# Is B more analogous to D or to E (electromagnetism)?

## Main Question or Discussion Point

In the book I'm reading on electromagnetic fields, the authors seem to associate the magnetic flux density B with the electric flux density D, and the magnetic field H with the electric field E.

Now, my question is, wouldn't it make more sense to say B is the analog to E? Because E and B are more fundamental in the sense that they are the fields you can actually measure by placing a charge in an electric field or a current in a magnetic field and measure the resulting force independently of the medium. With the Lorentz force law: $\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right)$.

On the other hand, D and H occur from Gauss's law and Ampere's law respectively, and are indirect results of charge distributions and current distributions, and cannot be "measured" directly. The surface integral of D over a closed surface is the total enclosed charge in the surface, and the line integral of H around a closed loop is the total current crossing the loop. So D and H are both related to charge density and current density.

In fact, I think B should have been called the magnetic field, and H the magnetic flux density.

Are there any arguments for choosing to call H the magnetic field and B the magnetic flux density?

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vanhees71
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2019 Award
You shouldne't use this book! It's very clear that there is only one electromagnetic field with field-components $\vec{E}$ and $\vec{B}$. In relativistic notation this is very clear, and the relativistic treatment is most natural for electromagnetism. There it's the antisymmetric Faraday tensor $F_{\mu \nu}$, where the time-space components make up the electric and the space-space components the magnetic components in the given frame of reference.

$\vec{D}$ and $\vec{H}$ are auxiliary fields introduced in macroscopic electrodynamics. In relativistic notation, they also form an antisymmetric tensor $H_{\mu \nu}$ with the time-space components given by the components of $\vec{D}$ and the space-components by those of $\vec{H}$.

You already are on the right track. The Lorentz force leads to the operational definition of the electromagnetic field by it's action on a test charge.

$\vec{D}$ and $\vec{H}$ are not the fields created by the total charge and current densities of the many-body system but only by the ones you consider as "free charges and currents". Usually these are the ones you add to a given many-body system (which is usually electrically neutral). These charge create additional electromagnetic fields which distort the charge-current distribution of the matter, giving rise to electric and magnetic polarization effects. The total field is always given by $\vec{E}$ and $\vec{B}$, which consist of the superposition of the electric and magnetic field created by the external charges and the fields due to the polarization effects.

The confusion in your textbook is as old as electrodynamics. Before the advent of special relativity and its mathematical complete formulation by Minkowski in 1908 the electromagnetic phenomena were simply not understood completely, and that's why we have strange conventions until today. While we define the dieelectric constant as $\vec{D}=\epsilon \vec{E}$ the magnetic permeability is defined the other way around via $\vec{B}=\mu \vec{H}$.

Additional confusion occurs because of the use of the SI units, which are very good for practical purposes in electrical engineering but pretty cumbersome from a theoretical point of view. You have different units for $\vec{E}$ and $\vec{B}$, $\vec{D}$ and $\vec{H}$, because in the SI you have an extra basic unit for charge (or in fact electric current). This requires to introduce an additional conversion factor between this artificial extra unit. Usually that's $\epsilon_0$ a kind of "dielectric constant of the vacuum", which in fact is only a conversion factor for units and no basic physical quantity as the speed of light (which in some sense, however, is also a conversion factor necessary because we use different units for lengths and times, but that's another story). The upshot is, I plead for the use of Gaussian (or even better Heaviside-Lorentz) units for electromagnetics in theoretical physics.

Lord Crc
That is a classic question for which there is no answer. Is E more like H or B? You can make a case either way. If you look at Lorentz force, then you can make a case for E relating to B per the equation already posted above. But it goes deeper.
If a dielectric capacitor and a ferrous cored inductor are energized, then the power sources both removed, we have some energy retention with both. D & B are the remnant quantities, while E & H vanish.
Also, when field encounter boundaries between 2 differing media, the normal components of D & B are equal, or for D with surface polarization, only differ by a constant. Also, the tangential components for E & H are equal. So E appears to behave more like H in these conditions.
But remember that B & H can only exist in solenoid form, i.e. closed loops, no start or end. D & E can exist as loops, but can also exist as segments with start and end. So in this case neither H nor B is the counterpart of E, nor D for that matter.
In vacuum, or air, things are simple. We don't need all 4 field quantities, 2 would suffice. In CGS system, in air, D=E & B=H. Without polarization B & H converge into one and the same thing. Is E the analogy of B or H is a moot question as B & H are one and the same. In MKS units, mu0 & epsion0 have units. In CGS they are 1 w/o units.

