Maxwell's Equations, Dielectrics, Auxiliary Fields D and H

[fog37]

Hello,
I am reviewing how classical EM problems are treated when dielectric materials are involved. Maxwell's equations relate the following vector fields: ##E(r,t)##, ##B(r,t)##, ##D(r,t)##, ##H(r,t)##, ##J(r,t)## and scalar field ##\rho(r,t)##. The two constitutive equations are also needed to relate ##E(r,t)##, ##B(r,t)##, ##D(r,t)##, ##H(r,t)## together via the material's permittivity and permeability.

The ultimate goal is solving for the two fields ##E(r,t)##, ##B(r,t)##. In the presence of dielectrics, the electric polarization ##P(r,t)## and magnetization ##M(r,t)## can be introduced instead of permittivity and permeability. ##P(r,t)## and ##M(r,t)## are vector fields generated by bound sources (bound charges and bound currents) .

In theory, we could solve EM problems without using the auxiliary fields ##D(r,t)##, ##H(r,t)## fields that are controlled by free charges and conduction currents only. For a linear dielectric material only, the fields ##E(r,t)## and ##D(r,t)## are related in two different but equivalent ways: $$D(r,t) =\epsilon_0 E(r,t) +P(r,t)$$ $$E(r,t)=\epsilon(r,t) D(r,t)$$

The electric permittivity ##\epsilon## can be a constant, a function, a tensor function, etc.

Why is it so convenient to use the electric displacement vector ##D(r,t)## to find ##E(r,t)##? According to ##E(r,t)=\epsilon(r,t) D(r,t)##, to find E(r,t) we need both ##D(r,t)## and knowledge of the electric permittivity ##\epsilon##. Is it easier to gain knowledge of ##\epsilon## than gaining knowledge of the electric polarization ##P(r,t)## and solve for ##E(r,t)## using ##D(r,t) =\epsilon_0 E(r,t) +P(r,t)##?

Fundamentally, I see three possible scenarios:

Scenario 1: solving for ##E(r,t)## and ##B(r,t)## using Maxwell's equations, the total sources (##J(r,t)## and ##\rho(r,t)##) and ##P(r,t)## and ##M(r,t)## (no presence of the auxiliary fields).

Scenario 2: solving for ##E(r,t)## and ##B(r,t)## using Maxwell's equations, the auxiliary fields##D(r,t)##, ##H(r,t)##, the total sources ##J(r,t)## and ##\rho(r,t)##, ##P(r,t)## and ##M(r,t)##.

Scenario 3: solving for ##E(r,t)## and ##B(r,t)## using Maxwell's equations, the auxiliary fields##D(r,t)##, ##H(r,t)##, the total sources ##J(r,t)## and ##\rho(r,t)##, the electric permittivity ##\epsilon## and the magnetic permeability ##\mu##.

Are these approaches all equivalent to each other? It seems that scenario 3 is the most convenient for some reason but I am not sure why...

Thank you!

 

[Meir Achuz]

How you solve things depends on the constitutive equations and the geometry.
You can't pick a method until these are known.

 

[fog37]

Thanks, I see. But I think knowing the functions ##M## and ##P## is a difficult task, more difficult than knowing ##\epsilon## and ##\mu##. Not sure why.

 

[rude man]

M and P are sort of ancillary parameters. They crop up when you study polariztion in detail but after that it's uncommon to work with them.

There are often good reasons for invoking D instead of E and H instead of B: A Maxwell equation is simply written as ## \nabla \cdot \mathbf D = \rho ##; the normal D vector is continuous across dielectric discontinuities, and Gauss's law is similarly simplified using D instead of E: ## \iint_S \mathbf D \cdot d\mathbf A = \iiint_V \nabla \cdot \mathbf D~ dV ##.

H is nice because Ampere's law is most simply stated as ## \oint \mathbf H \cdot \mathbf d \mathcal l = \mathbf I. ## Another Maxwell equation is most simply written as ## \nabla \times \mathbf H = \mathbf j + \frac {\partial \mathbf D} {\partial t} ##. And when you get to magnetic materials, H is an independent parameter with B the dependent, since ## \mu ## varies non-linearly with H while current and H relate linearly in isotropic material. But I've all but forgotten M and P

Some professors like to minimize the number of parameters whenever possible (e.g. Yale's professor R. Shankar) but I like the use of H and D. So much easier to remember the formulas.

 

[hutchphd]

Thanks, I see. But I think knowing the functions ##M## and ##P## is a difficult task, more difficult than knowing ##\epsilon## and ##\mu##. Not sure why.

