Maxwell stress tensor for a nonlinear media

[Hassan2]

Hi all,

It seems to me that the derivation of Maxwell stress tensor is independent of the permeability of the media or the nonliterary of its B-H relation. By this I mean that we use μ₀ in the equations rather than μ. Would you please confirm that?

 

[Andy Resnick]

AFAIK, the stress tensor contains E and D, B and H - so material properties are included. But there's still some disagreement about the proper form of the relevant equations:

http://en.wikipedia.org/wiki/Abraham–Minkowski_controversy

 

[Hassan2]

Many thanks.

In wikipedia the derivation is for vacuum. Of course when we want to calculate the total force on a body ( even ferromagnetic) we do the surface integration of Maxwell stress tensor in the air region, hence the material property is not involved.

If I understood correctly, the general case tensor which as you said contains E and D, B and H is called Minkowsky stress tensor.

I have a question about the application now. The tensor is discontinuous when we have different media so its divergence is not differentiable. Can we still use divergence theorem and reduce the volume integral to a surface integral for force calculation?

 

[Andy Resnick]

Interfaces (surfaces of discontinuity) can be handled straightforwardly. For example, see the Reynolds Transport Theorem. If there is a discontinuous change in the stress tensor, the dividing surface provides a 'jump condition', meaning the dividing surface has properties distinct from the bulk. In the context of electromagnetism, these most likely correspond to surface charges and currents.

Most of the material I have seen relates to magnetohydrodynamics (Alfvén discontinuity).

 

[Meir Achuz]

Hi all,

It seems to me that the derivation of Maxwell stress tensor is independent of the permeability of the media or the nonliterary of its B-H relation. By this I mean that we use μ₀ in the equations rather than μ. Would you please confirm that?

The medium must be linear to drive a Maxwell stress tensor.

 

[Andy Resnick]

The medium must be linear to drive a Maxwell stress tensor.

Why do you say that?

 

[Meir Achuz]

In the derivation, there is a grad(D.E) term with D held constant. This can become
(1/2)grad(D.E) only if the medium is linear.

 

[Andy Resnick]

Er... where did you see that derivation? It seems unnecessarily restrictive.

 

[Meir Achuz]

Pauli, Griffiths, and Jackson only derive T without a polarizable medium.
Panofsky & Phillips derive T only for linear media.
Franklin shows it can't be derived for nonlinear media.
Those are the only EM books I have at home.
Do you know of a derivation of T for nonlinear media?

 

[Hassan2]

I have see the following formula for entries of T( for magnetic field only):

$T_{ij}=B_{i}H_{j}-\delta_{ij} p_{em}$

where $p_{em}=\int BdH$

 

[Andy Resnick]

Pauli, Griffiths, and Jackson only derive T without a polarizable medium.
Panofsky & Phillips derive T only for linear media.
Franklin shows it can't be derived for nonlinear media.
Those are the only EM books I have at home.
Do you know of a derivation of T for nonlinear media?

Nonlinear magnetic medium:
http://pof.aip.org/resource/1/phfle6/v21/i3/p034102_s1?isAuthorized=no

Seems to allow for nonlinear constitutive relations, but only explicitly presents results for linear and quasi-linear materials:
http://www.google.com/url?sa=t&rct=...sg=AFQjCNGMEs4Zj2XhRbpZIFlFUWsirqlIIg&cad=rja

I wonder if we are talking about different kinds of nonlinearities- clearly, the polarization of the material P may depend nonlinearly on the field E (Eqn. 5 in the second reference) without causing any problems, and the material may also deform nonlinearly without causing any conceptual difficulty.

 

[Meir Achuz]

Nonlinear magnetic medium:
http://pof.aip.org/resource/1/phfle6/v21/i3/p034102_s1?isAuthorized=no

Seems to allow for nonlinear constitutive relations, but only explicitly presents results for linear and quasi-linear materials:
http://www.google.com/url?sa=t&rct=...sg=AFQjCNGMEs4Zj2XhRbpZIFlFUWsirqlIIg&cad=rja

I wonder if we are talking about different kinds of nonlinearities- clearly, the polarization of the material P may depend nonlinearly on the field E (Eqn. 5 in the second reference) without causing any problems, and the material may also deform nonlinearly without causing any conceptual difficulty.

Try not to say "clearly" when it is not clear that "the polarization of the material P may depend nonlinearly on the field E (Eqn. 5 in the second reference) without causing any problems, and the material may also deform nonlinearly without causing any conceptual difficulty." As far as I can see neither of your references derive the MST. They may use it for nonlinear materials (although I don't see where in either reference), but that is not justified.

The equation I wrote in my first post is simple, and shows the need for linearity. The standard equation in Hassan2's latest post also shows that linearity is required to get the (1/2)B.H that appears in the usual MST. If the MST is written as the integral BdH then linearity is not needed, but that MST would on the past history.

 

[Andy Resnick]

Fair enough, I'm willing to start the derivation: let's first just consider the E and D fields. The material polarization can be written as:

$$P_{i} = \chi^{1}E_{i} + \chi^{2}_{ij}E_{i}E_{j} + \chi^{3}_{ijk}E_{i}E_{j}E_{k}+...$$

There are probably more compact ways to write this, but in any case the field D = (E+P) or something like that. The stress tensor is defined as

$$T_{ij} = E_{i}D_{j}+B_{i}H{j}- 1/2 (ED+ BH)\delta_{ij}$$

so just plug-n-chug from there.

 

[Meir Achuz]

The derivation starts with dp/dt=\int[\rho E + jXB], and then derives
T=DE + BH -(1/2)[D.E+B.H]
You can't just write it down ithout deriving it.

