---
title: "Phasor E in free space, find f η Hs"
subject: "Electromagnetism"
chapter: 4
key: "semester_3/electromagnetism:phasor-e-find-frequency-h"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/phasor-e-find-frequency-h
site: BCT Engineering
---

# Phasor E in free space, find f η Hs

Electromagnetism · Chapter 4: Wave equation and Wave propagation

- Asked in: 2068 Chaitra, 2+2+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/phasor-e-find-frequency-h
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-4

## Question

The phasor component of electric field intensity in free space is given by E_s = (100∠45°) e^(−j50z) a_x V/m. Determine the frequency of the wave, wave impedance, H_s, and magnitude of E at z = 10 mm, t = 20 ps.

## Short answer

Frequency f = βc/(2π) = 2.39 GHz, wave impedance η = 377 Ω. Phasor H_s = 0.265∠45° e^{−j50z} a_y A/m. At z=10 mm and t=20 ps, instantaneous electric field magnitude is 83.3 V/m.

## Answer

Given: free space, E_s = (100∠45°) e^{−j50z} a_x V/m. So β = 50 rad/m, E₀ = 100 V/m, phase 45°, and the wave travels in +z.

**(a) Frequency**

```text
ω = β c = 50 × 3×10⁸ = 1.5×10¹⁰ rad/s
f = ω/2π = 1.5×10¹⁰/6.2832 = 2.387×10⁹ Hz ≈ 2.39 GHz
```

**(b) Wave impedance**

```text
η₀ = 120π = 377 Ω
```

**(c) H_s.** For travel along +z with E along a_x, H points along a_y (a_x × a_y = a_z):

```text
H_s = (E₀/η₀)∠45° e^{−j50z} a_y
    = (100/377) ∠45° e^{−j50z} a_y
    = 0.2653 ∠45° e^{−j50z} a_y   A/m
```

**(d) |E| at z = 10 mm, t = 20 ps.**

```text
E(z,t) = 100 cos(ωt − βz + 45°)
ωt = 1.5×10¹⁰ × 20×10⁻¹² = 0.3 rad
βz = 50 × 0.01 = 0.5 rad
45° = 0.7854 rad
phase = 0.3 − 0.5 + 0.7854 = 0.5854 rad
E = 100 cos(0.5854) = 100 × 0.8335 = 83.3 V/m  (a_x direction)
```

**Answer:** f = 2.39 GHz; η = 377 Ω; H_s = 0.265∠45° e^{−j50z} a_y A/m; E(10 mm, 20 ps) = 83.3 a_x V/m.

## Check yourself

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

1. E_s = 100 angle 45 e^(-j50z) a_x in free space. What is f?
   A. 15 GHz (taking w as f)
   B. 239 MHz
   C. f = beta*c/2pi = 2.39 GHz
   D. 50 MHz
   Answer: C. beta = 50 rad/m gives w = 50*3x10^8 = 1.5x10^10 rad/s, and f = w/2pi = 2.39 GHz. (Key point: Frequency)

2. What is H_s?
   A. (100/377) angle 45 e^(-j50z) a_x A/m
   B. (100/377) angle 45 e^(-j50z) a_y A/m
   C. 100*377 angle 45 e^(-j50z) a_y A/m
   D. (100/377) angle -45 e^(+j50z) a_y A/m
   Answer: B. For +z travel with E along a_x, H is along a_y (a_x x a_y = a_z) with amplitude E0/eta0 and the same phase. (Key point: Magnetic phasor)

3. What phase is used in E = 100 cos(...) at z = 10 mm, t = 20 ps?
   A. wt - bz + 45 degrees = 0.3 - 0.5 + 0.785 = 0.585 rad
   B. 0.3 + 0.5 + 0.785 = 1.585 rad
   C. 0.3 - 0.5 = -0.2 rad (omitting the 45 degrees)
   D. 0.5 - 0.3 + 0.785 = 0.985 rad
   Answer: A. For +z travel the phase is wt - bz plus the phasor angle. The result is E = 100 cos(0.585) = 83.3 V/m. (Key point: Phase at a point)

4. What is the wave impedance in this wave?
   A. 50 ohm, from beta
   B. 2.39 GHz-dependent value of about 100 ohm
   C. 377 ohm only at 45 degrees phase
   D. eta0 = 120 pi = 377 ohm, since the medium is free space
   Answer: D. In free space the wave impedance is eta0 = 377 ohm, independent of the frequency or the initial phase. (Key point: Wave impedance)

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