# Resource for learning about D and H

Greetings,

I learned E&M from Griffiths, and for the most part I understood everything - I am very comfortable using Maxwell's equations, at least in the E and B form.

However, the same is not true of D and H, which I really don't understand at all. The reason that I am having trouble learning these concepts from Griffiths may be that the concepts that lead up to them are introduced in fairly quick succession, so that I have to increase the abstraction in my thinking too quickly for me to grasp each level of abstraction.

For instance, in leading to the electric displacement D, Griffiths introduces the dipole moment p on page 149, which I am fine with. However, the polarization P is introduced on page 166, the bound surface and volume charge densities σb and ρb on page 167 and 168, and the electric displacement D on page 175, and I am having difficulty grasping these concepts.

Similarly, m is encountered on page 257, M on page 262, Jb and Kb on page 264, and H on page 269.

So, although for the most part I find all of Griffiths' textbooks to be superb resources, his E&M textbook is not working for me on these particular subjects. Does anyone know of some good resources (preferably online, eg., online lecture notes. I'm not looking to buy a whole new textbook) that explain the above concepts, perhaps more slowly or with a different approach than does Griffiths?

Thanks very much for any help that you can give.

-HJ Farnsworth

Jano L.
Gold Member
Feynman explains well in his Lecture (2nd volume). For a more detailed discussion, I know of this nice paper:

Cloete, J.H, Is B or H the fundamental magnetic field?, AFRICON, 1996., IEEE AFRICON 4th (Volume:1 )

http://dx.doi.org/10.1109/AFRCON.1996.563137

Feynman explains well in his Lecture (2nd volume). For a more detailed discussion, I know of this nice paper:

Cloete, J.H, Is B or H the fundamental magnetic field?, AFRICON, 1996., IEEE AFRICON 4th (Volume:1 )

http://dx.doi.org/10.1109/AFRCON.1996.563137

I haven't obtained the paper yet, will do so. Would you be kind enough to just give a brief review and state his conclusions? Thanks.

Claude

My simplest summary is:

E and B are simple in a vacuum but horrendously complicated inside materials that consist of atoms. Atoms are tiny charges and current loops with their own personal E and B fields, so the true E and B fields are extremely granular and messy. To deal with this, we forget that the materials are have atomic structure and say they are a continuum with some bulk material properties that adequately describe their net response to external E and B fields.

P and M are these properties.

However, putting P and M into Maxwell's equations make them ugly, so people invented two auxiliary fields D and H which are defined in such a way that, upon substitution, the resulting form of Maxwell's equations looks similar to their microscopic counterpart. div(D) = rho and curl(H) = J + dD/dt look very much like div(E) = rho/e and curl(B/mu) = J + d(E/e)/dt.

In order for the two forms of Maxwells equations to look similar, D and H must be defined in annoying weird ways:

D = e0*E + P
H = B/mu0 - M.

And rather than defining material relationships in an intuitive cause-and-effect way like P = f(E) and M = f(B), we have H and D doing that job too, e.g. D = e0*er*E.

Hey ho.

https://www.physicsforums.com/showpost.php?p=4487256&postcount=88

Please review the above post from recently. I bel;ieve I covered this topic fully. As to which is more basic, not only do I not know, but I ask why one of them should be basic. They could both be related to a singular quantity yet doscovered that is "fundamental". Who knows?

Claude

marcusl
Gold Member
E and B are considered to be the "fundamental" field quantities as mikeph explained.

E and B are considered to be the "fundamental" field quantities as mikeph explained.

Woud you mind explaining why these 2 are "fundamental" and not the other 2? So far nobody has put forth any concrete evidenct that suggests what you've stated nor counters the same. Please read my treatise per the above link. Would you please respond as to why you disagree with me.

My conclusion seems ir-refutable, but I may have overlooked something. My conclusion is that with what we presently know, there is no answer as to which is basic, which is derived. The future may change that via new doscoveries, but for now, I state that we cannot specifically say that E/D/B/H can be resolved into basis pair and derived pair. THanks.

