# D field

1. Sep 4, 2008

### fisico30

The D (displacement) vector has only the free charges as source.
I heard it is better to work with D than E, because E can be due to both free and induced charge, and we cannot easily know the bound charge distribution(?). But it seems that we can know the polarization P however...from which we can derive the bound charges.(first the egg or the chicken).
I also read that the D field makes Gauss law easier to use. How?

Q: What is the story? Can we not just use B and E since they are the primary fields that a charge would actually experience?

In summary:
* E_net is due to both free and bound charges.
* D is only due to free charges.
* The electric bound charges can be derived from P (and vice versa).
* The electric permittivity can be derived from P OR from the bound charges(and vice-versa).
* It is said that it is HARDER to know the bound sources than P or the permittivity. Not sure why....
* Maxwell equation can account for materials and only use B and E, IF we stick into them an expression for P(and M), OR just the bound charges,OR the material permittivity.
* Why bother with extra constitutive eqns, when we can just work with E and B, if we just insert P and M (or rho or epsilon) in M.E.?
* The constitutive eqns still demand knowledge of P and M (or,again, the bound rho or permittivity).
* Maybe Maxwell eqns then become too difficult to solve computationally with E and B...
* Using D (and H) we treat the material as if it was vacuum. We get D and then value of E_net at every point, adjusted by the local permittivity....

*D has field lines that behave differently from the field lines of E in materials (dielectric,ferroelectric...).

(Maxwell came up with D...because they did not know about atoms and P.
D=(eps_0)E_net *P, i.e., D always intrinsically takes P and bound charges into account....)

thanks!

2. Sep 4, 2008

### Defennder

Hi, I believe that it is D and not E which takes into account the contributing factor of the polarization of bound charges.

Gauss law: $$\oint_S \mathbf{D} \cdot d\mathbf{S} = Q$$. Writing it using E you have to divide the RHS by the constant of permitivitty of free space.

I assume Q here means "Question" and not "charge". Reason why we use D is because we want to differentiate between an applied E-field in a medium and the actual field which also accounts for polarised bound charges. Similarly B takes into account the effect of magnetisation due to "bound current" as opposed to "free current" when we are dealing with magnetic materials.

I believe it's the other way round.
I don't see how this can be done. P is the net polarisation vector per unit volume or dipole moment per unit volume. The local bound electric charge configuration cannot be derived from P since P is the net dipole moment of a huge collection of dipole moments, all of which differ from each other locally unless the medium is isotropic (I hope that is the correct term) and the elecric field is constant.
Only true in an isotropic material since the relationship between P and E there is linear.
It depends on what you mean by "know". div(D) gives you the volume charge density. So you need D which comes from E and P. As said earlier P is a linear vector function of E only if the material is isotropic. Otherwise there are no easy answers (and I don't know how to find them).
Don't understand what you mean by "account". Again note that there is no easy way out if the material is anisotropic both for D and B. You'll have to write out and solve a matrix equation for these materials.
Because E doesn't take into account the effect of bound charges. And B isn't that easily calculated since the applied H-field would not have the same effect in a magnetic material as B.

P.S. I'll appreciate if the experts could verify what I wrote above. I'm doing a second-year E&M course and certainly don't want to give wrong or misleading answers.

3. Sep 4, 2008

### fisico30

Thank you Defennder.

In a dielectric there are two E fields:
E_a=the externally applied electric field.
E_s= the secondary field due to the polarization P, which is due to the fact that there is E_a in the material.

So at a point in the material, a hypothetical test point charge would experience an electric field E_total, superposition of the two mentioned fields (also E_total<E_a since the E_s is opposite to E_a. But NOT ALWAYS)....

D is equal to the net field E_total (time epsilon_0) + P. It seems that it implicitly takes into account P. We know the source of D are only the free charges, but as you comment, D is somehow determined by P(which is caused by bound charges).
I guess a better perspective would be to look at the equation this way

E_total=(D/epsilon_0) - (P/epsilon_0), which shows how E_total is due to D and P (free and bound charges).......

4. Sep 5, 2008

### Andy Resnick

These are really good questions- they raise very fundamental and foundational issues in electrodynamics. Usually the fields D and H are introduced very quickly, usually simply as D = $\epsilon$E and H=$\mu$B, and then attention quickly shifts to P and the suseptibility $\chi$, thence to the index of refraction. E and D (and B and H) are shown to be interchangable, and for the overwhelming number of applications, this is fine. There is a subtlely when dealing with *moving* matter.

