Formula: Ohm's Law Electric Current Density    Electric field (E field)    Electrical Conductivity

Formula: Ohm's Law
Material in an Electric Field and Conductivity as a Second Rank Tensor
Material in an Electric Field and Conductivity as a Zero Rank Tensor

Electric Current Density

The electric current density is generally a three-dimensional vector and describes how much electric current traverses a given cross-sectional area: $$ \class{red}{\boldsymbol j} ~=~ \begin{bmatrix} \class{red}{j_{1}} \\ \class{red}{j_{2}} \\ \class{red}{j_{3}} \end{bmatrix} $$

Electric field (E field)

External electric field in which a conductive material is placed. This E-field generally has three components and it triggers an electric current: $$ \class{purple}{\boldsymbol{E}} ~=~ \begin{bmatrix} \class{purple}{E_{1}} \\ \class{purple}{E_{2}} \\ \class{purple}{E_{3}} \end{bmatrix} $$

Electrical Conductivity

The (specific) electrical conductivity describes how easily a material can conduct electric current.

In isotropic materials, conductivity is an ordinary number (zero rank tensor). In anisotropic materials, on the other hand, the conductivity is a matrix (second rank tensor): $$ \sigma ~=~ \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} $$

The reciprocal (or inverse) \( \sigma^{-1} \) of the electrical conductivity is the electrical resistivity: \( \rho = \sigma^{-1} \).

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