Maxwell Stress Tensor

1. Sep 26, 2008

Ben473

Would someone please be able to run me through the different components of the Maxwell Stress Tensor equation.

$$T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} E^2 \right)$$

I dont understand some of it and would be grateful if someone ran me through.

Thanks,

Ben

2. Sep 27, 2008

Andy Resnick

What part do you not understand? Or better, what part *do* you understand?

3. Sep 28, 2008

Ben473

Hi Andy,

What I fully understand is the Electric and Magnetic constants.
Im not so sure on the Kronecker Delta.
Everything else is a bit 'iffy'!

I presume that the i and j indices are unit vectors. Would I be right in saying this?

What I was really asking for (should have been more specific in the original post) was the different units the different omponents are measured in (e.g. Teslas etc.) as well as an example that ran through how to do the calculation.

Thanks,

Ben.

4. Sep 28, 2008

Andy Resnick

Ben,

I'm not really sure what the natural units of E and H are- 'E' can be Volts/meter, for example, but then I don't know what the correct unit for B is.

Graphically, I think of the stress tensor as the surface of a cube, with each face of the cube having three directions- 1 normal to the face, the other two tangential. The normal component is like a pressure, and correspond to T_ii (i = 1, 2, 3) while the other two components are like shear terms. It's easy to picture in Cartesian coordinates, anyway.

5. Sep 28, 2008

Staff: Mentor

i and j are just indices. $\delta_{ij} = 1$ if i = j, otherwise it equals 0. For example, $\delta_{22} = 1$ and $\delta_{13} = 0$. Therefore,

$$T_{22} = \epsilon_0 \left( E_2^2 - \frac{1}{2} E^2 \right) + \frac{1}{\mu_0} \left( B_2^2 - \frac{1}{2} B^2 \right)$$

$$T_{13} = \epsilon_0 E_1 E_3 + \frac{1}{\mu_0} B_1 B_3$$

(oops. I had to correct the second equation. Forgot about $\delta_{13} = 0$. )

Last edited: Sep 28, 2008
6. Sep 28, 2008

Ben473

Thanks Alex and JtBell.

That really helps. I think I understand it now.

But if I was to do this on a real life object, how would I work out the i and j indices?

Ben.

7. Sep 28, 2008

Staff: Mentor

Depends on what you need for a particular calculation. Indices 1,2,3 are the x,y,z components of $\vec E$ and $\vec B$. Often you deal with all nine combinations at once, in a matrix:

$$\left( {\begin{array}{*{20}c} {T_{11} } & {T_{12} } & {T_{13} } \\ {T_{21} } & {T_{22} } & {T_{23} } \\ {T_{31} } & {T_{32} } & {T_{33} } \\ \end{array}} \right)$$

Or have I missed the point of your question?

8. Sep 28, 2008

Ben473

Thanks,

I get it now.

Ben.

9. Sep 28, 2008

tiny-tim

Hi Ben and Andy!

B is in teslas or weber per metre² or volt-seconds per metre².

For more details, see electric units in PF Library.

10. Sep 28, 2008

Ben473

Thanks Tiny-Tim,

I had an inkling that this was the case, but i wasnt sure.

Ben.