Claude

marcusl
Gold Member
It is as mikeph says: E and B are the fundamental quantities, known to Maxwell before he synthesized E&M into his four equations. In a medium, on the other hand, the fields are modified by contributions from the atoms. Rather than having to carry the effects of bound charges around in the Maxwell equation $$\mathbf \nabla\cdot\mathbf E = 4\pi\left(\rho-\mathbf \nabla\cdot\mathbf P \right),$$ for instance, it is convenient to define a new effective field D such that the equation remains simple $$\mathbf \nabla\cdot\mathbf D = 4\pi\rho.$$ This is just a convenience however. If one were to look between the atoms in the medium, one would see E, not D. D includes a macroscopic average over the complicated microscopic fields and sources. Such averaging is common in condensed matter physics. Fields are often "coarse-grained" or averaged to capture, at a high level, the effects of a microscopic or low level scenario without having to carry around its complexity.

The way in which the averaging is performed is important, and Jackson devotes much of Ch. 6 of his Classical Electrodynamics (2nd ed.) to showing how to do it properly.

The argument for B vs. H is analogous. As Jackson says,

"We emphasize that the fundamental fields are E and B... The derived fields D and H are introduced as a matter of convenience in order to take into account in an average way the contributions to ρ and J of the atomic charges and currents."

Nobel laureate Mel Schwartz would rather get rid of H altogether. In his lovely book Principles of Electrodynamics he says

"At this point we must interject a small bit of philosophy. It is customary to call B the magnetic induction and H the magnetic field strength. We reject this custom inasmuch as B is the truly fundamental field and H is a subsidiary artifact. We shall call B the magnetic field and leave the reader to deal with H as he pleases."

Claiming that your personal argument is "irrefutable" is irrelevant. You won't sway the rest of the physics world, who consider E and B to be fundamental.

Last edited:
marcusl
Gold Member
Woud you mind explaining why these 2 are "fundamental" and not the other 2? So far nobody has put forth any concrete evidenct that suggests what you've stated nor counters the same. Please read my treatise per the above link.

Detailed 'concise' explanations? Where. Every person claiming E/B as basis vectors simply asserted the same as fact w/o support. They referenced high profile svientist quotes who merely stated the same as fact w/o any basis.
I gave concrete examples showing that E sometimes correlates to B, sometimes to H, and sometimes to neither. As far as scientists not accepting my position, unless they refute my examples solidly, their wordsare dust in the wind.
This question clearly takes all EE/physics people to the limits of their knowledge. Nobody knows the answer. B/D/E/H are all needed. We just don't know enough to say for sure which is more basic, which is derived. Those who think otherwise are merely dispkaying their prejudice. Physicists have had a love affair w/ B and regard H as a step child. This prejudice amazes me. W/o both, I couldn't do my job.
Claude

DrDu
I wouldn't even say that E and B are more fundamental than P and M as P and M are fullly equivalent to specifying the charge and current distribution in the medium. This becomes most clear in a relativistic formulation. where the 4-divergence of the current 4-vector vanishes (charge conservation). This restriction can be taken care off by writing the current as a divergence of a tensor, the (magnetisation-)polarisation tensor. In principle, this is not only a possibility to describe macroscopic media but can also be used to describe e.g. the electrodynamics of point charges.
I also want to stress, that in contrast to what can still be found in some books, polarisation is not linked to spatial averaging.

Last edited:
Physicists have had a love affair w/ B and regard H as a step child. This prejudice amazes me.

But H owes its existence to an approximation, B does not.

To me it's like saying that a fluid velocity is no more fundamental a quantity than "turbulent kinetic energy" in fluid dynamics problem. One will always be the real thing, the other just an approximation.

So how does H owe its exisance to an approximation? Take a loop with current. We have H and B. Wrap the loop around an iron core. H remains, B changes. They both are important.

Jano L.
Gold Member
Cabraham rightfully points out that in practice, both fields ##\mathbf{B},\mathbf{H}## are very important and none seems to be more basic. I agree with him as far as laboratory practice, or macroscopic EM theory goes.

However, the majority is also right. The reason so many people say ##\mathbf B## is more fundamental is they are judging this from the perspective of microscopic theory, i.e. a theory that takes into account existence of atoms and molecules, which in their eyes is more basic, or more fundamental.

