How to Determine Current Directions from an Electrical Conductivity Tensor?

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Discussion Overview

The discussion revolves around determining current directions from an electrical conductivity tensor, focusing on the mathematical properties of the tensor, particularly its eigenvalues and eigenvectors. Participants explore how these properties relate to current flow directions and the absence of flow in certain directions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests finding a coordinate system where the conductivity tensor is diagonal, indicating that the eigenvectors correspond to the directions of current flow and the eigenvalues indicate the magnitude of flow.
  • Another participant provides a specific example using a conductivity tensor matrix, noting that the eigenvalue of 0 corresponds to a direction of no flow, represented by the vector (1,1,1).
  • There is a discussion about the implications of having multiple eigenvectors corresponding to the same eigenvalue (3 in this case), raising the question of whether flow is greatest in all those directions.
  • A further clarification indicates that the eigenvectors corresponding to the eigenvalue of 3 satisfy the equation x + y + z = 0, suggesting a two-dimensional eigenspace where flow is uniform across that plane.
  • Another participant inquires about finding the electric field components related to the given electric conductivity tensor.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the implications of multiple eigenvectors for the same eigenvalue, indicating a lack of consensus on how to interpret the flow directions associated with those eigenvectors.

Contextual Notes

The discussion does not resolve the mathematical steps involved in determining the electric field from the conductivity tensor, nor does it clarify the assumptions underlying the interpretation of eigenvalues and eigenvectors.

jsmith12
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If I'm given the components of an electrical conductivity tensor, how can I work out the directions in which the current is biggest and the directions in which no current flows?
 
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Find a coordinate system in which the tensor can be written as a diagonal matrix. (Writing the original tensor as a matrix, your coordinate axes will be the eigenvectors of the original matrix, the flow along those axes the eigenvalues.) The flow will be greatest in the direction of the eigenvector corresponding to the largest eigenvalue, 0 in any eigenspace corresponding to 0 eigenvalue.
 
Ok thanks. But if I'm trying to work this out for the matrix

2 -1 -1
-1 2 -1
-1 -1 2

which has eigenvalues 0, 3, 3.
The 0 eigenvalue gives (1,1,1) so that's the direction of no flow, but with eigenvalue 3 you can get lots of different eigenvectors so is the flow greatest in all of those directions?
 
jsmith12 said:
Ok thanks. But if I'm trying to work this out for the matrix

2 -1 -1
-1 2 -1
-1 -1 2

which has eigenvalues 0, 3, 3.
The 0 eigenvalue gives (1,1,1) so that's the direction of no flow, but with eigenvalue 3 you can get lots of different eigenvectors so is the flow greatest in all of those directions?
The only eigenvectors corresponding to eigenvalue 3 all satisfy x+ y+ z= 0. That means that there is a 2 dimensional "eigenspace" of covering the plane x+ y+ z= 0. The flow is the same in all directions on that plane.
 
if the electric conductivity tensor is given how can i find the electric field with its related components
 

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