That's a very interesting question, and as far as I know, the problem of calculating the force applied to a charge distribution in matter by an electromagnetic field is still incompletely solved. It turns out that there are several different formulations of electrodynamics which provide identical results for matter-free space, but produce different results for the stress-energy tensor in matter- how much force is applied to a charge by the field. This has a lot of technological applications (piezoelecticity, for example).

The E and B fields are technically for matter-free space. It's important to note that electrodynamics is a continuum model: there are no such things as point charges or point particles. In regions of space containing ponderable matter, there are charge distributions or densities that can respond and move in the presence of an externally applied field. To be sure, we can write D$_{0}$ = $\epsilon_{0}$E$_{0}$ for matter-free space, but that's not all that interesting. D and H are considered to be the electric and magnetic fields within ponderable matter. P and M are yet other ways of writing down the field in ponderable matter.

I'll come back to this after answering:

Constitutive laws are an important part of physics, and it's not widely appreciated that the relations come from *outside* of physics. That is, constitutive relations cannot be derived from first principles, they are invented. They serve to allow comparison between theory and experiment.

A constitutive relation is simply a functional relationship between two objects. Maybe it would be more clear if instead of writing D = $\epsilon$E, I write D=D(E) and H=H(B). Now it should be clear that I can use either D or E in the equations- I can switch between the two arbitrarily, as long as the functional relationship is invertible (which is usually the case). The specific form of the relationship is up to me- maybe there a simple linear dependence, maybe the permittivity depends on frequency or the material is anisotropic (permittivity tensor) or both, maybe the relationship is nonlinear, depends on the material's history, etc. etc... It doesn't matter: once I write down a functional relationship between D and E, I can use either field.

Now we get to Maxwell's equations. Strictly speaking, Maxwell write down the laws for matter-free space. Minkowski postulated the fields D and H and wrote down a simple constitutive relationship between them. This formulation of electrodynamics (the Minkowski formulation) has produced excellent agreement with experiment for stationary polarizible and magnetizible media. To generalize the constitutive relations to moving media, it was originally assumed that the fields simply transform as we expect as per special relativity- D depends on H and vice versa- but this is not surprising: moving charge creates a magnetic field.

For 60+ years, it was thought that the electrodynamics of moving media was solved. In 1960, Fano, Chu and Adler wrote down a different formulation of electrodynamics that produced identical results for matter-free space, but produced large discrepancies in the stress-energy tensor for moving media. It turns out that there are at least 4 different formulations of electrodynamics, depending on which 4 fields (out of E, D, P, B, H, M) are considered primary. Again, use of constitutive relations allows one to move between formulations, but the calculated stress-energy tensor produces different results. There is no obivous reason why one field should be considered primary over another, so there is no clear way to solve this problem. I do have a source that claims to have reconciled all the different formulations, so maybe this is a moot point. Typically, the Minkowski formulation is used for everything except relativistic calculations, in which case the Chu formulation is used.

In terms of QED- the microscopic nature of matter and how it interacts with the electromagnetic field, I don't know enough to say how this discussion fits in- that is, if the Minkowski formulation is used or some other formulation, or if it matters at the microscopic scale.

5. Sep 5, 2008

### tiny-tim

Hi fisico30!

Yes …

unfortunately, the minus in that equation makes it look as if D is the "bigger" field …

for some reason, P is defined to be minus what one would expect!

But E is definitely the total field … one only has to look at where rhob and rhof come in the three versions of Maxwells equations to see that.

6. Sep 5, 2008

### fisico30

Wow, I knew I was gonna open pandora's box.
Thanks Mr.Resnick.(I guess I read to many amateurs books).

1) Constitutive equations: they are then not laws but expressions, relations between the various fields. They vary from material to material and need to be discovered by experiment. Then we place materials in categories based on the right const. relation.

If the medium is not moving, the electric polarization P is only a function of E, and the magnetic polarization M only a function of B.(and we always assume a not-moving medium, so D=D(E) and H=H(B).
The electromagnetic properties of materials can be affected by factors of mechanical and thermal nature. In that case, other quantities like pressure and temperature appear in the const. equations.
Another assumption is that everything happens instantaneously: one vector field time t affects the other vector field at the same time t. In reality, atomic and molecular structures have inertia, so the time variation of,say, polarization, follow with some delay the variations of the field E. In that case the constitutive equations should also involve time derivative. Media that follow const. equations with time derivative are called materials "with memory" (of what was happening to the electric field earlier).
Also (another approximation), we are always assuming that the value of the polarization P field at a point is due only to the E field at the same point. Sometimes, instead, what happens to the E field in the neighborhood of the point is also affecting P at that point.
(we then need spatial derivatives in the const equations).

