoids, etc. If I'm generating a mag field with a current, H is medium-independent and canonical.

Bottom line, if we wish to attract a charge to a wire loop, we cannot eliminate μ either way. The H field generated by a current loop is independent of μ. But the attractive force depends on μH since it depends on B. So a charge in the center of the loop will incur a force of e(vXB) = e(vXμH) = eμ(vXI/2R).

The permeability constant μ shows up no matter if you use H or B.

Likewise 2 parallel wires each carrying current incur attractive/repulsive force based on the product of the 2 currents, distance between them, and μ the medium.

If Lorentz force acting on a charge is more important to you than mag field surrounding a current loop, then it makes sense to use B as the basis quantity, then derive H as B/μ. Either method produces the correct answer. If the physics community prefers to regard B as the basis, no problems should be created by doing so.

I just want to emphasize that such a convention is arbitrary, one could just as well treat H as the basis. Depending on boundary conditions, like the ones I mentioned with ferrous cores having series and parallel boundaries, the quantity that is independent of media could be either, B or H.

It's no big deal, you can start at B basis, then derive H as B/μ. But you can do the opposite. If we carefully keep track of our variables, the answer should be the same either way.

One exception is when the medium is ferrous, operating at or near saturation. Then the relation B = μH is not linear any more. In such a case, the B-H curve must be examined, and graphical analysis can be used. B & H in this case, cannot be interchanged because μ is not constant.

I will elaborate. Comments/questions welcome.

Claude

 

[Jano L.]

If I'm generating a mag field with a current, H is medium-independent and canonical.

I do not know what do you mean by the word "canonical". Field H depends on the conduction currents which are well controlled and can be easily calculated as their function. Field B depends also on the properties of the medium and may be harder to calculate, especially in ferromagnetic material.

This does not imply any of them as more basic.

But the attractive force depends on μH since it depends on B. So a charge in the center of the loop will incur a force of e(vXB) = e(vXμH) = eμ(vXI/2R).

The force on charge is established to be e(vXB) in vacuum, where μ = μ0. In medium, I do not think one can simply take the same formula. The microscopic field in the medium varies on atomic length scale and most probably neither H nor B is sufficient to find the force.

I would say, in macroscopic theory, both fields are equally important and neither seems more basic. But in microscopic theory, which is more detailed than the macroscopic theory and explains it in more elementary notions, there is place only for one magnetic field. It used to be denoted by $\mathbf h$ but I think it should be written as $\mathbf b$.

 

[cabraham]

I do not know what do you mean by the word "canonical". Field H depends on the conduction currents which are well controlled and can be easily calculated as their function. Field B depends also on the properties of the medium and may be harder to calculate, especially in ferromagnetic material.

This does not imply any of them as more basic.


My point exactly. I've been saying the same since day 1. I concede to your point with no argument whatsoever. The current loop was brought up by me not to establish H as a basis, but rather to illustrate the futility of trying to establish either as more basic. We agree perfectly.

The force on charge is established to be e(vXB) in vacuum, where μ = μ0. In medium, I do not think one can simply take the same formula. The microscopic field in the medium varies on atomic length scale and most probably neither H nor B is sufficient to find the force.

I would say, in macroscopic theory, both fields are equally important and neither seems more basic. But in microscopic theory, which is more detailed than the macroscopic theory and explains it in more elementary notions, there is place only for one magnetic field. It used to be denoted by $\mathbf h$ but I think it should be written as $\mathbf b$.

Well then, we now have to form a consensus on just what is the most "elementary" .

 

[Jano L.]

The main reason I consider $\mathbf B$ as basic is that microscopic theories give it as a result of (some of many kinds of) averaging of the microscopic field $\mathbf b$. While $\mathbf H$ does not seem to be result of such direct averaging - there is no microscopic$\mathbf h$. Instead, $\mathbf H$ is defined only in macroscopic theory as the quantity that gives only part of total current density - the conduction current density due to mobile charges - via

$$
\nabla \times \mathbf H = \mathbf j_{\text{cond.}}
$$

 

[cabraham]

The main reason I consider $\mathbf B$ as basic is that microscopic theories give it as a result of (some of many kinds of) averaging of the microscopic field $\mathbf b$. While $\mathbf H$ does not seem to be result of such direct averaging - there is no microscopic$\mathbf h$. Instead, $\mathbf H$ is defined only in macroscopic theory as the quantity that gives only part of total current density - the conduction current density due to mobile charges - via

$$
\nabla \times \mathbf H = \mathbf j_{\text{cond.}}
$$

I understand, but I still need an answer for the following. Take a parallel plate cap w/ dielectric, ε>1, and excite it with an ac generator plus resistor. There is a sinusoidal I & V in the cap. The conduction current in the cap leads and plates and partial mag field intensity will obey the relation you posted above:

$$
\nabla \times \mathbf H = \mathbf j_{\text{cond.}}
$$

What about the mag field inside the dielectric per Ampere's Law, aka "displacement current"? It is:

curl H = dD/dt.

Is there a B inside the dielectric but no H present? I find that ir-reconcilable since the dielectric contains no ferrous material, hence B = μH. So even inside the dielectric, μ = μ₀, so that B & H co-exist. Why one would be the basis vs. the other seems pretty arbitrary unless I am missing something else not covered yet. The atomic structures are displacing in the dielectric. Electrons are moving towards one direction, with nuclear protons moving towards the opposite direction, both in a sinusoidal fashion. By definition these displacing charges constitute an ac current, and thus are surrounded by an ac mag field intensity, as well as density.

Inside the dielectric must exist both B & H in unison, inter-related per B = μ₀H, as long as no ferrous material is involved.

Again, I am having trouble understanding the line of demarcation between "mAcroscopic vs. mIcroscopic". Thanks.

Any comments would be appreciated.

Claude

 

[Jano L.]

Of course, both fields are present and non-zero in the dielectric. The distinction between macro and micro is that macroscopic field is that used in basic theory when we ignore the atomic structure of the material. The microscopic field varies randomly on the scale of molecules since it reflects their presence.

 

[mikeph]

I'm not sure if this is answered yet, but why are H and D different units?

Can't we define: epsilon*div(D) = free charge density?
Can't we define: curl(H)/mu - epsilon*dD/dt = free current density?

Can't we do the same for P and M with the bound charges and currents?

Then E = P + D and B = M + H

All have the same units.

What does this neglect?

 

[cabraham]

Of course, both fields are present and non-zero in the dielectric. The distinction between macro and micro is that macroscopic field is that used in basic theory when we ignore the atomic structure of the material. The microscopic field varies randomly on the scale of molecules since it reflects their presence.

Ok we are in agreement, but no closer to solving the question "mAcro vs. mIcro". You gave a definition, but I would like an example akin to my cap given above, showing that in a vacuum, or non-ferrous material, that just 1 quantity is needed. Thanks in advance.

Claude

P.S. In my cap example, the dielectric being non-ferrous would mean that 2 quantities are un-needed, only 1 being necessary. But it looks like we get 2 of them. How do we decide which is basis, which is derived?

 

[vanhees71]

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.

 

[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.