---
title: "Short note: Loss tangent"
subject: "Electromagnetism"
chapter: 4
key: "semester_3/electromagnetism:loss-tangent-short-note"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/loss-tangent-short-note
site: BCT Engineering
---

# Short note: Loss tangent

Electromagnetism · Chapter 4: Wave equation and Wave propagation

- Asked in: 2068 Chaitra, 2 marks
- Video: [Loss Tangent of a Medium: Basics and Relations with Intrinsic Impedance](https://www.youtube.com/watch?v=s2PbSUGpdQk) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/loss-tangent-short-note
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4

## Question

Write a short note on loss tangent.

## Short answer

Loss tangent tan δ = σ/(ωε) is the ratio of conduction to displacement current density. It classifies media: good dielectric if tan δ ≪ 1, good conductor if tan δ ≫ 1. It determines attenuation, complex permittivity, and phase shift between E and H.

## Answer

In a medium with conductivity σ and permittivity ε, Maxwell's equation is ∇×H = σE + jωεE = jω(ε − jσ/ω)E. So the medium has a complex permittivity ε_c = ε − jσ/ω.

**Loss tangent** is the ratio of the conduction current density to the displacement current density:

tan δ = |J_c| / |J_d| = σ / (ωε)

(δ is the loss angle, tan δ = σ/(ωε); it measures how far the medium departs from a perfect dielectric, for which δ = 0.)

Uses and meaning:
- It shows how lossy a medium is at a given frequency. It changes with frequency.
- **Good dielectric:** tan δ ≪ 1 (σ ≪ ωε); small attenuation, α ≈ (σ/2)√(μ/ε).
- **Quasi-conductor:** tan δ ≈ 1.
- **Good conductor:** tan δ ≫ 1 (σ ≫ ωε); α = β = √(πfμσ).
- The intrinsic impedance becomes complex when tan δ ≠ 0, so E and H are out of phase by an angle θ_η = δ/2 (E leads H).
- Example: dry soil (σ = 10⁻⁴ S/m, ε_r = 3) has tan δ = 1 at about 599 kHz.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is loss tangent?
   A. The ratio of displacement to conduction current, w eps/sigma
   B. The ratio of E to H in the medium
   C. The ratio of conduction current density to displacement current density, sigma/(w eps)
   D. The attenuation constant divided by phase constant squared
   Answer: C. tan delta = |J_c|/|J_d| = sigma/(w eps). (Key point: Definition)

2. Why does the medium have a complex permittivity?
   A. Because eps varies with position
   B. grad x H = jw(eps - j sigma/w)E, so eps_c = eps - j sigma/w
   C. Because mu becomes complex
   D. Because E and H are in phase
   Answer: B. Combining sigma E and jw eps E into one term gives the complex permittivity eps - j sigma/w. (Key point: Complex permittivity)

3. Which medium has tan delta much greater than 1, with alpha = beta = sqrt(pi f mu sigma)?
   A. Good conductor
   B. Good dielectric
   C. Quasi-conductor
   D. Perfect dielectric
   Answer: A. tan delta >> 1 (sigma >> w eps) means a good conductor. A good dielectric has tan delta << 1 and a quasi-conductor about 1. (Key point: Classification)

## Related in Electromagnetism

- [Loss tangent, α, β, η at 5 MHz](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Anon-magnetic-material-loss-tangent-5mhz) — Electromagnetism, chapter 4

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