Field in a cavity inside a homogeneous dielectric under uniform external field A small cavity (void) is created inside an otherwise homogeneous, isotropic dielectric subjected to a uniform electric field E. Will the electric field intensity inside the cavity equal the field in the surrounding dielectric?

Introduction / Context:
Voids or inclusions in dielectrics alter local electric fields, which is crucial in high-voltage insulation design and dielectric breakdown analysis. A classic problem asks whether the field inside a cavity equals the applied macroscopic field in the host material.

Given Data / Assumptions:

• Homogeneous, isotropic dielectric with relative permittivity ε_r.
• Small cavity (air or vacuum) introduced.
• Uniform external macroscopic field E in the dielectric before cavity creation.

Concept / Approach:

Boundary conditions at the cavity interface require tangential E to be continuous and normal D = ε E to be continuous. Because the cavity permittivity differs from that of the host (ε_cavity ≈ ε_0), the normal component of E must change across the interface to satisfy D_normal continuity. Therefore, the field inside the cavity generally differs from the far-field in the dielectric. For a spherical cavity in a uniform field, the internal field is E_in = 3 ε_r /(2 ε_r + 1) * E_0 for a cavity of permittivity ε_0 inside a medium of ε = ε_r ε_0; this is not equal to E_0 unless ε_r = 1.

Step-by-Step Solution:

Verification / Alternative check:

Analytical solutions for spherical/ellipsoidal cavities show closed-form factors relating E_in to E_out, confirming the inequality except in special limiting cases.

Why Other Options Are Wrong:

(a) contradicts boundary conditions. (c) is not generally true; even for a sphere, equality requires ε_r = 1. (d) Static/low-frequency limit does not enforce equality. (e) Non-sphericity changes the factor but not the basic inequality.

Common Pitfalls:

Assuming macroscopic averaging applies locally; local fields near permittivity discontinuities can be significantly higher or lower than the average field.

Final Answer:

False