Polarization–Field Relation for an Insulator For a linear, isotropic insulating material with relative permittivity εr, what is the correct expression for polarization P (dipole moment per unit volume) as a function of electric field E?

Introduction / Context:
In linear dielectrics, polarization is proportional to the applied electric field. The proportionality involves permittivity and susceptibility, relating the microscopic dipole alignment to macroscopic field variables. The correct algebra prevents double-counting εr when switching between D, E, and P relations.

Given Data / Assumptions:

• Linear, isotropic, homogeneous dielectric.
• Constitutive relations: D = ε E, with ε = εr ε0.
• Polarization defined by D = ε0 E + P.

Concept / Approach:

Combine D = ε E with D = ε0 E + P to get εr ε0 E = ε0 E + P ⇒ P = (εr − 1) ε0 E. This is equivalent to P = χe ε0 E with χe = εr − 1. Using this prevents common mistakes like setting P equal to ε E, which would misstate units and physical meaning.

Step-by-Step Solution:

Verification / Alternative check:

In vacuum, εr = 1 ⇒ P = 0 as expected; in strong dielectrics, εr ≫ 1 ⇒ P grows proportionally, matching physical intuition.

Why Other Options Are Wrong:

• Options A and D incorrectly multiply by εr without subtracting the vacuum part.
• Option B is the vacuum displacement term, not polarization.
• Option E contradicts observed behavior of real dielectrics.

Common Pitfalls:

Confusing D, E, and P; forgetting that P measures the excess over vacuum response.

Final Answer:

P = ε0 * (εr − 1) * E