One needs to be careful to keep segregated in your head the cases where the constitutive relations are simply linear (and so one can recover Maxwell Eguations mutatis mutandis) from cases where things like heresteresis and nonlinearity and life quite a bit more complicated

 

[vanhees71]

One should note that ##\vec{D}## and ##\vec{H}## are auxiliary fields which are introduced to organize the solution of Maxwell's equations. Their physical meaning is not a priory and generally clear but depends on how you split the total charges and currents into internal and external pieces, i.e., you can always shuffle the effects of the medium in electromagnetism between fields and sources. In a sense ##\vec{D}## and ##\vec{H}## are like "potentials" for the part of the microscopic charges and currents taken as "external" in your calculation. For details of this point of view see (open access):

https://doi.org/10.1088/1361-6404/ab009c

 

[fog37]

Thank you. Helpful discussion. I see your points. Here a quick summary for beginners like me.

As far as source terms, we can categorize them into free current/charge sources (due to freely moving charges) and induced bound current/charge sources.

Free Conduction Sources
Due to freely moving electric charges:

• Free volume charge ##\rho_f##
• Free surface charge ##\sigma_f##
• Free volume current density ##\bf{J}_f##
• Free surface current density ##\bf{K}_f##

Electric and Magnetic Bound Sources
Terms related to polarization ##\bf{P}##:

• Induced surface polarization charge density ##\sigma_p = - \bf{P} \cdot \bf{n} ## (true for all types of media)
• Induced volume polarization charge density ##\rho_p = - \nabla \cdot \bf{P}## (true for all types of media)
• Induced volume polarization current density ##\bf{J}_p## (equal to ##\frac {\partial \bf{E}}{\partial t}## only for linear media?)
• Induced volume polarization current density ##\bf{J}_p## ?
• Induced surface polarization current density ##\bf{K}_p## ?

Terms related to magnetization ##\bf{M}##:

• Induced surface magnetization charge density ##\sigma_m## ?
• Induced volume magnetization charge density ##\rho_m## ?
• Induced magnetic surface current density ##\bf{K}_m = \bf{M} \times \bf{n}## (true for all types of media)
• Induced magnetic volume current density ##\bf{J}_m = \nabla \times \bf{M}## (true for all types of media)

Equations

• 4 Maxwell's equations
• 1 Total charge continuity equation involving ##\rho_{total} = \rho_c + \rho_p + \rho_m##
• 2 Constitutive equations relating ##\bf{E}, \bf{B}, \bf{H}, \bf{D}## through the parameters ##\epsilon## and##\mu##

Any corrections?

 

[rude man]

Free Conduction Sources
Due to freely moving electric charges:

• Free volume charge ##\rho_f##
• Free surface charge ##\sigma_f##
• Free volume current density ##\bf{J}_f##
• Free surface current density ##\bf{K}_f##

Electric and Magnetic Bound Sources
Terms related to polarization ##\bf{P}##:

• Induced surface polarization charge density ##\sigma_p = - \bf{P} \cdot \bf{n} ## (true for all types of media)
• Induced volume polarization charge density ##\rho_p = - \nabla \cdot \bf{P}## (true for all types of media)
• Induced volume polarization current density ##\bf{J}_p## (equal to ##\frac {\partial \bf{E}}{\partial t}## only for linear media?)
• Induced volume polarization current density ##\bf{J}_p## ?
• Induced surface polarization current density ##\bf{K}_p## ?

• I think maybe you're a bit too concerned with the idea of compiling an exhaustive table of electric vectors.

So I'm going to cheat a bit and have you look at a page I scanned from my Resnick & Halliday textbook (see attached). Not exhaustive in terminology but really all you need to solve any problem in isotropic matter (in non-isotropic matter the D, P and E vectors can have different magnitudes & direcions; you will in all likelihood not encounter these unless you take advanced courses).

