# Maxwell stress tensor for a nonlinear media

by Hassan2
Tags: maxwell stress
 P: 396 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?
 Sci Advisor P: 5,410 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...ki_controversy
 P: 396 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?
P: 5,410

## Maxwell stress tensor for a nonlinear media

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).
HW Helper
P: 1,839
 Quote by 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?
The medium must be linear to drive a Maxwell stress tensor.
P: 5,410
 Quote by Meir Achuz The medium must be linear to drive a Maxwell stress tensor.
Why do you say that?
 Sci Advisor HW Helper P: 1,839 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.
 Sci Advisor P: 5,410 Er... where did you see that derivation? It seems unnecessarily restrictive.
 Sci Advisor HW Helper P: 1,839 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?
 P: 396 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 $p_{em}=\int BdH$
P: 5,410
 Quote by 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?
Nonlinear magnetic medium:
http://pof.aip.org/resource/1/phfle6...sAuthorized=no

Seems to allow for nonlinear constitutive relations, but only explicitly presents results for linear and quasi-linear materials:

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.
HW Helper
P: 1,839
 Quote by Andy Resnick Nonlinear magnetic medium: http://pof.aip.org/resource/1/phfle6...sAuthorized=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=j...rqlIIg&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.
 Sci Advisor P: 5,410 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.
 Sci Advisor HW Helper P: 1,839 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.
 Sci Advisor P: 5,410 I don't understand your objection- my definition of the stress tensor?
 Sci Advisor HW Helper P: 1,839 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.
 Sci Advisor P: 5,410 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 ?
 Sci Advisor HW Helper P: 1,839 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.

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