---
title: "Plane wave parameters from Hs"
subject: "Electromagnetism"
chapter: 4
key: "semester_3/electromagnetism:plane-wave-parameters-from-hs"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/plane-wave-parameters-from-hs
site: BCT Engineering
---

# Plane wave parameters from Hs

Electromagnetism · Chapter 4: Wave equation and Wave propagation

- Asked in: 2076 Chaitra, 1+2+1+2+2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/plane-wave-parameters-from-hs
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4

## Question

A uniform plane wave in free space is given by H_s = (250∠30°) e^(−j350z) a_x V/m. Determine the phase constant, frequency of the wave, intrinsic impedance, E_s and the magnitude of H at z = 25 mm and t = 4 ps.

## Short answer

Phase constant β=350 rad/m. Frequency f=ω/2π=βc/2π≈16.7 GHz; intrinsic impedance η₀=377 Ω. With H along a_x and propagation +a_z, E must point along −a_y: E_s = −94.25∠30° e^{−j350z} a_y kV/m. At z=25 mm, t=4 ps, |H| = 11.9 A/m.

## Answer

Given (the unit of H_s is A/m; V/m in the question is a misprint): H_s = (250∠30°) e^{−j350z} a_x A/m in free space (ε_r = μ_r = 1).

Comparing with e^{−jβz}: wave travels in +z, and H_s amplitude is 250 A/m with phase 30°. Take c = 3×10⁸ m/s.

**1. Phase constant**

```text
β = 350 rad/m
```

**2. Frequency**

```text
ω = β c = 350 × 3×10⁸ = 1.05×10¹¹ rad/s
f = ω / 2π = 1.05×10¹¹ / 6.2832 = 1.671×10¹⁰ Hz ≈ 16.7 GHz
```

**3. Intrinsic impedance**

```text
η₀ = 120π = 376.99 ≈ 377 Ω
```

**4. E_s.** The wave travels along +a_z, and E × H must point along +a_z. With H along a_x, E must point along −a_y, since (−a_y) × a_x = a_z. Magnitude E = η H = 377 × 250 = 94 248 V/m.

```text
E_s = −94.25∠30° e^{−j350z} a_y kV/m
    = 94.25∠210° e^{−j350z} a_y kV/m
```

**5. |H| at z = 25 mm, t = 4 ps.**

```text
H(z,t) = 250 cos(ωt − βz + 30°) a_x
ωt = 1.05×10¹¹ × 4×10⁻¹² = 0.42 rad
βz = 350 × 0.025 = 8.75 rad
30° = 0.5236 rad
phase = 0.42 − 8.75 + 0.5236 = −7.806 rad
cos(−7.806) = cos(7.806 − 2π) = cos(1.523) = 0.0476
H = 250 × 0.0476 = 11.9 A/m
```

**Answer:** β = 350 rad/m; f ≈ 16.7 GHz; η = 377 Ω; E_s = −94.25∠30° e^{−j350z} a_y kV/m; H(25 mm, 4 ps) ≈ 11.9 a_x A/m.

## Check yourself

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

1. In H_s = 250 angle 30 e^(-j350z) a_x, the phase constant and direction are:
   A. beta = 350 rad/m, wave travels in +z
   B. beta = 350 rad/m, wave travels in -z
   C. beta = 250 rad/m, wave travels in +z
   D. beta = 30 rad/m, wave travels in +x
   Answer: A. Comparing with e^(-j beta z): beta = 350 rad/m, and the minus sign means +z propagation. (Key point: Phase constant)

2. In free space the frequency is:
   A. f = beta c = 1.05 x 10^11 Hz
   B. f = beta c/(2 pi) = 16.7 GHz
   C. f = 350 Hz
   D. f = c/(2 pi beta) = 137 kHz
   Answer: B. w = beta c = 1.05 x 10^11 rad/s, so f = w/2 pi = 16.7 GHz. (Key point: Frequency)

3. With H along a_x and travel along +a_z, E is along:
   A. +a_y
   B. -a_x
   C. -a_y
   D. +a_z
   Answer: C. E x H must point along +a_z; (-a_y) x a_x = a_z, so E is along -a_y with magnitude eta H. (Key point: Direction of E)

4. How is |H| at a given z and t found?
   A. H = 250 cos(wt - beta z + 30 degrees), evaluating the phase in radians
   B. H = 250 at all points
   C. H = 250 e^(-350 z)
   D. H = 250 sin(beta z) only
   Answer: A. The time-domain field is 250 cos(wt - beta z + 30 degrees); at z = 25 mm and t = 4 ps the phase is -7.806 rad, giving about 11.9 A/m. (Key point: Instantaneous value)

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