# Relation between H and B fields, and D and E fields

1. Dec 7, 2006

### Repetit

Hey!

I am having some trouble understanding why the magnetic flux density B and the magnetic field strength H does not have the same units. I mean, as far as I understand, the H field is just the B field with the magnetic properties (magnetization) of the material taken into account? Isn't this correct? So why isn't the H field just the B field multiplied by some dimensionless scalar (or tensor)?

The same thing applies to the electric field E and the electric displacement D. If the D field is the electric field with the polarization taken into account, why do they not have the same units, and why is D not just E times some scalar value which depends on the polarizability of the material?

2. Dec 7, 2006

### Claude Bile

This is exactly how D is calculated! (although the polarisability is generally a matrix not a scalar).

The polarisability however has units, it is not a dimensionless quantity, so when we multiply some quantity by another, non-dimensionless (dimensioned?) quantity (i.e. multiplying E by polarisability), we must necessarily obtain a quantity that has different units.

So your question really boils down to "why does the polarisability have units?" In which case the answer should be fairly self-evident.

Claude.

3. Dec 8, 2006

### Meir Achuz

What you say is correct and is the basis for gaussian units. The use of SI units confuses the connection between the vectors E,D,B,H. SI was adopted about 50 years ago at an international congress, dominated (the hall was packed) by engineers who didn't understand the physics of electromagnetism.
Most elementary and intermediate texts use the confusing SI system, but working physicists more often use gaussian for their own calculations.
The 3rd ediltion of Jackson triles to go to SI, but even he as to use gaussian to complete the subject.

4. Dec 9, 2006

### dextercioby

The confusion arises only when one's taught SI units in high-school and follows a CGS-based book in college, or viceversa.

If one's being kept in the same "environment" along his academic studies (HS+ college), there's no confusion.

Daniel.

5. Dec 10, 2006

### Repetit

Thanks for the replies, good to know my thinking was right!

It seems strange however that E and D have different units in one system of units, and the same units in another one.

I mean, one could never define another system of units where a newton had the same units as coulomb, right?

6. Dec 10, 2006

### Meir Achuz

In gaussian units, charge and current are defined in absolute terms by the forces between two charges and between two wires.
Then there is no arbitrariness about their units.
1 stacoulomb=1 dyne-cm^2 from Coulomb's law.
1 statamp=1 sc/sec if the 1/c^2 is put into the force between two wires.
That connection (1/c^2) was measured by Weber and Kolrausch in 1856.
SI was developed 100 years ago because an engineer named Georgi thought that electric charge was a new unit, independent of force. Thus SI was originally called MKSA (A for ampere). Geogi did not know that there were other charges besides electric charge, and that after SR and QM physicists would realize that all charge (in terms of alpha) is dimensionless. Trying to give a dimension to an intrinsicly dimensionless quantity is why SI has a different unit for so many things (E,P,D,B,M,H) that should have the same unit. If SI can give units to empty space, then it would be equally sensible
(really nonesensical) to define the unit of charge as one newton if epsilonzero were given different arbitrary units than it has in SI.

7. Dec 11, 2006

### octol

Sorry don't fully follow here. I'm only used to the SI-system so this sounds interesting, can you please elaborate a little? Why is charge dimensionless for example?

8. Dec 11, 2006

### Meir Achuz

In QED, the electron charge appears in the combination e^2/(hbar c).
(It would be divided by 4piepsilonzero in SI). Putting numbers in in ANY system gives the dimensionless number 1/137.036. This is called alpha and is the dimensionless charge value.

9. Dec 13, 2006

### rbj

you can define reality in terms of Planck Units and then every quantity measured really is unitless and dimensionless. (and the Gravitational constant, Speed of Light, and Planck's constant are all just the number 1, removing those scaling factors from the equations of physical law.)

from an SI point-of-view, D is measuring flux. think of this in terms of the natural meaning of an inverse-square action. we have a point charge Q "emmiting" a total quantity of flux that is the same as the amount of charge Q. these lines of flux are emmitted equally in all directions in 3 dimensional space.

the flux density is the density of these lines of flux per unit of area that is perpendicular to the lines of flux (perpendicular to the line connecting Q to that unit area). a sphere has a surface area of $4 \pi r^2$ so this total flux of Q is distributed equally among that entire surface area. that means that that the flux density, D, is the total flux, Q, divided by the total surface area $4 \pi r^2$ or

$$D = \frac{Q}{4 \pi r^2}$$

or in vector form (assuming Q is at the origin)

$$\mathbf{D} = \frac{Q}{4 \pi | \mathbf{r} |^3} \mathbf{r}$$

so flux density, D, is the physical quantity with dimension of charge per area or charge per square of length [Q][L]-2.

