# Material in an Electric Field and Conductivity as a Second Rank Tensor

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A conductive material is in an external electric field $$\class{purple}{\boldsymbol{E}}$$, generating an electric current density $$\class{red}{\boldsymbol{j}}$$ in the material.

In this case the material is anisotropic. This property is characterized by an electrical conductivity $$\sigma$$ as a 2nd rank tensor (a matrix). Thus, the current density $$\class{red}{\boldsymbol{j}}$$ generally depends on all three components $$\class{purple}{E_1}$$, $$\class{purple}{E_2}$$, $$\class{purple}{E_3}$$ of the E-field: $$\begin{bmatrix} \class{red}{j_{1}} \\ \class{red}{j_{2}} \\ \class{red}{j_{3}} \end{bmatrix} ~=~ \begin{bmatrix} \sigma_{11}\class{purple}{E_1} + \sigma_{12}\class{purple}{E_2} + \sigma_{13}\class{purple}{E_1} \\ \sigma_{21}\class{purple}{E_1} + \sigma_{22}\class{purple}{E_2} + \sigma_{23}\class{purple}{E_1} \\ \sigma_{31}\class{purple}{E_3} + \sigma_{32}\class{purple}{E_3} + \sigma_{33}\class{purple}{E_3} \end{bmatrix}$$

In an anisotropic material, the current density does not point in the same direction as the electric field. The charges therefore do not flow exactly in the direction of the E-field.

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