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.

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:

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