Claude

robphy
Homework Helper
Gold Member
At a more abstract level (sometimes called "pre-metric electromagnetism" or "pre-metric electrodynamics"), one can see that these are 4 different geometrical objects.
Following Bill Burke and Fred Hehl (and others who advocate the use of differential forms in electromagnetism),
E is a 1-form and H is a "twisted" [odd] 1-form (think about something associated with a line-integral)
B is a 2-form and D is a "twisted" [odd] 2-form (think about something associated with a flux-integral)

The equations of electrodynamics relate some of these quantities.
By using additional structure, like a volume-form or a metric, you can begin to establish other relations [and thus analogies].

Eventually, the symmetries of 3-dimensional Euclidean space allow you to interpret these as [polar and axial] vector fields.
(Part of the pre-metric view is trying to understand what physics exists independent of the metric...
and thus tells us a something about what the metric does here [and possibly elsewhere].)

vanhees71 and cabraham
vanhees71
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2019 Award
This analysis is using the 3D vector formalism. I was talking about the relativistic structure of the theory, which is more physical than the non-covariant formalism. In the latter approach it's clear which components belong together, and it's clearly $(\vec{E},\vec{B})$ and $(\vec{D},\vec{H})$. Hehl's book, however, is really great.

This analysis is using the 3D vector formalism. I was talking about the relativistic structure of the theory, which is more physical than the non-covariant formalism. In the latter approach it's clear which components belong together, and it's clearly $(\vec{E},\vec{B})$ and $(\vec{D},\vec{H})$. Hehl's book, however, is really great.
Only if you exclude polarization behavior. In the case of non-unity values for permeability/permittivity, then E behaves like H, D like B, no getting around that. In SI units with or without polarization, the Lorentz force law describing influence of fields on particles relates E to B. Those who insist this is the general relation only observe Lorentz force while excluding other important observations.
It's moot anyway. In polarized media, all 4 are needed, neither is more basic. Without polarization, the B & H vectors are the same entity in CGS units, and related by a constant in SI units. Any relation involving B can be expressed in mu*H, and vice versa. Moot question for sure.

Claude

vanhees71
Gold Member
2019 Award
Polarization is an emergent phenomenon, described in an effective theory of microscopic electrodynamics (strictly speaking you need even QED, although phenomenologically and qualitatively you can go pretty far with classical electromagnetism) of many-body systems. Of course, the simple constitutive laws are also only valid as effective models since they are derived from linear-response theory, with the external fields considered as a weak perturbation. In any case, from a fundamental point of view, it's pretty clear, what's a "fundamental" and what's a "derived" (auxiliary) quantity.

Of course, it's totally irrelevant which units one uses. I only think CGS or rationaliced CGS units are more intuitive from a fundamental point of view, because it reflects better which quantities belong together dimensionwise than it's the case in the SI.

robphy
Homework Helper
Gold Member
Generally speaking, the differential forms approach is 4- (and possibly n-)dimensional and independent of the metric.
E and B are components of a 2-form (usually called F ["Faraday tensor" in MTW]).
H and D are components of a twisted 2-form (usually called G [vanhees71's $H_{\mu\nu}$]).
Although these k-forms can be considered as forms in 3-D,
they are first born as forms in 4-D (or generally n-D).
B is a 2-form in 4-D and that E is a 1-form in 4-D ($E \wedge t$ is a 2-form in 4-D).
The decomposition into components is observer (that is, $t$-) dependent, analogous to decomposing the 4-momentum into components.
$F=B+E\wedge t$ and $G=D-H\wedge t$.
In the presence of a metric and volume form in 4-dimensions in a vacuum, G is the Hodge-dual of F.
Generally, one needs a "constitutive equation" to relate G to F.

Part of my earlier posting is to suggest that the choice of additional structure
[say a field equation or a metric-volumeForm] is one way of establishing a relationship or analogy.

So, (as vanhees71 says) B and E are related because they are the components of F...
and, after using the properties of metric and volume form, become vector fields in 3-D.

However, because of the similarity of F and G [although dF is identically zero],
one can say that B and D are analogous as [flux] 2-forms, then (as before) as vector fields in 3-D.

