Electrical Resistivity and Conductivity - Tensor Equations For Anisotropic Materials

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:

    Artists, whatever their medium, make selections from the abounding materials of life, and organize these selections into works that are under the control of the artist.... In relation to the inclusiveness and literally endless intricacy of life, art is arbitrary, symbolic and abstracted. That is its value and the source of its own kind of order and coherence.
    Jane Jacobs (b. 1916)