---
title: "Lossless dielectric, find ω η E"
subject: "Electromagnetism"
chapter: 4
key: "semester_3/electromagnetism:lossless-dielectric-h-given-find-e"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/lossless-dielectric-h-given-find-e
site: BCT Engineering
---

# Lossless dielectric, find ω η E

Electromagnetism · Chapter 4: Wave equation and Wave propagation

- Asked in: 2072 Chaitra, 2+2+4 marks
- Video: [Numerical Problems on Electromagnetic Wave Propagation part 3](https://www.youtube.com/watch?v=eM1p6eVgOnI) — M Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/lossless-dielectric-h-given-find-e
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4

## Question

A lossless dielectric material has σ = 0, μ_r = 1, ε_r = 4. An electromagnetic wave has magnetic field expressed as H = −0.1 cos(ωt − z) a_x + 0.5 sin(ωt − z) a_y A/m. Find: (a) angular frequency (ω) (b) wave impedance (η) (c) E.

## Short answer

From β = 1 rad/m, ω = β c / √(μ_r ε_r) = 1.5×10⁸ rad/s. Wave impedance η = η₀/√ε_r = 188.5 Ω. Using E = η H × a_z gives E = 94.25 sin(ωt − z) a_x + 18.85 cos(ωt − z) a_y V/m.

## Answer

Given: σ = 0, μ_r = 1, ε_r = 4, and

H = −0.1 cos(ωt − z) a_x + 0.5 sin(ωt − z) a_y A/m.

The phase is (ωt − z), so the wave travels in +z with phase constant β = 1 rad/m.

**(a) Angular frequency.** β = ω√(με) = (ω/c)√(μ_r ε_r) = (ω/c) × 2.

```text
ω = β c / 2 = 1 × 3×10⁸ / 2 = 1.5×10⁸ rad/s     (f = 23.9 MHz)
```

**(b) Wave impedance**

```text
η = η₀ √(μ_r/ε_r) = 377 / 2 = 188.5 Ω
```

**(c) E.** For a wave along a_z, E = η H × a_z (equivalently H = a_z × E / η).

```text
a_x × a_z = −a_y,   a_y × a_z = a_x

E = 188.5 [ −0.1 cos(ωt − z)(−a_y) + 0.5 sin(ωt − z)(a_x) ]
  = 188.5 [ 0.5 sin(ωt − z) a_x + 0.1 cos(ωt − z) a_y ]
  = 94.25 sin(ωt − z) a_x + 18.85 cos(ωt − z) a_y   V/m
```

Check: E·H = E_xH_x + E_yH_y = (94.25 sin)(−0.1 cos) + (18.85 cos)(0.5 sin) = −9.425 sin cos + 9.425 sin cos = 0 ✓ (E ⟂ H). Also 94.25/0.5 = 18.85/0.1 = 188.5 = η for each component pair.

**Answer:** ω = 1.5×10⁸ rad/s; η = 188.5 Ω; E = 94.25 sin(1.5×10⁸t − z) a_x + 18.85 cos(1.5×10⁸t − z) a_y V/m.

## Check yourself

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

1. H has phase (wt - z) in a lossless dielectric with mu_r = 1, eps_r = 4. How is w found?
   A. w = beta*c = 3x10^8 rad/s (ignoring the dielectric)
   B. beta = 1 rad/m, so w = beta*c/sqrt(mu_r*eps_r) = 1.5x10^8 rad/s
   C. w = beta*c*2 = 6x10^8 rad/s
   D. w = beta = 1 rad/s
   Answer: B. The phase (wt - z) gives beta = 1 rad/m. In the dielectric the speed is c/2, so w = beta*c/2 = 1.5x10^8 rad/s. (Key point: Angular frequency)

2. What is the wave impedance of this lossless dielectric?
   A. 188.5 ohm
   B. 377 ohm
   C. 754 ohm
   D. 94.25 ohm
   Answer: A. eta = eta0*sqrt(mu_r/eps_r) = 377/2 = 188.5 ohm. (Key point: Wave impedance)

3. For a wave travelling along +z, which relation gives E from H?
   A. E = eta a_z x H
   B. E = H / eta
   C. E = eta (H . a_z) a_z
   D. E = eta H x a_z
   Answer: D. E = eta H x a_z (equivalent to H = a_z x E / eta). Using a_z x H would reverse the sign of E. (Key point: Finding E from H)

## Related in Electromagnetism

- [9.4 GHz plane wave in dielectric](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aplane-wave-9-4ghz-dielectric) — Electromagnetism, chapter 4, asked 2×
- [Plane wave in non-magnetic medium](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aplane-wave-non-magnetic-e-given) — Electromagnetism, chapter 4
- [Dielectric boundary at y=0, find fields](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adielectric-boundary-y-d1-find-region2) — Electromagnetism, chapter 2
- [Poynting numerical, wave in free space](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Apoynting-numerical-em-wave-free-space) — Electromagnetism, chapter 4
- [Phasor E and H of wave in −y direction](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aplane-wave-minus-y-phasor-expressions) — Electromagnetism, chapter 4
- [Derive E and H of plane wave](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Auniform-plane-wave-derive-e-h) — Electromagnetism, chapter 4, asked 5×

## Related in other subjects

- [Magnetic material for electromagnetic relays](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Arelay-magnetic-material) — Electrical Engineering Material, chapter 4
- [EM wave in dielectric from Maxwell's equations](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Amaxwell-em-wave-dielectric-velocity) — Engineering Physics, chapter 9, asked 2×
- [Soft magnetic material, Faraday's laws](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Asoft-magnetic-faraday-laws) — Electrical Machines, chapter 1
- [Lossless impedance check; Foster I and Cauer I](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alossless-impedance-four-functions-2073) — Filter Design, chapter 4