now the electrostatic field, E, is the physical quantity that represents how much force per unit charge that there is applied to a test charge, q, placed at position r from the main charge Q (which is at the origin). from the Coulomb Force equation we know that

$$F = \frac{1}{4 \pi \epsilon_0} \frac{Q q}{r^2}$$

or in vector form (assuming Q is at the origin)

$$\mathbf{F} = \frac{1}{4 \pi \epsilon_0} \frac{Q q}{| \mathbf{r} |^3} \mathbf{r}$$

and the E field is

$$E = \frac{F}{q} = \frac{Q}{4 \pi \epsilon_0 r^2}$$

or in vector form (assuming Q is at the origin)

$$\mathbf{E} = \frac{\mathbf{F}}{q} = \frac{Q}{4 \pi \epsilon_0 | \mathbf{r} |^3} \mathbf{r}$$

electrostatic field, E, is the physical quantity with dimension of force per charge or mass-length per time-squared per charge: [M][L][T]-2[Q]-1.

so comparing the equations, what is the relationship between flux density, D, and electric field strength, E?

Last edited: Dec 13, 2006
10. Dec 13, 2006

### rbj

actually, Meir, we need to be clear about a couple of things. the most common name for this quantity is the Fine-structure constant and it is proportional to the square of the elementary charge, e. the other thing is that the expression

$$\alpha = \frac{e^2}{\hbar c}$$

is only dimensionless (and currently believed to be 1/137.03599911) in unit systems that define the unit charge so that the Coulomb Force constant $1/(4 \pi \epsilon_0)$ is 1 and disappears from the equations of physical law, namely the Coulomb force law (sorta like defining the unit force so that the constant C in $F = C dp/dt$ is 1 and goes away). this is an arbitrary human decision. in any system of units, the general expression for the Fine-structure constant is

$$\alpha = \frac{e^2}{\hbar c (4 \pi \epsilon_0)}$$ .

(i know you know this, Meir, but i think in introducing this to someone, one should not say simply "e^2/(hbar c)".)

one thing also to point out is that this Fine-structure constant really is the square of the elementary charge when measured in Planck units:

$$\alpha = \frac{e^2}{\hbar c (4 \pi \epsilon_0)} = \left( \frac{e}{q_P} \right)^2$$

and can be thought of, in a world of natural units where all these scaling constants in the field equations go away, and given a constellation of charged bodies all with fixed numbers of protons and electrons in these charged bodies, that $\alpha$ represents the strength of the electromagnetic action.

Last edited: Dec 13, 2006
11. Dec 13, 2006

### rbj

BTW, Meir, i know that this is an old thread (and a different thread, but one of the issues are the same), i must say that i agree with you and marcusl fully about this. "H" should be called "flux" and "B" should be called "field".

12. Dec 14, 2006

### Meir Achuz

I am puzzled by your post, since I assume you read mine that you quote, and I think we are in almost complete agreement.
1. The term "fine structure constant" is the historical designation because it was first noted in atomic spectrocopy 100 years ago. Now that we know that alpha is a standard ratio between many numbers in physics, it is probably time to not limit it to fine structure, and just call it alpha, but that is not an important point.
2. I did mean to imply that alpha had somewhat different algebraic forms
in different systems of units. That is why I explicity mentioned the division by
fourpiepsilonzero in SI. I did not mention, but should have that it is simplest in the form of natural units used today by most HE theorists, where
alpha=e^2. Fortunately, I know of no one (you and I included) who redefines alpha to be anything other than 1/137. e^2, on the other hand, has many different values in different unit systems.
3. I am glad we both agree that "$\alpha$ represents the strength of the electromagnetic action". I would only add: whatever system of units is used.

13. Dec 14, 2006

### Meir Achuz

I think this post, compared my sentence "What you say is correct and is the basis for gaussian units." presents the case for using gaussian rather than SI units.

14. Dec 14, 2006

### rbj

but the name is there. it's what you use to look up the concept, even if it is now known that this concept sorta trancends the fine-structure splitting.

i know i'm swimming against the trend, but i still think that the general expression (not just for SI) should be

$$\alpha = \frac{e^2}{\hbar c (4 \pi \epsilon_0)}$$

and then just note that $4 \pi \epsilon_0$ gets set to 1 in some systems of units because of the manner that the unit charge is defined in those systems of units. personally, i think it's much more natural to define charge so that $\epsilon_0 = 1$ (and, for gravitation set $4 \pi G = 1$) and then (with $c = \hbar = 1$) you get $\sqrt{4 \pi \alpha} = e$ which i think is the more natural and salient dimensionless number for which \alpha is derived.

that's true, but hard to conceptualize without thinking in terms of natural units. the gravitational attraction between two planets exceeds the electrostatic repulsion if they happened to be equally charged by a few elementary charges. without going to natural units, comparing the magnitude of actions from different fundamental forces is like comparing apples to oranges. electromagnetics operates on charge and gravitation (in the Newtonian sense) operates on mass. when you compare the two actions, how much charge do you use in comparison to how much mass?