Finally, as is noted above, B and H can be related by the "constitutive equation" (Hodge dual in vacuum in 3-D, 4-D)
(involving $\mu$ and $\epsilon$, which may be tensorial--not necessarily constants).

The strength of these relationships could be measured by the number of structures [not to be taken for granted] needed to make the relationship.

cabraham and vanhees71
Khashishi
Nowadays, everybody means B when they say the magnetic field and H is hardly used at all. Anywhere that H is needed we just write B/mu.

Nowadays, everybody means B when they say the magnetic field and H is hardly used at all. Anywhere that H is needed we just write B/mu.
H is used when designing inductors, transformers, solenoids, motors, and generators, as well as antenna & waveguides, and transmission lines. The losses in the ferrous core can be estimated only by observing the full B-H curve. H is as important as B, no more no less. Without both, how does one estimate power lost in core due to eddy current & hysteresis? Likewise for power inductors, and relays/solenoids. I use H about as much as B. Don't ask me to choose which I feel is more important because I can't choose.

Claude

Khashishi
Oh. I guess it's more important in engineering then.

Nowadays, everybody means B when they say the magnetic field and H is hardly used at all.
If you ask engineers about B, most of them will say it is called a "magnetic induction"

Jano L.
Gold Member
Now, my question is, wouldn't it make more sense to say B is the analog to E? Because E and B are more fundamental in the sense that they are the fields you can actually measure by placing a charge in an electric field or a current in a magnetic field and measure the resulting force independently of the medium.
That is one way to understand the letters E,B. Today, most people are taught EM force as being determined by fields that are designed E,B: $\rho\mathbf E + \rho \mathbf v \times \mathbf B$.

Lorentz used to express the force density with letters D,H : $\rho \mathbf D + \rho \mathbf v \times \mathbf H$.

There is a school of thought (Chu's formulation) which uses E,H for this purpose: $\rho \mathbf E + \rho \mathbf v\times \mu_0 \mathbf H$.

In fact all these conventions are right because Lorentz's D,H meant our E,B and Chu's $\mu_0 \mathbf H$ si the same as $\mathbf B$ in vacuum.

All this is just an example of the fact that different people find different notation (units, ... ) suit their needs. The needs of people differ so does their notation and units.

What is important is that we understand and can communicate what E,D,B,H mean in a specific context.

It makes sense to say B is analog of E in the sense they give force in the current major convention for the Lorentz formula. It also makes sense to say H is analog of E in the context of Maxwell's equations for EM waves in empty space.

If no specific context is considered, the important thing is E,D,B,H are all different fields with different meaning. They are all necessary (or need to be replaced by other fields, say polarization and magnetization) to formulate the macroscopic theory.

In microscopic theory (e.g. point charged particles) only two fields seem sufficient. Today it is common to designate them as e,b or E,B but Lorentz used d,h. It is just a convention.

cabraham
In the book I'm reading on electromagnetic fields, the authors seem to associate the magnetic flux density B with the electric flux density D, and the magnetic field H with the electric field E.

Now, my question is, wouldn't it make more sense to say B is the analog to E? Because E and B are more fundamental in the sense that they are the fields you can actually measure by placing a charge in an electric field or a current in a magnetic field and measure the resulting force independently of the medium. With the Lorentz force law: $\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right)$.

On the other hand, D and H occur from Gauss's law and Ampere's law respectively, and are indirect results of charge distributions and current distributions, and cannot be "measured" directly. The surface integral of D over a closed surface is the total enclosed charge in the surface, and the line integral of H around a closed loop is the total current crossing the loop. So D and H are both related to charge density and current density.

In fact, I think B should have been called the magnetic field, and H the magnetic flux density.

Are there any arguments for choosing to call H the magnetic field and B the magnetic flux density?
1st bold quote: If we looked only at the Lorentz force law, observing the influence of fields upon charges then yes E and B would form the basis set as you mentioned since the forces are independent of media. But that is just one set of conditions. How do we say just what is "fundamental?"

Let us look at the "fundamental form" for generating magnetic field with a current loop. The Biot-Savart law gives the *H* field as I/2R, for a loop radius "R" with a current "I". If we wish to express in B, not H, then B = mu*I/2R. When charges generate fields, the H is independent of medium, whereas B is not. In MKS units, the force on a charge by a group of 1 or more charges is F = Q1*Q2/(4*pi*epsilon*r^2), Coulombs law. Expressing this in terms of fields gives E = Q/(4*pi*epsilon*r^2), and D = (4*pi*r^2), so that D is independent of medium, not so for E.

