Maxwell Stress Tensor: Explained for Ben

[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 don't understand some of it and would be grateful if someone ran me through.

Thanks,

Ben

 

[Andy Resnick]

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

 

[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.

 

[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.

 

[jtbell]

Im not so sure on the Kronecker Delta.

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

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 $\delta_{13} = 0$. )

 

[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.

 

[jtbell]

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?

 

[Ben473]

Thanks,

I get it now.

I really appreciate your help.

Ben.

 

[tiny-tim]

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 …

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.

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.

 

[Ben473]

Thanks Tiny-Tim,

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

Ben.