# Maxwell stress tensor for a nonlinear media

1. May 9, 2012

### 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?

2. May 10, 2012

### Andy Resnick

3. May 10, 2012

### 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?

4. May 10, 2012

### 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).

5. May 11, 2012

### Meir Achuz

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

6. May 11, 2012

### Andy Resnick

Why do you say that?

7. May 11, 2012

### 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.

8. May 11, 2012

### Andy Resnick

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

9. May 11, 2012

### 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?

10. May 11, 2012

### 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 $p_{em}=\int BdH$

Last edited: May 12, 2012
11. May 11, 2012

### Andy Resnick

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

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.

Last edited by a moderator: May 6, 2017
12. May 12, 2012

### Meir Achuz

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.

Last edited by a moderator: May 6, 2017
13. May 12, 2012

### 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.

14. May 12, 2012

### 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.

Last edited: May 12, 2012
15. May 12, 2012

### Andy Resnick

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

16. May 12, 2012

### 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.

17. May 12, 2012

### 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 ?

18. May 13, 2012

### 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.

19. May 13, 2012

### 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?

20. May 13, 2012

### 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 lets 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.