In this more refined view, there is only one EM field, however we denote it. Operationally, the field is described by two vectors, electric and magnetic, that are defined by the Lorentz formula for force acting on a test particle:

$$\mathbf F = q (\textbf{microscopic electric vector}) + q\mathbf v\times (\textbf{microscopic magnetic vector}).$$

The matter of notation of these microscopic quantities was never quite settled, especially whether use uppercase or lowercase letters, but disregarding that the most modern convention is that the electric vector is denoted by ##\mathbf e## and the magnetic vector is denoted by ##\mathbf b##.

These microscopic vectors are defined by the equation

$$\mathbf F = q\mathbf e + q\mathbf v \times \mathbf b$$

and are thus unambiguous. There is no useful quantity that would be denoted by ##\mathbf h## in this approach.

Why we define ##\mathbf b## and not ##\mathbf h## by the above equation?

Whether we denote the magnetic vector ##\mathbf b## or by ##\mathbf h##, its divergence has to vanish, since there are no magnetic poles:

$$\nabla \cdot (\textbf{microscopic magnetic vector}) = 0.$$

Now since from macroscopic theory we know ##\nabla \cdot \mathbf B = 0##, it is most logical to denote

$$\textbf{microscopic magnetic vector}~~ by~~~ \mathbf b$$

since then we have

$$\nabla \cdot \mathbf b = 0 ~~~(*).$$

The Gauss law for magnetic field ##\mathbf B##
$$\nabla \cdot \mathbf B = 0$$

can be derived by averaging both sides of the equation (*) over space, plane or over statistical ensemble (depends). Then, the field ##\mathbf B## is understood as an average of the microscopic field ##\mathbf b##:

$$\mathbf B = \langle \mathbf b\rangle.$$

The field ##\mathbf H## is defined only on the macroscopic level by

$$\mathbf H = \frac{1}{\mu_0}\mathbf B - \mathbf M,$$
since ##\mathbf M## has only macroscopic meaning; on the microscopic level, magnetic moment density is not very useful, much more useful is microscopic electric current density ##\mathbf j## that creates magnetic fields.

This nice recovery of macroscopic theory would not seem possible if we denoted the microscopic magnetic vector by ##\mathbf h## ; we would be lead to magnetic monopoles, which have not been discovered.

DrDu
The field ##\mathbf H## is defined only on the macroscopic level by

$$\mathbf H = \frac{1}{\mu_0}\mathbf B - \mathbf M,$$
since ##\mathbf M## has only macroscopic meaning; on the microscopic level, magnetic moment density is not very useful, much more useful is microscopic electric current density ##\mathbf j## that creates magnetic fields.

The article by Hirst I cited gives many applications where the microscopic magnetization is very usefull, like magnetic neutron scattering.

Cabraham rightfully points out that in practice, both fields ##\mathbf{B},\mathbf{H}## are very important and none seems to be more basic. I agree with him as far as laboratory practice, or macroscopic EM theory goes.

However, the majority is also right. The reason so many people say ##\mathbf B## is more fundamental is they are judging this from the perspective of microscopic theory, i.e. a theory that takes into account existence of atoms and molecules, which in their eyes is more basic, or more fundamental.

In this more refined view, there is only one EM field, however we denote it. Operationally, the field is described by two vectors, electric and magnetic, that are defined by the Lorentz formula for force acting on a test particle:

$$\mathbf F = q (\textbf{microscopic electric vector}) + q\mathbf v\times (\textbf{microscopic magnetic vector}).$$

The matter of notation of these microscopic quantities was never quite settled, especially whether use uppercase or lowercase letters, but disregarding that the most modern convention is that the electric vector is denoted by ##\mathbf e## and the magnetic vector is denoted by ##\mathbf b##.

These microscopic vectors are defined by the equation

$$\mathbf F = q\mathbf e + q\mathbf v \times \mathbf b$$

and are thus unambiguous. There is no useful quantity that would be denoted by ##\mathbf h## in this approach.

Why we define ##\mathbf b## and not ##\mathbf h## by the above equation?

