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Lect12Waves

# Lect12Waves - ECE 3030 Electromagnetic Fields and Waves...

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1 ECE 3030 Electromagnetic Fields and Waves Fall 2009 Lecture 12 2009/9/23 Electromagnetic Waves Wave Equation In Free Space Plane Waves 1 Swartz and Rana 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 12: Instructor: Dr. Wesley E. Swartz Energy and Power λ c c z x y E H c Basic Wave Motion Consider a wave moving in the +x direction. v x () t x a , 2 Swartz and Rana 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 12: Fundamental relation for wave motion: 1-D wave equation: v f = v = velocity of wave propagation λ = wavelength of the wave f = frequency of the wave T = period = 1/ f 2 2 2 2 2 , 1 , t t x a v x t x a = Electromagnetic Wave Motion (1) Time varying electric and magnetic fields are coupled. This coupling is responsible for the propagation of electromagnetic waves. Deriving the electromagnetic wave equations, Assume free space 0 = = H E o o μ ρ ε t E J H t H E o o + = × = × 0 0 J 3 Swartz and Rana 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 12: Assume free space: , = = 2 2 t E t H t H E o o o o = × = × −∇ = × × 2 2 t E E o o = × × 2 2 2 1 t E c E = × × m/s 10 3 1 8 × = o o c Electromagnetic Wave Motion (2) . 2 2 2 1 t E c E = × × Equation for an E&M wave traveling at speed c in free space. 2 2 2 2 1 t E c E E = 0 2 2 2 2 1 t E c E = 0 = = E o Use the vector Identity: F F F 2 = × × 4 Swartz and Rana 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 12: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 3 1 2 1 1 t E c z E y E x E t E c z E y E x E t E c z E y E x E z z z z y y y y x x x x = + + = + + = + + The wave equation is essentially three equations stacked together – one for each component of the E- field Wave must also satisfy: 0 z E y E x E 0 E z y x o = + + = Electromagnetic Wave Motion (3) The H-field also satisfies a similar wave equation. Start from Maxwell’s Equations: and Assume free space: t E J H t H E o o + = × = × 0 = = J 2 2 t H t E t E H o o o o = × = × = × × 2 H 5 Swartz and Rana 06/5/31 Electromagnetic Fields and Waves – Fall 2009 Lecture 12: ( ) 2 t H o o = × × 2 2 2 1 t H c H = × × 2 2 2 2 t H c 1 H H = 0 0 = H o 2 2 2 2 1 t H c H = General Solutions of the Electromagnetic Wave Equation Assume there is only an x -component of the E-field:

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Lect12Waves - ECE 3030 Electromagnetic Fields and Waves...

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