 

[Andy Resnick]

I don't understand your objection- my definition of the stress tensor?

 

[Meir Achuz]

In physics you can't just 'define' things you have to derive them.
Read a textbook or work it out yourself. I've wasted too much time on this.

 

[Andy Resnick]

Hang on- I am honestly trying to understand what you are claiming. Are you saying the Maxwell stress tensor is not

T_ij=E_iD_j+B_iH_j−1/2(ED+BH)δ_ij ?

 

[Meir Achuz]

One more try.
If you look at a textbook, you will see that it DERIVES the MST, and does not just define it out of the air. Your 'definition' cannot be derived for a nonlinear material.
For the case given by Hasan2 in post #10, $$\int{\bf B\cdot dH}$$ only equals
$$\frac{1}{2}{\bf B\cdot H}$$ for a linear material.

 

[Andy Resnick]

That's not true, either- let's start with

$$F = q(E + v x B)$$, which in the continuum approximation goes to

$$F = \rho E + J x B$$

using Maxwell's equations for ponderable media to replace the charge and current densities, we get

$$F = (\nabla\bullet D) E + (\nabla \times H - \frac{\partial D}{\partial t}) \times B$$

and then going through the usual steps we get the Maxwell stress tensor I wrote previously. Gauss's law, Faraday's Law, Ampere's law. and all the other intermediate steps do not require the medium to be linear- or do you claim that nonlinear optics somehow violates Maxwell's laws?

 

[Hassan2]

Thank you both.

In my opinion, in the second equation above, $J$ is not "free current" only, but the sum of free current and material current( other wise your equation doesn't give reluctant forces). Thus the following Maxwell equation holds:

$\nabla \times B = \mu_{0}J$

Now let's focus on the static case and for the magnetic field only



$f=\frac{1}{ \mu_{0}}\nabla \times B \times B$

Would you please derive MST from the above equation? It seems to me that the material properties are not involved at all.

 

[Andy Resnick]

Kind of- the correct formulation of Ampere's law in the presence of matter is:

$$\nabla \times H = J + \frac{\partial D}{\partial t}$$

so your expression for f is something like

$$\nabla \times H \times B$$

Now you need a way to relate H and B- for an isotropic linear material, B = μH. However, the allowable class of constitutive relations is much more general; accordingly we write B =μ_0(H + M) where M is the magnetization (in parallel with D = ε_0(E+P) for the electric field).

In any case, leaving D, E, H, and B separate and going through the usual steps:
http://www.google.com/url?sa=t&rct=...zaWJCQ&usg=AFQjCNFl4AQ4B29kEj3cTJKtl_TsXVZ4MA

the Maxwell Stress Tensor in the presence of matter is given by Eqn 22.21 in the above reference.

Does this help?

 

[Hassan2]

Thanks for the reference. It also proves your point.

But it states that the defined tensor is for the force on free charges only. It means we can't use it to calculate the body force on a piece of iron in a static magnetic field. Now I'm confused as I need to calculate such forces in my finite element code. Some papers do use a tensor ( in post #10) for that and call it Maxwell stress tensor too.

 

[Andy Resnick]

I'm not sure how to respond- clearly the magnetization of iron is well-defined, so the relationship between B and H is straightforward. But as I mentioned earlier, there is still some disagreement about the correct form of the stress tensor. Maybe these will help:

http://www.sciencedirect.com/science/article/pii/002072259290023A

http://www.google.com/url?sa=t&rct=...upzOCQ&usg=AFQjCNEO5Rle29vykcXw8_AoBRhwUQLKsA

http://www.google.com/url?sa=t&rct=...sg=AFQjCNH_GpX1obyUf1OQqs1PpTqXUvPzDA&cad=rja

 

[Meir Achuz]

Kind of- the correct formulation of Ampere's law in the presence of matter is:

$$\nabla \times H = J + \frac{\partial D}{\partial t}$$

so your expression for f is something like

$$\nabla \times H \times B$$

Now you need a way to relate H and B- for an isotropic linear material, B = μH. However, the allowable class of constitutive relations is much more general; accordingly we write B =μ_0(H + M) where M is the magnetization (in parallel with D = ε_0(E+P) for the electric field).

In any case, leaving D, E, H, and B separate and going through the usual steps:
http://www.google.com/url?sa=t&rct=...zaWJCQ&usg=AFQjCNFl4AQ4B29kEj3cTJKtl_TsXVZ4MA

the Maxwell Stress Tensor in the presence of matter is given by Eqn 22.21 in the above reference.

Does this help?

I don't know why I persist, but for one more try: Eq. (22.48) assumes a linear medium.

 

[Meir Achuz]

Thanks for the reference. It also proves your point.

But it states that the defined tensor is for the force on free charges only. It means we can't use it to calculate the body force on a piece of iron in a static magnetic field. Now I'm confused as I need to calculate such forces in my finite element code. Some papers do use a tensor ( in post #10) for that and call it Maxwell stress tensor too.

For iron, you would have to use the form you showed $$\int{\bf B\cdot dH}$$,
but you don't know B(H) for a piece of iron. I think you are better off using
$${\bf F}=-\int d^3r\nabla\cdot{\bf M}+\oint{\bf dS\cdot M}$$.
Try to find M from the geometry and the boundary conditions.

 

[Hassan2]

It is common to use B_H curves obtained from measurement and the integration is done numerically. Since a piece of iron with dimensions of about 10 mm comprises of a large number of magnetic domains with various geometry and random orientations, I think obtaining the microscopic M is practically impossible.

In your equation, What is F? The right hand side seems to be the sum of volume and surface magnetic charge. It is a scalar so it can't be force.

 

[berkeman]

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