---
title: "Plane wave in non-magnetic medium"
subject: "Electromagnetism"
chapter: 4
key: "semester_3/electromagnetism:plane-wave-non-magnetic-e-given"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/plane-wave-non-magnetic-e-given
site: BCT Engineering
---

# Plane wave in non-magnetic medium

Electromagnetism · Chapter 4: Wave equation and Wave propagation

- Asked in: 2073 Shrawan, 1+5+2 marks
- Video: [Plane wave porpagating through a medium | Numerical example](https://www.youtube.com/watch?v=m8b7is7t15k) — 100 Million Thoughts
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/plane-wave-non-magnetic-e-given
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4

## Question

A uniform plane wave in a non-magnetic medium has E = 50 cos(10⁸t + 2z) a_y V/m. Find: (i) the direction of propagation; (ii) phase constant β, wavelength λ, velocity v_p, relative permittivity ε_r, intrinsic impedance η; (iii) H.

## Short answer

Propagation direction is −a_z. Phase constant β = 2 rad/m, wavelength λ = 3.14 m, velocity v_p = 5×10⁷ m/s, relative permittivity ε_r = 36, intrinsic impedance η = 62.8 Ω. Magnetic field H = 0.796 cos(10⁸t + 2z) a_x A/m.

## Answer

Given: E = 50 cos(10⁸t + 2z) a_y V/m in a non-magnetic medium (μ_r = 1). So ω = 10⁸ rad/s, E₀ = 50 V/m.

**(i) Direction of propagation.** The phase is (ωt + 2z), with a plus sign between t and z, so the wave travels in the **−z direction** (−a_z).

**(ii) Parameters.**

```text
β = 2 rad/m                         (coefficient of z)
λ = 2π/β = 2π/2 = π = 3.142 m
v_p = ω/β = 10⁸/2 = 5×10⁷ m/s
```

Non-magnetic: v_p = c/√ε_r, so

```text
ε_r = (c/v_p)² = (3×10⁸ / 5×10⁷)² = 6² = 36
η = η₀/√ε_r = 377/6 = 62.83 Ω
```

**(iii) H.** For a wave travelling along a_k = −a_z, H = (1/η) a_k × E:

```text
a_k × E = (−a_z) × a_y = a_x          (since a_z × a_y = −a_x)
H = (50/62.83) cos(10⁸t + 2z) a_x
  = 0.796 cos(10⁸t + 2z) a_x   A/m
```

Check: E × H = a_y × a_x = −a_z ✓ (travelling in −z).

**Answer:** (i) −a_z; (ii) β = 2 rad/m, λ = 3.14 m, v_p = 5×10⁷ m/s, ε_r = 36, η = 62.8 Ω; (iii) H = 0.796 cos(10⁸t + 2z) a_x A/m.

## Check yourself

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

1. For E = 50 cos(10^8 t + 2z) a_y, the wave travels:
   A. In +z
   B. In -z, because the signs of wt and beta z are the same
   C. In +y
   D. In -y
   Answer: B. A plus sign between t and z in the phase means propagation in the -z direction. (Key point: Direction of propagation)

2. Reading off w = 10^8 and beta = 2 gives:
   A. lambda = 3.14 m, v_p = 5 x 10^7 m/s
   B. lambda = 1 m, v_p = 2 x 10^8 m/s
   C. lambda = 6.28 m, v_p = 10^8 m/s
   D. lambda = 3.14 m, v_p = 2 x 10^8 m/s
   Answer: A. lambda = 2 pi/beta = pi m and v_p = w/beta = 5 x 10^7 m/s. (Key point: Wave parameters)

3. For a non-magnetic medium with v_p = 5 x 10^7 m/s:
   A. eps_r = 6, eta = 62.8 Ω
   B. eps_r = 36, eta = 62.8 Ω
   C. eps_r = 36, eta = 2262 Ω
   D. eps_r = 6, eta = 377 Ω
   Answer: B. eps_r = (c/v_p)^2 = 36 and eta = 377/sqrt(36) = 62.8 Ω. (Key point: Permittivity and impedance)

4. For travel along -a_z with E along a_y, H is along:
   A. +a_x with magnitude 0.796 A/m
   B. -a_x
   C. +a_z
   D. +a_y
   Answer: A. H = (1/eta) a_k x E = (-a_z) x a_y /eta = +a_x/eta, magnitude 50/62.83 = 0.796 A/m. (Key point: Direction of H)

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