Image »Material in an Electric Field and Conductivity as a Zero Rank Tensor« download
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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 isotropic. This property is characterized by a electrical conductivity \( \sigma \) as a 0th rank tensor (a number). Thus, the E-field is simply scaled by the number \( \sigma \) to obtain the current density: $$ \begin{bmatrix} \class{red}{j_{1}} \\ \class{red}{j_{2}} \\ \class{red}{j_{3}} \end{bmatrix} ~=~ \begin{bmatrix} \sigma \class{purple}{E_{1}} \\ \sigma \class{purple}{E_{2}} \\ \sigma \class{purple}{E_{3}} \end{bmatrix} $$
In an isotropic material, the current density points in the same direction as the electric field. The charges therefore flow in the direction of the E-field.
