Cavity in a Dielectric – Condition for Uniform Field From a homogeneous, isotropic dielectric of relative permittivity εr, a small volume element is removed to create a cavity. To keep the electric field homogeneous across the cavity boundary, which relation between D_i (electric flux density inside the cavity) and D_0 (electric flux density in the surrounding dielectric) must hold?

Introduction / Context:When a small cavity (e.g., air void) is created within a uniform dielectric that is under a uniform electric field, boundary conditions at the cavity interface determine how electric field E and electric flux density D distribute inside and outside the cavity. Engineers often need a quick criterion for maintaining a homogeneous field (no distortion) in conceptual analysis.

Given Data / Assumptions:

• Linear, isotropic dielectric of relative permittivity εr.
• Small cavity with permittivity approximately ε0 (air/vacuum), inside a region initially under uniform E.
• Goal: require E to be homogeneous across the interface for this idealized condition.

Concept / Approach:

For linear media, D = ε E. If we require the same electric field E both inside the cavity and in the surrounding dielectric (homogeneous E), then: outside, D0 = εr ε0 E; inside, Di = ε0 E. Eliminating E gives D0 = εr * Di. This relation ensures field continuity consistent with the idealized “no distortion” condition. Although, strictly, a real cavity in a dielectric perturbs the field, this relation captures the condition for equal E across the boundary in the simple model.

Step-by-Step Solution:

Verification / Alternative check:

Using boundary condition on normal D without free charge, D_normal is continuous. If ε differs, equal E requires D0/Di = εr, consistent with the above relation.

Why Other Options Are Wrong:

• D0 = Di: would imply εr = 1 (same medium), not a cavity in a dielectric.
• Di = εr * D0: inverts the correct proportionality; would force E_inside ≠ E_outside.
• “Either (a) or (b)”: only one relation maintains equal E.

Common Pitfalls:

Confusing continuity of normal D (true with no free surface charge) with continuity of E; mixing up which side has εr factor.

Final Answer:

D_0 = εr * D_i