Formula: Ohm's Law
$$\class{red}{\boldsymbol{j}} ~=~ \sigma \, \class{purple}{\boldsymbol{E}}$$
$$\class{red}{\boldsymbol{j}} ~=~ \sigma \, \class{purple}{\boldsymbol{E}}$$
$$\class{purple}{\boldsymbol{E}} ~=~ {\sigma}^{-1} \, \class{red}{\boldsymbol{j}}$$
Electric Current Density
$$ \class{red}{\boldsymbol j} $$ Unit $$ \frac{ \mathrm A }{ \mathrm{m}^2 } $$
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)
$$ \class{purple}{\boldsymbol E} $$ Unit $$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$
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
$$ \sigma $$ Unit $$ \frac{1}{ \mathrm{\Omega} \, \mathrm{m} } = \frac{ \mathrm{s}^3 \, \mathrm{A}^2 }{ \mathrm{m}^3 \, \mathrm{kg} } $$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: \( \rho = \sigma^{-1} \).