Imperfect dielectrics and equivalent circuit — assertion–reason Assertion (A): A capacitor with an imperfect (lossy) dielectric can be modeled as a capacitance in parallel with a resistance. Reason (R): For imperfect dielectrics, the dielectric constant is complex with real and imaginary parts (ε* = ε′ − jε″).

Introduction / Context:
Real dielectrics exhibit losses due to molecular relaxation, conduction, and dipolar friction. Capturing this behavior in circuits and material models requires acknowledging complex permittivity and equivalent loss elements.

Given Data / Assumptions:

• Linear time-harmonic operation.
• Imperfect dielectric with finite loss tangent (tan δ).
• Small-signal regime so linear superposition applies.

Concept / Approach:
Complex permittivity ε* = ε′ − jε″ represents both stored and dissipated energy in a dielectric. The conductance associated with ε″ can be modeled as a resistor in parallel with the ideal capacitor (same voltage across both), yielding the familiar “RC in parallel” equivalent for dielectric losses at a given frequency.

Step-by-Step Solution:
Write displacement current density: J = jωε*E = jωε′E + ωε″E.Interpret jωε′E as reactive (capacitive) current and ωε″E as in-phase (resistive) loss current.Equivalent circuit: a capacitor C (from ε′) in parallel with a resistor R (representing losses from ε″).

Verification / Alternative check:
The loss tangent tan δ = ε″/ε′ equals the ratio of resistive to reactive currents in the parallel model, confirming equivalence.

Why Other Options Are Wrong:
(b) denies the causality link; however, the complex permittivity directly motivates the parallel RC model. (c) and (d) contradict known dielectric behavior. (e) rejects both facts.

Common Pitfalls:

• Confusing series and parallel loss models; both can be used but the complex ε interpretation maps naturally to a parallel conductance.
• Assuming ε″ represents conduction only; it encapsulates multiple microscopic loss mechanisms.

Final Answer:
Both A and R are true and R is correct explanation of A