Understanding of Maxwell's Stress Tensor

• vwishndaetr
In summary, the conversation discusses the stress tensor and its components, questioning what is being squared in the squared electric and magnetic field terms. It is determined that the terms represent the dot product of the vector field with itself, or simply the sum of the squares of the individual components. The individual asking the questions acknowledges feeling silly for asking, but still wants clarification.
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.

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

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

Born2bwire said:
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.

vwishndaetr said:
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

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

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.

1. What is Maxwell's stress tensor?

Maxwell's stress tensor is a mathematical tool used in the study of electromagnetism. It is a second-order tensor that describes the stress, or force per unit area, exerted on a material by an electromagnetic field.

2. How is Maxwell's stress tensor derived?

Maxwell's stress tensor is derived from Maxwell's equations, which are a set of equations that describe the behavior of electric and magnetic fields. The stress tensor is derived by taking the Maxwell stress tensor, which describes the stress on individual charges, and extending it to describe the stress on a continuous medium.

3. What does the stress tensor tell us about electromagnetic fields?

The stress tensor tells us about the direction and magnitude of the stress exerted by an electromagnetic field on a material. It can also give us information about the flow of energy and momentum within the electromagnetic field.

4. How is Maxwell's stress tensor used in practical applications?

Maxwell's stress tensor is used in various applications, such as in the design of electrical machines and devices, in the development of materials for electromagnetic shielding, and in the study of plasma physics. It is also used in the construction of models for electromagnetic phenomena, such as the propagation of electromagnetic waves.

5. Are there any limitations to Maxwell's stress tensor?

While Maxwell's stress tensor is a powerful tool for understanding the behavior of electromagnetic fields, it does have some limitations. It assumes a linear, isotropic material and does not account for any non-linear or anisotropic effects. It also does not take into account quantum effects, which are important at very small scales.

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