# Defination of E D and P fields

1. May 12, 2010

### omri3012

Hallo everyone,

in electrostatic we have a linear realation between E D and P, i do understand that P
is related to the polaryzation density but what about E and D? which of those represent the
overall field?

2. May 12, 2010

### Andy Resnick

'E' is the electric field in free space, 'D' is the electric field within ponderable matter. Relating 'E' and 'D' requires a constitutive relation (for example, D = eE), and 'P' is taken to be the contribution to 'D' from the matter responding to 'E'.

It can be a little confusing....

3. May 12, 2010

### omri3012

from your answer i dont understand what the diffrence between D and P?

4. May 12, 2010

### matonski

P is the dipole moment per unit volume.

E is the electric field. Its field lines start and end on all charges, including the charges in the dipoles that make up P.

D is like E, except that the lines of D only start and end on free charges, and not on the charges that make up P.

5. May 12, 2010

### Andy Resnick

'D' is the *total* electric field within a material, 'P' is that portion that comes from the material in response to the 'E' field.

For simple linear materials:

D = eE
P = D - e_0E

6. May 14, 2010

### DrDu

That is at best an approximation for substances which can be thought to consist of independent molecules or atoms.
The polarization derives fundamentally from the conservation of charge of the substance. Hence it is possible to introduce the polarization magnetization tensor whose divergence yields the charge current vector. This tensor is not uniquely defined. E.g. in optics one sets M=0 and P=int j dt.
The relation is somewhat analogous to the relation of the magnetic vector potential to the electromagnetic field tensor.

7. May 14, 2010

### matonski

What are examples of substances where this approximation does not hold true?

8. May 16, 2010

### DrDu

I meant that the electrons are strictly confined to these atoms and molecules. There are many examples of substances where the electrons are delocalized, e.g. semicoductors, metals.

9. May 17, 2010

### Jano L.

Hi,
does anybody know any textbook or article that treats this question of polarization in detail?

1) What is the correct definition of polarization?
Standard textbooks say that P is the density of dipole moment. But referred to what? Dipole moment of a charged body depends on the reference point.

2) And then, how to prove the relation div P = -rho?
Textbooks say that the polarized body with density P creates the same potential as a body with no polarization at all but with the charge density of -div P. So what we have is in fact equality between two integrals. But we know very well that from this we can not always infer the equality of integrands, right? As a contra-example, the dipole moment of the body with respect to the origin of coordinates is

$$\boldsymbol p = \int \boldsymbol r \rho dV = \int \boldsymbol P dV$$

Clearly the integrands are not the same.

And furthermore, this argument with potential works only in static situation; but we use the Maxwell's equations with polarization for optic and radiative phenomena...

10. May 18, 2010

### DrDu

A definition of the polarization and magnetization which goes beyond the dipole approximation and which is especially suitable for the discussion of molecular systems can be found e.g. in these lecture notes of Claude Cohen Tannoudji (in French):
http://www.phys.ens.fr/cours/college-de-france/1986-87/cours9/cours.pdf
Or in Molecular Quantum Electrodynamics of D. P. Craig,T. Thirunamachandran; Dover Publications.
These definitions can be taken over to solid systems only if the solid can be thought to be consisting of appreciably non-overlapping entities to which the above decomposition can be applied.
When this is not the case, one usually sets M=0 and $$P=\int_{-\infty}^t j dt$$, see, e.g., Electrodynamics of continuous media, Lev Davídovich Landau,Evgenij M. Lifšic.

Last edited by a moderator: Apr 25, 2017
11. May 18, 2010

### Jano L.

Thank you DrDu, I will read it carefully. But on the first glance, the method of Tannoudji is quite suspicious - why do we need to use artificial line of dipoles and Fourier's transformation? That is not very convincing. Simple relations should reflect simple physical picture. Besides, when we introduce the dipole line instead of the charge at the end of $${\boldsymbol r}_{\alpha}$$, we manufacture a new charge $$-q_\alpha}$$ at the beginning of $${\boldsymbol r}_{\alpha}$$, but it was not there before...

12. May 18, 2010

### Count Iblis

Hmmm, all the textbooks I've read call D an artificial mathematical quantity. E is the real physical macroscopic electric field in the medium. I.e. the real electric field will be rapidly fluctuating on the atomic scale, but if you average out the short distance fluctuations, you get the so-called coarse grained macroscopic electric field, which is the E field.

The E field contains contributions from induced dipole moments. When doing computations, you don't a priori know what the dipole moment density P is. It is possible to subtract this contribution from E and then you obtain the D field. The usual arguments then lead to equations like D = epsilon E, and

Div D = rho_free

which together with boundary conditions allow you to compute the electric field in dielectric media.

13. May 19, 2010

### DrDu

The introduction of D and P is independent from any coarse graining or macroscopic averaging.
This obviously does not mean that it is not usefull to consider coarse grained P and D, but that these can be calculated from the microscopic quantities, see e.g. S. Adler, Phys Rev (1962) vol 126, pp. 413.
In fact, P it is only a reformulation of charge conservation, see e.g.:
http://arxiv.org/abs/physics/9907046
or R.G. Woolley, Int J Quantum Chem, (1999), Vol 74., pp. 531, who also discusses the multipolar hamiltonian which is quite a standard at least in molecular physics.
Especially, polarization does not equal the dipole moment density in general and Jano L. realized quite correctly that the latter approximation is ambiguous as it depends on the choice of the reference points.

14. May 19, 2010

### omri3012

15. May 19, 2010

### DrDu

Ok omri, are there still open questions then?
The E field is still the true E field (eventually one considers only some average values of it), while the polarization is an alternative way to describe the electric charges and currents of the medium.

16. May 19, 2010

### Jano L.

Count Iblis,
that most textbooks say D and H are secondary is true, however it is not very clear why. In the matter, there is no way to measure the fields E and B, so all of these four quantities get the same reality. Furthermore, EDBH still survives everywhere, equations look nice and also Maxwell used them. If I remember correctly, he had an idea that even the free charges are just the result of nonuniform polarization of vacuum, described by D: $$\rho_{free} = div D$$.
When I read the last part of your message, I recalled an amusing result: did you know that in a linear dielectric $$D = \epsilon E$$ implies $$\rho = \epsilon_r \rho$$, so that $$\rho = 0$$ always? The whole argument have a meaning just for nonlinear or anisotropic media...