Tensor Equations For Anisotropic Materials
Some materials are anisotropic, meaning they have different properties in different directions. For example, a crystal of graphite consists microscopically of a stack of sheets, and current flows very easily through each sheet, but moves much less easily from one sheet to the next.
For an anisotropic material, it is not generally valid to use the scalar equations
For example, the current may not flow in exactly the same direction as the electric field. Instead, the equations are generalized to the 3D tensor form
where the conductivity σ and resistivity ρ are rank-2 tensors (in other words, 3×3 matrices). The equations are compactly illustrated in component form (using index notation and the summation convention):
The σ and ρ tensors are inverses (in the sense of a matrix inverse). The individual components are not necessarily inverses; for example σxx may not be equal to 1/ρxx.
Read more about this topic: Electrical Resistivity And Conductivity
Famous quotes containing the word materials:
“What is most interesting and valuable in it, however, is not the materials for the history of Pontiac, or Braddock, or the Northwest, which it furnishes; not the annals of the country, but the natural facts, or perennials, which are ever without date. When out of history the truth shall be extracted, it will have shed its dates like withered leaves.”
—Henry David Thoreau (1817–1862)