---
title: "Plane wave from E and H, find β λ η μr εr"
subject: "Electromagnetism"
chapter: 4
key: "semester_3/electromagnetism:plane-wave-from-e-and-h"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/plane-wave-from-e-and-h
site: BCT Engineering
---

# Plane wave from E and H, find β λ η μr εr

Electromagnetism · Chapter 4: Wave equation and Wave propagation

- Asked in: 2075 Asoj, 2+1+2+2+1 marks
- Video: [Solved Problem - 1 | UNIFORM PLANE WAVE |](https://www.youtube.com/watch?v=v5TRBx_oS5I) — Vijaya Academy 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/plane-wave-from-e-and-h
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4

## Question

Let E(z, t) = 1800 cos(10⁷πt − βz) a_x V/m and H(z, t) = 3.8 cos(10⁷πt − βz) a_y A/m represent a uniform plane wave propagating at a velocity of 1.4×10⁸ m/s in a perfect dielectric. Find (a) β (b) λ (c) η (d) μ_r (e) ε_r.

## Short answer

From ω=10⁷π rad/s and v=1.4×10⁸ m/s, β=ω/v=0.224 rad/m, λ=v/f=28 m. Intrinsic impedance η=E/H=473.7 Ω. Then ε_r = 1/(ηvε₀)=1.70 and μ_r = η/(vμ₀)=2.69.

## Answer

Given: E = 1800 cos(10⁷πt − βz) a_x V/m, H = 3.8 cos(10⁷πt − βz) a_y A/m, v = 1.4×10⁸ m/s, perfect dielectric.

ω = 10⁷π = 3.1416×10⁷ rad/s, so f = ω/2π = 5×10⁶ Hz.

**(a) Phase constant**

```text
β = ω/v = 3.1416×10⁷ / 1.4×10⁸ = 0.2244 rad/m
```

**(b) Wavelength**

```text
λ = 2π/β = v/f = 1.4×10⁸ / 5×10⁶ = 28 m
```

**(c) Intrinsic impedance**

```text
η = E/H = 1800/3.8 = 473.7 Ω
```

**(d) μ_r and (e) ε_r.** Use η = √(μ/ε) and v = 1/√(με). Multiplying: η v = 1/ε, and dividing: η/v = μ.

```text
ε = 1/(η v) = 1 / (473.7 × 1.4×10⁸) = 1.508×10⁻¹¹ F/m
ε_r = ε/ε₀ = 1.508×10⁻¹¹ / 8.854×10⁻¹² = 1.70

μ = η/v = 473.7 / 1.4×10⁸ = 3.384×10⁻⁶ H/m
μ_r = μ/μ₀ = 3.384×10⁻⁶ / 1.2566×10⁻⁶ = 2.69
```

Check: v = c/√(μ_r ε_r) = 3×10⁸/√(2.69 × 1.703) = 3×10⁸/2.14 = 1.40×10⁸ m/s ✓.

**Answer:** β = 0.224 rad/m; λ = 28 m; η = 473.7 Ω; μ_r = 2.69; ε_r = 1.70.

## Check yourself

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

1. With w = 10^7 pi and v = 1.4 x 10^8 m/s, beta is:
   A. 0.224 rad/m
   B. 4.46 rad/m
   C. 2.24 rad/m
   D. 0.0224 rad/m
   Answer: A. beta = w/v = 3.1416 x 10^7/1.4 x 10^8 = 0.2244 rad/m. (Key point: Phase constant)

2. How is the intrinsic impedance found from the given fields?
   A. eta = H/E
   B. eta = E/H = 1800/3.8 = 473.7 Ω
   C. eta = E x H
   D. eta = 377 Ω
   Answer: B. For a plane wave E/H = eta, giving 473.7 Ω. (Key point: Impedance)

3. To find mu and epsilon from eta and v, use:
   A. mu = eta v, epsilon = eta/v
   B. mu = eta/v, epsilon = 1/(eta v)
   C. mu = v/eta, epsilon = eta v
   D. mu = 1/(eta v), epsilon = eta/v
   Answer: B. eta = sqrt(mu/epsilon) and v = 1/sqrt(mu epsilon) give mu = eta/v and epsilon = 1/(eta v). (Key point: Using eta and v)

4. The resulting values are:
   A. mu_r = 1.70, eps_r = 2.69
   B. mu_r = 2.69, eps_r = 1.70
   C. mu_r = 1, eps_r = 2.14
   D. mu_r = 2.14, eps_r = 2.14
   Answer: B. mu_r = 3.384 x 10^-6/1.2566 x 10^-6 = 2.69 and eps_r = 1.508 x 10^-11/8.854 x 10^-12 = 1.70. (Key point: Relative constants)

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