Tensor Equations for Anisotropic Materials

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SUMMARY

The discussion centers on tensor equations for anisotropic materials, specifically the relationships defined by J=σE and E=ρJ. The participant questions the validity of the expression J_i=σ_{ii}E_i and its physical implications. It is confirmed that J_i=σ_{ij}E_j is the correct representation, utilizing Einstein summation convention, where repeated indices imply summation. The participant seeks clarification on whether J_i can be expressed as a combination of J_i=σ_{ij}E_j and σ_{ii}E_i.

PREREQUISITES
  • Understanding of tensor notation and operations
  • Familiarity with anisotropic material properties
  • Knowledge of classical electromagnetism principles
  • Proficiency in Einstein summation convention
NEXT STEPS
  • Study the derivation of tensor equations in anisotropic materials
  • Explore the physical significance of the conductivity tensor σ in electromagnetism
  • Learn about the implications of anisotropic versus isotropic materials
  • Investigate advanced topics in electromagnetism, such as Maxwell's equations in tensor form
USEFUL FOR

Students and professionals in physics, particularly those focusing on electromagnetism and material science, will benefit from this discussion on tensor equations and their application to anisotropic materials.

KayDee01
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Hi,
I am in the middle of revising for and a classical electromagnetism exam, and I've hit a wall when it comes to tensor equations.

I know that for anisotropic materials: J=σE and E=ρJ
And that in component form the first equation can be written as J_i = σ_{ij} E_j

What I'm wondering is, does J_i=σ_{ii}E_i make mathematical sense and if so does it make physical sense when applied to anisotropic materials. The second part of my question is, if this does make sense, is the final value of J_i written as: J_i=σ_{ij} E_j+σ_{ii}E_i or is there another way to combine the two values of J_i?
 
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The component form equation means that J_i = \sum_j \sigma_{ij} E_j, the notation is known as Einstein summing convention. You don't bother to write down the summation sign, but just implicitly assume that the indices repeated on one side of the equation are summed over.
 

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