# Material in an Electric Field and Conductivity as a Zero 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 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.

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