Dielectric losses and loss tangent (tan δ) Assertion (A): Dielectric power losses are proportional to tan δ (the loss tangent) for a given frequency, field, and permittivity. Reason (R): The loss tangent is defined by tan δ = εr'' / εr', where εr' and εr' denote the real and imaginary parts of the complex relative permittivity εr.
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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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EBoth are false
Answer
Correct Answer: Both A and R are true but R is not correct explanation of A
Explanation
Introduction / Context:Dielectric materials in AC fields dissipate power as heat due to molecular relaxation and conduction. This dissipation is commonly characterized by the loss tangent, tan δ, which links the in-phase and quadrature components of polarization with respect to the electric field. Understanding what tan δ is and how it relates to power loss is essential in capacitor design, RF substrates, and insulation systems.
Given Data / Assumptions:
- Complex relative permittivity εr* = εr′ − j εr″ (engineering sign convention).
- Applied sinusoidal field with angular frequency ω and RMS magnitude E_rms.
- Linear, isotropic material; negligible magnetic loss.
Concept / Approach:
Dielectric loss power density is p_loss = ω ε0 εr′ tan δ * E_rms^2, where tan δ = εr″ / εr′. Holding ω, εr′, and E_rms fixed shows p_loss ∝ tan δ, which justifies the assertion (A). The reason (R) correctly states the definition of tan δ, but it does not by itself demonstrate proportionality of power loss to tan δ; that proportionality requires the explicit power expression, not only the definition.
Step-by-Step Solution:
Write εr* = εr′ − j εr″.Define tan δ = εr″ / εr′.Use p_loss = ω ε0 εr′ tan δ * E_rms^2 → shows proportionality to tan δ at fixed ω, εr′, E_rms.Verification / Alternative check:
Equivalent circuit (capacitor with series/parallel resistance) leads to the same proportionality for small loss angles, reinforcing the relationship between tan δ and power dissipation.
Why Other Options Are Wrong:
(a) claims the definition alone explains proportionality; it does not. (c) and (d) incorrectly negate one of the true statements. (e) is incorrect since both statements are true.
Common Pitfalls:
Mixing sign conventions or confusing εr″/εr′ with εr′/εr″; ensuring consistency avoids errors in calculating losses.
Final Answer:
Both A and R are true but R is not correct explanation of A