Lect14PolarPlane

# Lect14PolarPlane - ECE 3030 Electromagnetic Fields and...

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1 ECE 3030 Electromagnetic Fields and Waves Fall 2009 Lecture 14 2009/9/28 Polarization States of Plane Waves More Complex Math Linea 1 Swartz and Rana 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 14: Instructor: Dr. Wesley E. Swartz Linear Circular Elliptical E x y E x y x y Prelim 1 and Problem Assignments Prelim Exam tomorrow night, starting at 7:30 in Phillips 101 Electrostatics and electroquasistatics. Charge interactions with metals. Charge interactions with dielectrics. Limited Magnetostatics. No virtual work. No Demo Section this week. No Homework due tomorrow 3 Swartz and Rana 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 14: No Homework due tomorrow There will be Workshop 5 Sections: Problem 8.14 Problem 9.2 Problem 9.3a Homework 5 (due Tuesday, October 6) : Problem 8.15 Problem 9.3b,c Review: Maxwell’s Equations for Phasors Time-harmonic E and H-fields are given as: () ( ) [] t j e r E t r E ω Re , = ( ) t j e r H t r H Re , = ( ) t j e r t r ρ Re , = ( ) t j e r J t r J Re , = Maxwell’s equations in terms vector phasors of time-harmonic fields are then: 4 Swartz and Rana 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 14: ( ) ( ) r r E o ε = 0 r H o = μ r H j r E o = × r E j r J r H o + = × Gauss’ Law: Gauss’ Law for the Magnetic Field: Faraday’s Law: Ampere’s Law: Review: Phasors When you see a “ j ” like in the following: What does the j mean? ( ) t j e r E t r E Re , = The j means many things! Here is a list 5 Swartz and Rana 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 14: ) 2 / j exp( e j 2 / j π = = Here is a list: Complex notation, as is real and imaginary parts. The expression is valid only for sinusoidal variations in time and space. The j implies a phase shift. And note: ) j exp( e 1 j = = Review: Plane Wave Phasors and the Complex Poynting Vector For a plane wave the E-field and H-field phasors are: r k j o e E n ˆ r E = r k j o o e E n ˆ k ˆ r H × = η E H k Ω = 377 o o o * 6 Swartz and Rana 06/5/25 Electromagnetic Fields and Waves – Fall 2009 Lecture 14: The complex Poynting vector: o o o o E k E n k n r H r E r S t r S 2 ˆ ˆ ˆ ˆ Re 2 1 Re 2 1 Re 2 1 , 2 2 * = × × = × = = ( ) ( ) ( ) r H r E r S × = The time-average power per unit area is one-half of the real part of the complex Poynting vector, and for plane waves: Review: Calculations with Complex Notation, an In Class Exercise Consider a plane wave with E-field of amplitude E o and pointing in a direction 45-degrees w.r.t. the x-axis (as shown) and traveling in the + z - direction.

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Lect14PolarPlane - ECE 3030 Electromagnetic Fields and...

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