Internal microscopic electric field in dielectrics: For solid or liquid insulating materials placed in a uniform external field E, how does the local internal field Ei at an atomic site compare with E?

Introduction / Context:
The microscopic field acting on an atom or molecule embedded in a dielectric differs from the macroscopic applied field due to local field corrections (Lorentz field) and polarization of the surrounding medium. Understanding this distinction is crucial for deriving relations between macroscopic permittivity and molecular polarizability (e.g., Clausius–Mossotti relation).

Given Data / Assumptions:

• Linear, isotropic dielectric with relative permittivity ε_r.
• Atom located inside an imagined spherical cavity (Lorentz construction).
• Uniform external macroscopic field E.

Concept / Approach:

In the Lorentz model, the local internal field at the atom is Ei = E + P/(3ε0), where P is the polarization vector. Since P = ε0(ε_r − 1)E in linear media, the correction term is positive for ε_r > 1, implying Ei exceeds the applied macroscopic field. This enhancement drives larger microscopic dipole moments than one would estimate from E alone and underpins the link between molecular polarizability and bulk permittivity.

Step-by-Step Solution:

Verification / Alternative check:

Clausius–Mossotti: (ε_r − 1)/(ε_r + 2) = Nα/(3ε0) assumes the same local field Ei, confirming the consistency of Ei > E in ordinary dielectrics.

Why Other Options Are Wrong:

• Ei = E neglects local polarization; only holds for ε_r = 1 (vacuum).
• Ei < E would require negative correction, not applicable to typical solids/liquids with ε_r > 1.
• “Always zero” is unphysical; insulators transmit displacement fields.
• “Equal or less” contradicts Lorentz field result.

Common Pitfalls:

• Confusing macroscopic average field with microscopic local field—these differ in condensed matter.

Final Answer:

Ei > E