• nduced surface magnetization charge density ##\sigma_m## ?
• Induced volume magnetization charge density ##\rho_m## ?
• Induced magnetic surface current density ##\bf{K}_m = \bf{M} \times \bf{n}## (true for all types of media)
• Induced magnetic volume current density ##\bf{J}_m = \nabla \times \bf{M}## (true for all types of media)

• Here things get rough. These are not commonly used terms. There is no such thing as "magnetization charge".
• 
  
• There are however close analogies between electric and magnetic vectors:
• ## \bf E \leftrightarrow \bf H ##
• ## \bf D \leftrightarrow \bf B ##
• ## \bf P \leftrightarrow \bf M ##
• ## \bf B = \mu_0 (\bf H + \bf M) ##
• etc. Again, look at the attached file. BTW, the textbook uses the less encountered ## \kappa_m ## and ## \kappa ## for ## \mu_0 ## and ##\epsilon_0 ## resp.
• As for the rest, I would emphasize only your

• [*]2 Constitutive equations relating ##\bf{E}, \bf{B}, \bf{H}, \bf{D}## through the parameters ##\epsilon## and##\mu##
  [*]

• Hope that helps.
• SORRY, FORGOT THE ATTACHMENT

 

[rude man]

I could have added two other basic equations for electricity:

## \sigma = \bf D \cdot \hat n ## ( ## \hat n ## = normal to surface)
## \bf j = \sigma \bf E ## (Ohm's law for fields; ## \bf j ## = current density

 

[vanhees71]

Thank you. Helpful discussion. I see your points. Here a quick summary for beginners like me.

As far as source terms, we can categorize them into free current/charge sources (due to freely moving charges) and induced bound current/charge sources.

Free Conduction Sources
Due to freely moving electric charges:

• Free volume charge ##\rho_f##
• Free surface charge ##\sigma_f##
• Free volume current density ##\bf{J}_f##
• Free surface current density ##\bf{K}_f##

Electric and Magnetic Bound Sources
Terms related to polarization ##\bf{P}##:

• Induced surface polarization charge density ##\sigma_p = - \bf{P} \cdot \bf{n} ## (true for all types of media)
• Induced volume polarization charge density ##\rho_p = - \nabla \cdot \bf{P}## (true for all types of media)
• Induced volume polarization current density ##\bf{J}_p## (equal to ##\frac {\partial \bf{E}}{\partial t}## only for linear media?)
• Induced volume polarization current density ##\bf{J}_p## ?
• Induced surface polarization current density ##\bf{K}_p## ?

Terms related to magnetization ##\bf{M}##:

• Induced surface magnetization charge density ##\sigma_m## ?
• Induced volume magnetization charge density ##\rho_m## ?
• Induced magnetic surface current density ##\bf{K}_m = \bf{M} \times \bf{n}## (true for all types of media)
• Induced magnetic volume current density ##\bf{J}_m = \nabla \times \bf{M}## (true for all types of media)

Equations

• 4 Maxwell's equations
• 1 Total charge continuity equation involving ##\rho_{total} = \rho_c + \rho_p + \rho_m##
• 2 Constitutive equations relating ##\bf{E}, \bf{B}, \bf{H}, \bf{D}## through the parameters ##\epsilon## and##\mu##

Any corrections?

I'm not sure what you mean by ##\rho_m## and ##\vec{j}_m##. There are no magnetic charges in standard Maxwell theory, and so far no such thing as a magnetic monopole has been discovered (except as quasi particles in some materials like "spin ice").

Then you should be aware that you have to be careful about double counting. Take the magnetic sector as an example. You can work either with magnetization ##\vec{M}## or with the equivalent current ##\vec{J}_m=\vec{\nabla} \times \vec{M}##. The point is that in the definition of ##\rho_b## and ##\vec{j}_b## (usually handled in terms of the "fields" ##\vec{P}## and ##\vec{M}##) on the one hand and ##\rho_f## and ##\vec{j}_f## (usually taken as "sources"). You can equivalently shuffle these contributions of the medium back and forth among them, changing with it also what you define as ##\vec{D}## and ##\vec{H}##. The physically observable phenomena of course stay the same at the end.

It's of course also changing the constitutive relations. In the usually used linear-response theory you have different permittivity, permeability, and electric-conductivity relations, but the final result for observables stays again the same (within the approximation made and model used).

 

[fog37]

Thank you.

I guess one of my dilemmas is: when solving a problem with Maxwell'equations, is it more convenient to work with the electric permittivity ##\epsilon## instead of ##\bf{P}## (same reasoning goes for ##\mu## and ##\bf{M}##). Is the permittivity ##\epsilon## easier to experimentally determine than ##\bf{P}## and mathematically more convenient?

 

[fog37]

No matter how complex a material is, my understanding is that the following relations do not hold for any material: $$ \bf{D} = \epsilon_0 \bf{E} + \bf{P}$$ $$ \bf{H} = \frac {\bf{B}}{\mu_0} - \bf{M}$$
(for example, there are materials called bianisotropic for which ##\bf{D}## depends on both ##\bf{E}## and ##\bf{H}##).