15. Dec 14, 2006

### robphy

From a more abstract viewpoint, H and B aren't even the same type of geometrical object. H is a [twisted-]1-form in space, which is associated with a line-integral, and B is a 2-form in space, which is associated with a surface-integral. Similarly, E is a 1-form and D is a [twisted-]2-form.

http://arxiv.org/abs/physics/0407022 (see pictures on page 7)

16. Dec 14, 2006

### rbj

fine. but why stop there? continuing the "let's get rid of scaling constants" impetus, why not go all the way to Planck Units? personally, i think the $4 \pi$ constants should also be lost, also, when using Gauss's law for either electrostatics or for gravitation and then you get field equations with no scaling constants except for an occasional "2".

17. Dec 14, 2006

### marcusl

rbj, there's reason to call B "field," but please don't call H "flux" because this is a different quantity as I had noted earlier in the thread you quoted. Magnetic flux is defined by
$$\Phi = \int \vec{B} \cdot d\vec{A}$$
and has units of Webers (SI) or Maxwells (Gaussian/cgs).

It might be better to join authors such as E. Weber and Julius Stratton who call H "magnetic intensity."

18. Dec 15, 2006

### rbj

okay, i meant to call H "flux density" like we call D. i know we call B "flux density" because

$$\Phi = \int \vec{B} \cdot d\vec{A}$$

but, if E is "field" (because it is related to intensity of effect) and D is "flux density" (because it is related to the density of how much of the source of the effect is present at some point), to be consistent, shouldn't they have named "H" as "magnetic flux density" and "B" as "magnetic field"? isn't calling "B" a flux density an historical accident?

19. Dec 15, 2006

### marcusl

No, because E and B are the fundamental field quantities, while D and H are derived [Jackson] "as a matter of convenience to take into account in an average way the contributions ... of atomic charges and currents." That is why E and B should be called fields, and why Mel Schwartz doesn't bother to even name H in his book. Furthermore, D is almost universally called Electric Displacement, and only rarely "dielectric flux density."

A survey of E&M books on my shelf shows B is called
Magnetic Induction -- by Smythe, Stratton, Jackson, Reitz & Milford
Magnetic Field -- Schwartz, Weber
alternately
Magnetic Flux Density -- Weber, Jackson

H is called
Magnetic Field or Field Intensity -- Smythe, Stratton, Jackson
Magnetic Intensity -- Weber, Reitz & Milford
unnamed -- Schwartz

Can you elaborate on this for those of us who are differential-geometrically-challenged? I am lost by even the introduction (for instance the authors state that E&M is independent of gravitation, yet the field tensor F in eqs. (2)-(3) would seem to depend explicitly on g).

20. Dec 15, 2006

### robphy

In that paper, eqs (2) and (3) are a set of metric-dependent equations that appear in the Einstein paper referenced there. Actually in each set, the metric appears explicitly only in the "constitutive" equation relating the $$\cal F$$ tensor and the tensors $$F$$ (in 2) and $$\phi$$ (in 3). The story of formulating electromagnetism without a metric starts in part IV. A key idea is that there are two independent field tensors that capture most of electromagnetism. In the presence of a metric or other structure, the two tensors are related... possibly to the point where they are not easily distinguishable.

Here is a more accessible reference: http://www.ee.byu.edu/forms/forms-home.html [Broken]

Last edited by a moderator: May 2, 2017
21. Dec 16, 2006

### rbj

i'm in violent agreement. D and H are defined and there is no $\epsilon_0$ or $\mu_0$ needed as constants of proportionality as would be the case for E and B (because all of the physical quantities have defined units - i know that the unit of current and then charge are defined so that $\mu_0$ is a fixed and exact value).

well, it might be the difference of treatment between different disciplines. in my EE-based fields book "Hayt: Engineering Electromagnetics", D is "electrostatic flux density" and what comes out of the Gauss's Law integration is total "flux" which is proportional to the enclosed charge. if it is D that you're integrating of an enclosing surface, then D is defined so that this constant of propotionality is 1.

r b-j

22. Mar 17, 2008

### Philip Wood

When Maxwell was making his wonderful synthesis of the phenomena of electromagnetism, he regarded the fields as disturbances in a medium (the ether) which was thought to be present even in 'empty' space. He needed two different vectors related to stress and strain (for electric fields) and two different (rotational) quantities for magnetic fields. Now that we no longer believe in the ether, we surely need only one electric field vector and one magnetic field vector for electromagnetism in a vacuum.

What about when media are present? For Maxwell there was no fundamental distinction between electromagnetism in a material medium and in the ether - all that needed to be done was to use different values for the constants, eg. permittivity. [I'm considering only isotropic media.]