Anyway, it is really arbitrary, using E & H as basis vectors, or E & B as basis gives correct answers as long as we remember the nature of permittivity and permeability. For dielectrics w/ hysteresis, or ferrous material with the same, we cannot regard D as a constant times E, nor as B equaling a constant times H. In polarized media, we must account for all 4.
2nd bold quote: H is the intensity and B is the density. Recall Faraday's law which is expressed in B, which equals area A, times flux $\phi$. Take a transformer core driven by an ac voltage source where a section of the material is filed down. Say the core cross section is 1.0 cm square, and we file 0.1 cm of one edge in 1 location. The area of the core is 1.0 sq cm everywhere except the filed down section where it is 0.9 sq cm.
The flux does not change since the xfmr is driven by an ac voltage source, it is dictated by Faraday's law. But in the filed down section the flux density is 1/0.9, or 1.111 times greater since the same flux exists in a smaller area.
This can be affirmed in a lab. If the material has a saturation density limit of 15 kgauss, or 1.5 tesla, and the input magnetizing current and secondary voltage is measured with ac input voltage adjusted for 1.5T flux density, secondary unloaded, then a portion of the core is filed down to two thirds its original area, the following happens. Input magnetizing current increases drastically since the core is now saturated. But the secondary voltage does not increase, it actually drops a little since the increased primary magnetizing current incurs a larger IR drop, reducing emf impressed on core.
Since the flux per area, or density, increased due to reduced area, B went up, but $\phi# did not, so secondary voltage remained almost the same. The magnetizing primary current is related to H, which increases drastically. Due to core saturation more intensity is required for the increased flux density. It is clear that B is the density while H is the intensity. Did I help? Claude vanhees71 Science Advisor Gold Member 2019 Award I don't know, why you insist on these wrong statements. There's a very clear meaning of the various electromagnetic field components in macroscopic electromagnetics (which is an approximation to the fundamental electromagnetics, which can be fully understood only with relativistic many-body theory). So again:$\vec{E}$and$\vec{B}$are the three-dimensional notation (i.e., electrodynamics written in terms of SO(3) vectors and tensors) for the components of the antisymmetric Faraday tensor$F_{\mu \nu}$in Minkowski space (4D notation), and$\vec{D}$and$\vec{H}$(3D notation) are emergent auxilliary quantities which also are the components of an antisymmetric tensor,$H_{\mu \nu}$(4D notation). The phenomenological theory in fully relativistic covariant form goes back to Minkowski and is the final answer to the problems about electromagnetic interactions in arbitrarily moving matter, as far as linear-response theory is sufficient as an approximation. As stressed above, a full account of this model can be only given within relativistic many-body QED. A very good source for this is (although it doesn't give the fully relativistic theory) S. R. de Groot, L.G. Suttorp, Foundations of Electrodynamics, North Holland (1972) BTW: All this is independent of the units you use. It's more convenient to work in CGS (preferrably in Heaviside-Lorentz units) than in the SI (with its fourth unit Ampere for the electric current), but it doesn't change the physics, and there's a one-to-one conversion from one system of units to the other, using the conversion factors$\epsilon_0$and$\mu_0$of the SI. Which statement is wrong? The generation of fields with currents, force on particles by fields, or the density/intensity example? Please explain why each or all of those are wrong. Is B the density, H the intensity? If not please elaborate. Does a wire loop with current have the B or the H that is media independent? You call my statements wrong then offer no reason or proof. Halliday-Resnick text, a classic, states that B is chosen but arbitrarily. As I stated, your method should give correct answers in any unit system. The original question was if E is the analogy of B or H, and I gave illustrations where it could be either or neither. What is wrong with my filed down transformer illustration? Please explain why I am "wrong"? Best regards. Claude vanhees71 Science Advisor Gold Member 2019 Award It's absolutely clear that$\vec{E}$and$\vec{B}$belong together and not any other arbitrary combinations. Given constitutive equations of macroscopic electromagnetics you can use of course any combinations you like in your calculations, but this doesn't change the fundamental structure of electromagnetics, which reads \partial_{\mu} F^{\mu \nu}=\frac{1}{c} j^{\nu}, \quad \partial_{\mu} (^{\dagger}F)^{\mu \nu}=0, where$j^{\mu}=(c \rho,\vec{j})$is the microscopic charge-current density vector. The macroscopic Maxwell equations are derived from these fundamental equations using statistical physics ("coarse graining" and usually "linear response"; of course for strong fields you have to go further to non-linear models (e.g., in "non-linear optics")). ShayanJ Gold Member I don't know, why you insist on these wrong statements. There's a very clear meaning of the various electromagnetic field components in macroscopic electromagnetics (which is an approximation to the fundamental electromagnetics, which can be fully understood only with relativistic many-body theory). So again: E⃗ \vec{E} and B⃗ \vec{B} are the three-dimensional notation (i.e., electrodynamics written in terms of SO(3) vectors and tensors) for the components of the antisymmetric Faraday tensor FμνF_{\mu \nu} in Minkowski space (4D notation), and D⃗ \vec{D} and H⃗ \vec{H} (3D notation) are emergent auxilliary quantities which also are the components of an antisymmetric tensor, HμνH_{\mu \nu} (4D notation). The fact that its E and B that appear in the Faraday tensor is itself a convention because I can simply replace all Bs by $\mu H$ and adjust constants in the equations such that nothing changes. Its our choice to associate E with B or H. Its only a convention. Only if nature suggests a natural connection between E and B, we can say its not a convention but I see no such natural connection coming from nature herself because we can rewrite all equations in terms of E and H instead of E and B. So what you say can't be a reason for choosing E and B as a natural fundamental pair. In fact this has no meaning! cabraham vanhees71 Science Advisor Gold Member 2019 Award It's not only convention! In nature there's one electromagnetic field, that's it.$\vec{D}$and$\vec{H}$are auxiliary fields used in our computations of the electromagnetic field in media in an effective theory, which can be derived from statistical physics (first done in the classical realm by Lorentz in the early 20th century). ShayanJ Gold Member It's not only convention! In nature there's one electromagnetic field, that's it.$\vec{D}$and$\vec{H}## are auxiliary fields used in our computations of the electromagnetic field in media in an effective theory, which can be derived from statistical physics (first done in the classical realm by Lorentz in the early 20th century).
Yeah. But that only means we need two fields in vacuum and four fields in matter. In vacuum we can Either use E and B or $\frac{D}{\epsilon_0}$ and $\mu_0 H$ and equations can be manipulated to make everything as beautiful as with equations with E and B.
In matter, all four fields are present inevitably(in genera) and then its even easier to treat E and H as the fundamental pair.(Or maybe D and B!)

