In an anisotropic medium, the current density ( $J$ ) and the electric field ( $E$ ) are not always parallel because the conductivity tensor scales the field's components differently depending on their orientation. This means that a straightforward relationship like Ohm's law in isotropic materials does not hold universally. The two vectors only become perfectly aligned when the electric field is applied either parallel or perpendicular to the material's principal axis of conductivity, which highlights that current flows in the same direction as the electric field only when the field is aligned with the material's inherent symmetries.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Relationship between $E$ and $J$ : In an anisotropic medium, the current density $J$ is not necessarily parallel to the electric field $E$. The conductivity tensor acts as a linear transformation, rotating and scaling the electric field vector to produce the current density vector.
• Decomposition of Vectors: By breaking down the electric field $E$ into components parallel ( $n$ ) and perpendicular ( $t$ ) to the principal axis, we can easily see how each component is scaled by a different value: $\sigma_0+\lambda$ for the parallel component and $\sigma_0$ for the perpendicular component.
• General Misalignment: In general, for an anisotropic medium with $\lambda \neq 0$, the vectors $E$ and $J$ are not parallel. The angle between them depends on the orientation of $E$ relative to the principal axis $n$.
• Special Cases: $E$ and $J$ are only parallel in two specific scenarios:
• When $E$ is parallel to the principal axis $n$ (i.e., $\alpha=0, \pi$ ).
• When $E$ is perpendicular to the principal axis $n$ (i.e., $\alpha=\pi / 2,3 \pi / 2$ ). This demonstrates that for an anisotropic material, the current flows in the same direction as the applied field only when the field is aligned with one of the material's inherent axes of symmetry.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium

🧄Proof and Derivation-1

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