Later Lorentz brilliantly explained the differences between what was observed in media and what was observed in a vacuum, in terms of the bound electrons in media. [These electrons behave in accord with the basic laws of electromagnetism, so the explanation has a pleasing economy and 'closure'.] Where media are involved it is quite useful to have two electric vectors, D and E, and two magnetic vectors, H and B. Using D and H we can write Maxwell's equations in terms of 'free charges' (that is those which aren't merely inherent in the medium), so that the equations look much like Maxwell's equations for a vacuum (though these need just E and B). We can then 'forget about' the medium, knowing that for isotropic media, D and H are related quite simply to E and B.

I find it annoying (as, perhaps the original question-asker did) that, in the mks system, D was not defined as its present definition divided by epsilon-0 and that H was not defined as its present definition multiplied by mu-0. If these changes were made, then in a vacuum D and H would be the same vectors as E and B respectively. For media, Maxwell's equations written in terms of free charges (and free current densities), and E, D, B, H, would then give the vacuum equations as a special case simply by replacing D with E and H with B. There'd be no need to mess about with the mu-0's and epsilon-0's in the equations.

What has happened, unfortunately, is that the old ether-based conceptions are not completely dead, but exert a malign influence. Thus I believe that some electrical engineers, and perhaps some physicists, regard B and H, and D and E in a medium as somehow analogous to stress and strain. They do not have the Lorentz viewpoint. Worse still, I think this back-feeds to the vacuum case, where D and E, B and H are unfortunately sometimes still regarded as fundamentally different (like stress and strain or cause and effect).

This is, I suggest, why, even in the modern mks system, we are stuck with different units for D and E and for H and B. D and H were, I suspect, deliberately defined to be different quantities from E and B, even in a vacuum. This reflected a pre-Lorentzian, essentially ether-based, picture of electromagnetism.

Last edited: Mar 17, 2008
23. Mar 24, 2008

### Philip Wood

Apology in advance. What follows is a total rewrite of my last reply. I was expecting to be able to replace the last reply using the edit button, but button had disappeared! [Why?]

B and E are the fundamental field vectors as defined by the ‘Lorentz force’, F = q(E + v x B) on a charge q moving with velocity v.

When materials are present it is useful to think of E as the sum of E due to charges which don’t form an integral part of the material itself, and E due to charges that do. By using another vector, D, as well as E, we can write Maxwell’s equations in a way which doesn’t involve the charges integral to the material. The nice thing is that, for a large number of dielectric materials, E and D simply differ by a constant factor (the ‘permittivity’). So, for macroscopic purposes, we don’t need to calculate from first principles the contribution to E of the inherent charges in the material, as long as we know the value of the permittivity. Lorentz sorted all this out in the 1890s, and showed that the effects of the inherent charges are taken care of by using the (simply related) D and E vectors.

The same sort of thing was done for magnetic fields, resulting in a vector H, additional to B, for use when magnetic materials are present.

In a vacuum, there is clearly no need for the ‘supplementary’ vectors D and H, as there are no inherent charges or magnetic dipoles to take care of. [I’m not considering QED.] This is why I find it annoying that (e.g. in mks) D and H are not defined in such a way that, in a vacuum, they simply become E and B respectively. Instead, they are defined such that they become (epsilon-0)E and B/(mu-0) in which epsilon-0 and mu-0 are universal constants, with units, making it seem as if D and B are doing fundamentally different things from E and B respectively, even in a vacuum.

The frustrating thing is that it is probably no accident that D and H are defined to be different from E and B even in a vacuum. It stems, I think, from outdated concepts going back to the 1860s and Maxwell’s wonderful synthesis of the phenomena of electromagnetism. For Maxwell, who didn’t know about electrons, there was no fundamental distinction between electromagnetism in a material medium and in empty space. This was because he thought of empty space itself as full of a hard-to-detect medium, the ‘ether’. The same equations applied both to empty space and materials; you just needed different values of permittivity and permeability constants.

Now here is the key point: Maxwell and his pre-Lorentz followers needed two vectors for electric fields and two vectors for magnetic fields, for ‘empty’ space as well as for materials. The ‘extra’ two vectors weren’t thought of as coming into their own just for materials. Instead, they derived from Maxwell’s original attempts to model electromagnetic fields as altered states of a space-filling mechanical medium. In the case of electric fields they are related to stress and strain.

I believe that this stress and strain, stimulus and response, conception is still held by some engineers, and perhaps by some physicists. It is pre-Lorentzian, and essentially ether-based. It is, I think, partly responsible for why we are stuck with different units for E and D and for B and H.

None of this is to deny that certain fundamental theories of electromagnetism may require more than just an E and a B (or a single e-m field tensor) to describe the electromagnetic state of a point in empty space. All I’m arguing is that the reasons for using D and H when materials are present gives no support at all for needing these extra vectors in a vacuum.