Whether we denote the magnetic vector ##\mathbf b## or by ##\mathbf h##, its divergence has to vanish, since there are no magnetic poles:

$$\nabla \cdot (\textbf{microscopic magnetic vector}) = 0.$$

Now since from macroscopic theory we know ##\nabla \cdot \mathbf B = 0##, it is most logical to denote

$$\textbf{microscopic magnetic vector}~~ by~~~ \mathbf b$$

since then we have

$$\nabla \cdot \mathbf b = 0 ~~~(*).$$

The Gauss law for magnetic field ##\mathbf B##
$$\nabla \cdot \mathbf B = 0$$

can be derived by averaging both sides of the equation (*) over space, plane or over statistical ensemble (depends). Then, the field ##\mathbf B## is understood as an average of the microscopic field ##\mathbf b##:

$$\mathbf B = \langle \mathbf b\rangle.$$

The field ##\mathbf H## is defined only on the macroscopic level by

$$\mathbf H = \frac{1}{\mu_0}\mathbf B - \mathbf M,$$
since ##\mathbf M## has only macroscopic meaning; on the microscopic level, magnetic moment density is not very useful, much more useful is microscopic electric current density ##\mathbf j## that creates magnetic fields.

This nice recovery of macroscopic theory would not seem possible if we denoted the microscopic magnetic vector by ##\mathbf h## ; we would be lead to magnetic monopoles, which have not been discovered.

In the link above I already covered the Lorentz equation. You're saying that the rationale for defining B as bais is twofold. Lorentz force law states that force acting on a charge is related directly to B independent of media, whereas the same forc e expressed in terms of H would require inclusion of μ, the permeability of said media.

I would concur that this in itself makes a compelling case for arguing B as basic, H as derived. But that is akin to standing in one location, declaring that the earth I stand on is stationary, and the sun & mood revolve around the earth and myself. Including data obtained from other sources invalidates such a narrow view of science.

If the criteria for determining basis vector is the independence of force acting on charges wrt media, then I have already stated that B is the basis. But I also covered that just as fields exert forces on charges, that charges in motion (or still) also generate fields. A loop or coil carrying a current generates a mag field.

Is the field generated by a current loop independent of media when expressed with B, or with H? Of course the answer is H. If we consider the charged particle generating a field in a media, the H vector is independent of media, not so with B, hence H is the basis, B is derived as:

B = μ0H + M. The equation you gave above can be expressed in a form where either B or H is the isolated variable. Honestly I am amzed how trivial these arguments are.

Regarding polarization, it has been long known that the remnant quantities, i.e. the quantities retained after external energy source is removed, are D and B. If a capacitor with a non-air dielectric is energized, as well as an inductor with ferromagnetic core, the energy in these L & C components is given per the D-E and B-H curves.

When the external power source is removed, the cap and inductor retain energy known as remnance. When power is removed, it is E & H that vanish. What remains is D and B. This makes a strong case for correlation of E with H, and D with B.

I have stated again and again that which is the basis, and which mag vector, H or B, correlates with E, is contentious. Lorentz force law says E goes with B and D with H, atomic/molecular polarization says E goes with H, D with B. Then boundary relations in series connected media correlate D with B, whereas parallel media display correlation of E with H.

So how does the physics community "majority" settle the dispute. Easy. They ARBITRARILY choose the Lorentz force law as the SOLE criteria for determining the basis vectors. Since E & B exert forces on charges that are independent of media, then E & B are the basis vectors under this specific condition.

Rather than examine or consider other conditions which lead to different conclusions, they close the discussion and declare victory. Oh well, it isn't really that important. They are not creating harm by propagating their prejudiced opinions. Which one is more basic has little to no relevance anyway.

Claude believes that although the question is unsolvable, it is moot as well. Which "comes first" is pretty academic. Good day to all and enjoy the weekend.

Claude

Jano L.
Gold Member
The article by Hirst I cited gives many applications where the microscopic magnetization is very usefull, like magnetic neutron scattering.

Fair enough, it is useful. What I meant is that it is not fundamental. It is derived from microscopic current density and it is not unique - there are many different equivalent magnetizations. Current density is unique.

As Hirst says,

In the final analysis, the magnetization is only a
device for encoding information about the current den-
sity, and clarification of this encoding can have no im-
pact on quantities that could be discussed in terms of the
current density itself.

