Constitutive relation D = ε0 εr E — scope of validity State whether the following is true or false: “The scalar relation D = ε0 εr E applies only to isotropic, linear dielectrics. For anisotropic media, ε must be treated as a tensor.”

Introduction / Context:
Constitutive relations link electric flux density D and electric field E. In many practical cases, a scalar permittivity ε = ε0 εr suffices, but anisotropic materials require a more general description to capture direction-dependent response.

Given Data / Assumptions:

• Linear response (no field-strength dependence of coefficients).
• Homogeneous materials considered for simplicity.
• Possibility of anisotropy (crystals, composites).

Concept / Approach:
In isotropic materials, permittivity is the same in all directions and D = ε0 εr E with εr a scalar. In anisotropic media, the proportionality is direction-dependent, and the correct form is D = ε̄ · E, where ε̄ is a second-rank tensor (matrix). Consequently, D need not be parallel to E in anisotropic materials, and components couple through off-diagonal terms in ε̄.

Step-by-Step Solution:
Identify isotropy → scalar ε suffices → D parallel to E.Identify anisotropy → tensor ε̄ needed → D = ε̄ · E, generally not collinear.Hence the statement is true as written.

Verification / Alternative check:
Electro-optic crystals (e.g., calcite) exhibit birefringence precisely because ε̄ is anisotropic; wave propagation depends on direction, confirming tensorial permittivity.

Why Other Options Are Wrong:
“False” disregards well-established anisotropy. “True only in vacuum” is incorrect; many isotropic materials satisfy the scalar relation. “D and E are always parallel” fails in anisotropic media. Frequency limitations are not the main distinction here; linearity and isotropy are.

Common Pitfalls:

• Assuming isotropy for all dielectrics; many crystals and engineered composites are anisotropic.
• Ignoring coordinate transformation properties of tensors.

Final Answer:
True