The simplest type of dielectric material is:

• Linear
• Homogeneous (no spatial variation)
• Isotropic ( no variation with direction)
• Without memory (field at instant ##t## does not depend on field at previous instants)
• Temporally nondispersive (properties are not frequency dependent)
• Spatially non dispersive (for ex., polarization ##\bf{P}## at a spatial point depends only on field at that same point and not on the field values at the surrounding points)
• Time steady (constant in time)
• Stationary (not moving relative to the observer)

In that case, the parameters ##\epsilon## and ##\mu## are simply constants.

I wondering about a specific material case, one whose permittivity ##\epsilon (t)## is a function of time.
Taking the Fourier transform of ##\epsilon (t)## gives ##\epsilon (\omega)##, which makes the permittivity appear to be a function of frequency ##\omega##. Does that mean that a time-varying material is automatically a dispersive material?
I know there are materials which are frequency dependent without being time-varying...Glass, for example, is dispersive but its properties do not change with time.

 

[vanhees71]

Sure, it depends on the material, how the constitutive relations look in detail. A very good book on macroscopic electrodynamics is Landau&Lifshitz vol. 8.

 

[rude man]

Thank you.

I guess one of my dilemmas is: when solving a problem with Maxwell'equations, is it more convenient to work with the electric permittivity ##\epsilon## instead of ##\bf{P}## (same reasoning goes for ##\mu## and ##\bf{M}##). Is the permittivity ##\epsilon## easier to experimentally determine than ##\bf{P}## and mathematically more convenient?

I don't kow about experimental ease but the answer to your first question is 'yes'.

 

[rude man]

No matter how complex a material is, my understanding is that the following relations do not hold for any material:
$$ \bf{D} = \epsilon_0 \bf{E} + \bf{P}$$ $$
\bf{H} = \frac {\bf{B}}{\mu_0} - \bf{M}$$

I think you meant to say "some materials"?

The simplest type of dielectric material is:

• Linear
• Homogeneous (no spatial variation)
• Isotropic ( no variation with direction)
• Without memory (field at instant ##t## does not depend on field at previous instants)
• Temporally nondispersive (properties are not frequency dependent)
• Spatially non dispersive (for ex., polarization ##\bf{P}## at a spatial point depends only on field at that same point and not on the field values at the surrounding points)
• Time steady (constant in time)
• Stationary (not moving relative to the observer)

In that case, the parameters ##\epsilon## and ##\mu## are simply constants.

Not all items in your list preclude constant ## \epsilon ## and ## \mu ##.
In any case you are wandering beyond an introductory physics.

[/QUOTE]

 

[fog37]

Thanks rude man. Yes, I meant "some materials" for the relation ##D = \epsilon_0 E + P## which is only applicable to linear material since ##D## is directly proportional to ##E##.

However, the different relation ##D= \epsilon E## also shows both vectors to be linearly related. But using that relation, I guess the two vectors may not be linearly related depending on ##\epsilon##...

 

[DrDu]

I just stumbled by chance into this thread which is by now a little bit old, but due to the importance of the subject, I want to comment nevertheless.
The point behind polarization P and magnetization is that they arise as a mathematical consequence of local charge conservation. If ##j=(\rho,j_x,j_y,j_z)^T## is the charge-current density four vector, then its divergence vanishes, which expresses charge conservation.
Now a vector, whose divergence vanishes, can be expressed as the derivative of an antisymmetric tensor ##\Pi##, the tensor of polarization and magnetization by Stokes theorem.
Namely, it's components ##\Pi_{0i}=P_i## and the ##\Pi_{ij}=\epsilon_{ijk}M_k##.
Obviously the four components of the charge-current density vector are overparametrized in terms of the 6 components of the polarization-magnetization tensor as this tensor is only fixed up to a tensor whose derivative vanishes. So, for example in optics, it is customary to express ##j## in terms of only ##P## with no magnetization. This is possible as all frequency dependent magnetic effects can be encoded in ##P## alone. This includes also the mentioned bianisotropic materials.
A further important point is that these relations do not involve any need for taking macroscopical averages.
The main advantage of using ##P## instead of ##j## is that for ##P## approximately local material equations often hold.
If a separation of charges is meaningful, so that each component is conserved on its own (like free and bound charges), it sometimes makes sense to express only the bound charges in terms of the polarization vector. In other applications, also free charges are treated via the equivalent polarization, for example in the Lindhard model of the electron gas.