The analysis demonstrated that the electric susceptibility ($\chi_j^i$) in an anisotropic medium is a rank two tensor. In such media, the polarization vector $\mathbf{P}$ and the electric field vector $\mathbf{E}$ are linearly related by $P^i = \epsilon_0 \chi_j^i E^j$ using the summation convention. The proof relies on showing that $\chi_j^i$ adheres to the mixed-tensor transformation law for a rank two tensor: $\chi_j^{\prime i} = \frac{\partial x^{\prime i}}{\partial x^k} \frac{\partial x^l}{\partial x^{\prime j}} \chi_l^k$. This was achieved by substituting the known contravariant vector transformation laws for $\mathbf{P}$ and $\mathbf{E}$ into the defining constitutive relation and isolating the transformation factor for $\chi_j^i$.
Relationship in Anisotropic Media: In a non-isotropic (anisotropic) medium, the polarization $\mathbf{P}$ is linearly related to the electric field $\mathbf{E}$ by the electric susceptibility $\chi_{j}^{i}$, expressed as $P^{i} = \epsilon_{0} \chi_{j}^{i} E^{j}$ (using the summation convention).
Electric Susceptibility as a Tensor: The main goal of the document is to show that the electric susceptibility $\chi_{j}^{i}$ is a rank two tensor.
Demonstration via Transformation Law: This is proven by demonstrating that $\chi_{j}^{i}$ transforms according to the tensor transformation law when moving from an unprimed to a primed coordinate system.
Vector Transformation Laws: The proof relies on the transformation laws for contravariant vectors:
Derivation Result: By substituting the unprimed relation $P^k=\epsilon_0 \chi_l^k E^l$ and the inverse transformation for $E^l$ into the transformation for $P^{\prime i}$, the components in the primed system are found to be:
$P^{\prime i} = \epsilon_{0} \left( \frac{\partial x^{\prime i}}{\partial x^{k}} \frac{\partial x^{l}}{\partial x^{\prime j}} \chi_{l}^{k} \right) E^{\prime j}$
Tensor Transformation Rule: Comparing this to $P^{\prime i}=\epsilon_0 \chi_j^{\prime i} E^{\prime j}$ yields the transformation law for $\chi_j^i$ :
$\chi_{j}^{\prime i} = \frac{\partial x^{\prime i}}{\partial x^{k}} \frac{\partial x^{l}}{\partial x^{\prime j}} \chi_{l}^{k}$
This is the mixed-tensor transformation law for a rank two tensor, thus confirming that electric susceptibility is a rank two tensor.
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