Understanding of Maxwell's Stress Tensor

[vwishndaetr]

Was a bit fuzzy as to whether this better fit HW or here, but since there really is no question associated with it, figured this made a bit more sense.

I have a couple basic questions about the stress tensor:

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

For the component electric and magnetic fields (denoted with "i" and "j" indices), they are what they are. What ever the indice at the time, that particular component fills it in. But for both the squared Electric and Magnetic Field terms, what is being squared? Is it the magnitude of the net field squared?

Might be a bit silly to be dealing with Tensors and asking such silly questions, but still want to know.

Thanks.

 

[Born2bwire]

It implicitly means the dot product of the vector field with itself.

$$E^2 = \mathbf{E}\cdot\mathbf{E}$$

 

[vwishndaetr]

It implicitly means the dot product of the vector field with itself.

$$E^2 = \mathbf{E}\cdot\mathbf{E}$$

So it'd be,

$${E^2} = \sqrt {{E_x}^2 + {E_y}^2 + {E_z}^2} \cdot \sqrt {{E_x}^2 + {E_y}^2 + {E_z}^2}$$

I feel so dumb asking this. Thanks again.

 

[Born2bwire]

So it'd be,

$${E^2} = \sqrt {{E_x}^2 + {E_y}^2 + {E_z}^2} \cdot \sqrt {{E_x}^2 + {E_y}^2 + {E_z}^2}$$

I feel so dumb asking this. Thanks again.

Or more simply just

$${E^2} = {E_x}^2 + {E_y}^2 + {E_z}^2$$

 

[sardar]

I can clarify that the Maxwell's stress tensor is a mathematical representation of the stress exerted by an electromagnetic field on a material medium. It is a tensor because it has components in three dimensions and can be transformed according to the laws of tensor algebra.

In the equation you provided, the components of the stress tensor are determined by the electric and magnetic fields at a given point in space. The terms involving the squared electric and magnetic fields represent the magnitude of the fields squared, as you correctly stated. This is because the stress exerted by the fields on the material is directly proportional to their magnitude.

It is not silly to ask questions about tensors, as they can be quite complex and require a deep understanding of mathematical concepts. It is important to have a clear understanding of the equations and terms involved in order to accurately interpret the results. I hope this helps clarify your understanding of the Maxwell's stress tensor.