cabraham
vanhees71
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2019 Award
There are no "four fields". There's only one electromagnetic field, at least as far as we know today.

zoki85
ShayanJ
Gold Member
There are no "four fields". There's only one electromagnetic field, at least as far as we know today.
I was talking in terms of non-manifestly-covariant EM!
But as I said before, even Faraday's tensor can be written in terms of E and H or D and B. We just need to adjust constants in the equations properly.

vanhees71
Gold Member
2019 Award
I give up :-(.

ShayanJ
Gold Member
I give up :-(.
I was going to give up too. But it seems to me the very fact that we can't reach a conclusion means it has no physical meaning which pair we choose as fundamental, it makes no difference to physics! Otherwise you could persuade us by demonstrating that difference. Here, you're just conjecturing the existence of such a difference.

cabraham
Jano L.
Gold Member
I was going to give up too. But it seems to me the very fact that we can't reach a conclusion means it has no physical meaning which pair we choose as fundamental, it makes no difference to physics! Otherwise you could persuade us by demonstrating that difference. Here, you're just conjecturing the existence of such a difference.
vanhees71 is talking about microscopic theory for finite density charges in vacuum - there, two vector fields e,b are sufficient to describe force density. One could use d,h for this purpose (Lorentz did), but it does not matter which letters are used, the important thing is two of them are sufficient.

Shyan refers to the fact that Maxwell's equations in common EM course do use four vector fields E,D,B,H. One could replace two of these by P,M. But one cannot eliminate them to arrive at 2 fields E,B or D,H only. In macroscopic theory where only free charges and free currents are controllable directly, 4 field quantities are needed to describe the state of the field.

So there is no disagreement. There are only two distinct theories. One could say macroscopic theory is a result of the microscopic theory so eventually only 2 microscopic fields e,b are needed, but this does not imply that only E,B are needed in macroscopic theory.