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″).
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ABoth A and R are true and R is correct explanation of A
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BBoth A and R are true but R is not correct explanation of A
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CA is true but R is false
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DA is false but R is true
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ENeither A nor R is true
Answer
Correct Answer: Both A and R are true and R is correct explanation of A
Explanation
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