2
I always and erroneously thought that E and B are for space that contains matter. In vacuum, with only free charges, D has some functional form, as well as E. In space other than vacuum, with the same free charges, the D field does not change, but the E field does (it gets modified by the permittivity, therefore it depends in the material). The only way to change D is to change the free charges. If we change the bound(material) charges, D could not even care...
I guess I still don't get what the advantage is in using 4 vectors (D,E,H,B) instead of just 2...
But I will!
WE just need to stick with E and B and use an electric and magnetic permittivity function in the equations that take into account for allllll the material charges. We do not even need t otalk about P and M....

Thanks!!

thanks again

7. Sep 6, 2008

### Defennder

Hi, Dr Resnick, thanks for your very insightful post. I'm haven't learned any advanced formulation of E&M, but I did a quick search on Google and here it says that the Minkowskian formulation was developed to be compatible with relativity, so why is the Chu formulation used instead?

8. Sep 6, 2008

### atyy

They work for space that contain charged matter, but then you must treat all charges "fundamentally". Or you can hide your ignorance about what exactly every charge is doing in the "constitutive relations". Something similar happens in the definition of the refractive index. We can think that a medium slows light down according to its "refractive index". Or we can think of light going through a medium "fundamentally", and that light does not slow down at all in the vacuum between atoms. The slowing down is apparent and is due to the time taken for light to be absorbed and re-emitted by each atom, (HallsofIvy's post #4 at https://www.physicsforums.com/showthread.php?t=253789). The "fundamental" treatment of light going through a medium is really complicated (and actually needs quantum mechanics), so luckily we can hide our ignorance by defining a "refractive index" and pretending that light actually slows down. Like the constitutive relations, the refractive index is something that must be experimentally measured for each material. Actually, the refractive index and the constitutive relations are related, because the refractive index can be calculated (at least approximately) from a material's dielectric constant.

As Andy Resnick pointed out, in general D(E) can complicated. If we write this complicated function of E everywhere in the Maxwell matter equations, they will look nothing like Maxwell's fundamental equations. So D is defined to make the matter equations look like the fundamental equations, just to make them easy to remember. It's a psychological trick, since one still has to look up the constitutive relation.

9. Sep 6, 2008

### atyy

10. Sep 6, 2008

### fisico30

I think I am starting to get it atyy. Thank you.
Let's see if it is true (correct me please).

-The influence of a EM field on a material could be described by the charge unbalance it creates (bound charges), OR by polarization functions P and M, or by permittivities (mu and epsilon). The order is from very"fundamental" to less fundamental.
The less fundamental the closer to experiments: we can find the parameter, like the permittivity, by practical experiment.

-In numerical electromagnetism, since computers are solving abstract, theoretical, problems (that we can set up as complex as we want), using D,H,B,E or just E and B does not seem to make a difference, unless we prefer to keep a certain functional similarity between maxwell equations in vacuum and in matter (but supplementing them with const. eqns).

- For those examples inside materials, where H is going opposite to B, or D behaves strangely from E: how do you regard them? Is it somehow insightful to know about those differences or are they just artifact due to D and H (not corresponding to a reality). I mean, do they give a particular, different, useful perspective?

thanks

11. Sep 7, 2008

### Andy Resnick

That's ok- call it terminology. A constitutive law/relation can be something like "The present state of a material cannot depend on the future". Expressed mathematically, we obtain the Kramers-Kronig relations between the real and imaginary parts of the (for example) permittivity.

12. Sep 7, 2008

### Andy Resnick

It's not- but thanks for calling that one to my attention.

My reference is a book from the 80's, "Electrodynamics of moving media", by Haus, IIRC... it's not here in front of me.

Yet more evidence that the problem remains unsolved.

13. Sep 7, 2008

### Andy Resnick

The Minkowski formulation is compatible with special relativity- it is assumed that the fields transform normally. But the essential difference between the various formulations is in calculating the stress-energy tensor (for EM fields within moving matter). More than that, I can't say- I don't actively work in that field.