Polarization of a homogeneous linear dielectric For a homogeneous, linear, isotropic dielectric in an electric field E, is the polarization (dipole moment per unit volume) given by P = ε0 * (εr − 1) * E?

Introduction / Context:
Relating the macroscopic polarization P to the applied electric field E is fundamental in capacitor design, dielectric spectroscopy, and electrostatics. The simplest case is a linear, isotropic, homogeneous (LIH) dielectric.

Given Data / Assumptions:

• Medium is linear (response proportional to field), isotropic (direction-independent), and homogeneous (space-uniform).
• Relative permittivity εr is constant over the field range of interest.
• SI units with ε0 as the vacuum permittivity.

Concept / Approach:
The constitutive relation for a LIH dielectric is D = ε * E, where ε = εr * ε0. Also, D = ε0 * E + P by definition. Eliminating D gives:

ε0 * E + P = εr * ε0 * EP = ε0 * (εr − 1) * E.

This links the macroscopic polarization density to the applied field via the excess permittivity over vacuum.

Step-by-Step Solution:
Start with D = ε * E.Use D = ε0 * E + P.Subtract ε0 * E from both sides: P = (ε − ε0) * E.Replace ε by εr * ε0 to get P = ε0 * (εr − 1) * E.

Verification / Alternative check:
In materials with susceptibility χe, we have P = ε0 * χe * E and εr = 1 + χe, which is identical to the expression above.

Why Other Options Are Wrong:
Anisotropy would replace scalar εr by a tensor, but the scalar LIH relation remains correct for isotropic media. Vacuum has εr = 1, giving P = 0, not a special case invalidating the relation. Frequency restrictions are not inherent to this linear constitutive law (though εr can be dispersive in practice).

Common Pitfalls:
Confusing D with P or forgetting the vacuum contribution ε0 * E when relating D and E.

Final Answer:
True