Fall2023 ECE3120 Homework3 Solutions

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University of Colorado, Colorado Springs *

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3120

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Electrical Engineering

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Oct 30, 2023

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1/5 ECE3120 (Fall 2023) Homework #3 Solution September 21, 2023 1. The electric field of a certain plane electromagnetic wave is given by ? ? = 0; ? ? = 0; ? ? = 2.0 cos [𝜋 × 10 15 (? − ? 𝑐 )], with ? = 3.0 × 10 8 ?/? and all quantities in SI units. The wave is propagating in the positive ? direction in free space. Write expressions for the components of the magnetic field of the wave. Solution: ? ⃗ (?, ?) = 𝑅?[? ⃗ ̃ (?)? ?𝜔𝑡 ] = 𝑅?[?̂2.0? −? 𝜋10 15 𝑥 𝑐 ? ?𝜔𝑡 ] ∴ ? ⃗ ̃ = ?̂2.0? −? 𝜋10 15 𝑥 𝑐 𝐻 ⃗ ̃ = 1 𝜂 ? ̂ × ? ⃗ ̃ = 1 𝜂 ? ̂ × ? ⃗ ̃ = 1 𝜂 ? ̂ × ?̂2.0? −? 𝜋10 15 𝑥 𝑐 = − 1 𝜂 ? ̂2.0? −? 𝜋10 15 𝑥 𝑐 ∴ 𝐻 ⃗ (?, ?) = 𝑅? [𝐻 ⃗ ̃ (?)? ?𝜔𝑡 ] = 𝑅?[− 1 𝜂 ? ̂2.0? −? 𝜋10 15 𝑥 𝑐 ? ?𝜔𝑡 ] = − 2 377 ? ̂cos[𝜋10 15 (? − ? 𝑐 )] 2. (a) In free space we have ? ⃗ = ? 0 ? ̂sin(𝜔? − ??) . Find ? ⃗⃗ , ? ⃗ , and 𝐻 ⃗ . (b) In free space we have 𝐻 ⃗ = 𝐻 0 ? ̂? −?(𝜔𝑡+??) . Determine ? ⃗ . Solution: (a) ? ⃗ (?, ?) = ? 0 ? ̂ sin(𝜔? − ??) = 𝑅? [? ⃗ ̃ (?)? ?𝜔𝑡 ] = ? 0 ? ̂ cos (𝜔? − ?? − 𝜋 2 ) = 𝑅?[? 0 ? ̂? −?(??+ 𝜋 2 ) ? ?𝜔𝑡 ] ∴ ? ⃗ ̃ = ? 0 ? ̂? −?(??+ 𝜋 2 ) = −?? 0 ? ̂? −??? ( ∵ ? −? 𝜋 2 = ??? 𝜋 2 − ???? 𝜋 2 = −?) 𝐻 ⃗ ̃ = 1 𝜂 ? ̂ × ? ⃗ ̃ = 1 𝜂 ?̂ × ? ⃗ ̃ = 1 𝜂 ?̂ × (−?? 0 ? ̂? −??? ) = 1 𝜂 ? ̂?? 0 ? −???

2/5 ∴ 𝐻 ⃗ (?, ?) = 𝑅? [𝐻 ⃗ ̃ (?)? ?𝜔𝑡 ] = 𝑅?[ 1 𝜂 ? ̂?? 0 ? −??? ? ?𝜔𝑡 ] = 𝑅?[ 1 𝜂 ? ̂?? 0 {cos(𝜔? − ??) + ????(𝜔? − ??)}] = 𝑅?[ 1 𝜂 ? ̂? 0 {? cos(𝜔? − ??) − ???(𝜔? − ??)}] = − 1 𝜂 ? ̂? 0 ???(𝜔? − ??) = − 1 377 ? ̂? 0 ???(𝜔? − ??) ? ⃗⃗ (?, ?) = 𝜀 0 ? ⃗ (?, ?) = 𝜀 0 ? 0 ? ̂ sin(𝜔? − ??) ? ⃗ (?, ?) = 𝜇 0 𝐻 ⃗ (?, ?) = − 𝜇 0 377 ? ̂? 0 ???(𝜔? − ??) (b) 𝐻 ⃗ = 𝐻 0 ? ̂? −?(𝜔𝑡+??) = 𝑅? [𝐻 ⃗ ̃ (?)? −?𝜔𝑡 ] ? ⃗ ̃ = −?? ̂ × 𝐻 ⃗ ̃ 𝐻 ⃗ ̃ (?) = 𝐻 0 ? ̂? −??? ? ⃗ ̃ (?) = −??̂ × 𝐻 0 ? ̂? −??? = −?𝐻 0 ? ̂? −??? ? ⃗ (?, ?) = 𝑅?[−?𝐻 0 ? ̂? −??? ? −?𝜔𝑡 ] = −?𝐻 0 ? ̂ cos(𝜔? + ??) = −377𝐻 0 ? ̂ cos(𝜔? + ??)

3/5 3. A medium is characterized by 𝜎 = 0, 𝜇 = 2𝜇 0 , and 𝜀 = 5𝜀 0 . If 𝐻 ⃗ = 2 cos(𝜔? − 3?) ?̂ (?/?), find 𝜔 and ? ⃗ . Solution: 4. The electric phasor of a uniform plane wave is given by ? ⃗ ̃ = ? ̂10? ?0.2? (𝑉/?). If the phase velocity of the wave is 1.5 × 10 8 ?/? and the relative permeability of the medium is 𝜇 𝑟 = 2.4, find the below. (a) Wavelength (b) Frequency ? of the medium (c) Relative permittivity of the medium (d) Magnetic field, 𝐻 ⃗ (?, ?) Solution:

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