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.