Jano L.
Gold Member
Lorentz force law states that force acting on a charge is related directly to B independent of media, whereas the same force expressed in terms of H would require inclusion of μ, the permeability of said media.

No, you misunderstand what I meant. The microscopic force is not to be "related directly to ##\mathbf B## independent of media". It is given by microscopic magnetic field ##\mathbf b## acting on the particle whether in medium or isolated. It is universal. In microscopic theory, medium is made of particles and does not have any special role.

The same force cannot be expressed with ##\mathbf H##, not even with inclusion of its permeability ##\mu##. ##\mu\mathbf H## gives only macroscopic field ##\mathbf B##, and only in nonmagnetic materials. It cannot give us microscopic field ##\mathbf b##.

Is the field generated by a current loop independent of media when expressed with B, or with H? Of course the answer is H.

In general, neither ##\mathbf B## nor ##\mathbf H## vector is independent of the presence of the medium. The ##\mathbf H## vector is function not only of the conduction currents in wires, but also of the magnetization currents in magnets and other materials. In static cases, the vector ##\mathbf H## satisfies not only

$$\nabla \times \mathbf H = \mathbf J_c$$
with conduction current density ##\mathbf J_c##, but also

$$\nabla \cdot \mathbf H = -\nabla\cdot \mathbf M,$$

with magnetization ##\mathbf M##. How strong ##\mathbf M## is depends on the kind and shape of the medium (think of the H field of permanent magnets).

No, you misunderstand what I meant. The microscopic force is not to be "related directly to ##\mathbf B## independent of media". It is given by microscopic magnetic field ##\mathbf b## acting on the particle whether in medium or isolated. It is universal. In microscopic theory, medium is made of particles and does not have any special role.

The same force cannot be expressed with ##\mathbf H##, not even with inclusion of its permeability ##\mu##. ##\mu\mathbf H## gives only macroscopic field ##\mathbf B##, and only in nonmagnetic materials. It cannot give us microscopic field ##\mathbf b##.

In general, neither ##\mathbf B## nor ##\mathbf H## vector is independent of the presence of the medium. The ##\mathbf H## vector is function not only of the conduction currents in wires, but also of the magnetization currents in magnets and other materials. In static cases, the vector ##\mathbf H## satisfies not only

$$\nabla \times \mathbf H = \mathbf J_c$$
with conduction current density ##\mathbf J_c##, but also

$$\nabla \cdot \mathbf H = -\nabla\cdot \mathbf M,$$

with magnetization ##\mathbf M##. How strong ##\mathbf M## is depends on the kind and shape of the medium (think of the H field of permanent magnets).

What I inferred with the current loop was that the amp-turns per path length, i.e. Ampere's Law, defines the mag field intensity H surrounding the wire. If the media is non-ferrous, then μ = μ0, so that B and μH are indeed equivalent. Yes I realize that in ferrous media this simple linear relation cannot be invoked, I've stated such since my first post.

So let me summarize. I think we have universal consensus that if our universe consisted of free charges w/o any atomic structures and energy levels, that 4 field vectors would not be necessary, but rather only 2. There seems to be a strong tendency in the physics community to conclude that these 2 vectors are E & B. I say that it would work, but so would E & H, or D & B, etc. Every argument presented supporting B as more basic merely asserts that H does not come into play unless polarization occurs.

But if we examine a current loop w/o polarization, i.e. μ = μ0, it is academic and irreducible to say that I/2R = H = B/μ0 at the center. I've examined many texts from EE & physics background and none of them resolve the question as to which quantity is more basic sans polarization. Halliday-Resnik simply state that B is chosen, yet not for any scientific reason. They also state that it is pointless to argue which comes first.

I have a physics E&M text where the author illustrate field around a bar magnet including internal fields. He demonstrates that B is more basic and H is derived to account for polarization. However I redrew his diagram and I believe he erred. If viewers on this forum would like me to post it I will do so.

Moderators - I will give full credit to the author of the book and publisher to avoid copyright issues. I will then explain where I feel the author erred. Let me know. Thanks.

Claude

Last edited: