Dielectric loss and complex permittivity at low frequency A dielectric has ε′ = 2.1 and loss tangent tanδ = 5 × 10^-4 at 100 Hz. What is the imaginary part of its complex relative permittivity ε″ at 100 Hz?

Introduction / Context:
Loss in dielectrics at low frequency is often represented using a complex relative permittivity: εr* = ε′ − j ε″. The ratio ε″/ε′ is the loss tangent tanδ. Converting between ε′, tanδ, and ε″ is a routine step in capacitor design and insulation diagnostics.

Given Data / Assumptions:

• Real part ε′ = 2.1 (dimensionless).
• Loss tangent tanδ = 5 × 10^-4 at 100 Hz.
• Small-loss approximation is valid at such a low tanδ.

Concept / Approach:
By definition, tanδ = ε″ / ε′. Therefore, ε″ = ε′ * tanδ. Substitute the given values to compute ε″ directly.

Step-by-Step Solution:
Write relation: ε″ = ε′ * tanδ.Substitute: ε″ = 2.1 * (5 × 10^-4).Multiply: 2.1 * 5 = 10.5, so ε″ = 10.5 × 10^-4 = 1.05 × 10^-3.Report ε″ with appropriate scientific notation.

Verification / Alternative check:
Since tanδ is much less than 1, ε″ ≪ ε′, which is consistent with 1.05 × 10^-3 compared to 2.1.

Why Other Options Are Wrong:
2.1 × 10^-3 doubles the correct value; 5 × 10^-3 and 1.05 × 10^-2 are an order of magnitude too high.

Common Pitfalls:
Confusing ε″ with absolute imaginary permittivity (ε0 ε″); mixing radians and degrees (not relevant here); arithmetic slips in scientific notation.

Final Answer:
1.05 × 10^-3