Lecture13 - 1 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lecture 13 Electromagnetic Waves in Free Space In this lecture you will

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Unformatted text preview: 1 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lecture 13 Electromagnetic Waves in Free Space In this lecture you will learn: • Electromagnetic wave equation in free space • Uniform plane wave solutions of the wave equation • Energy and power of electromagnetic waves ECE 303 – Fall 2007 – Farhan Rana – Cornell University Basic Wave Motion v λ x Consider a wave moving in the + x-direction: The wave travels a distance equal to one wavelength in one time period Basic relation for wave motion: v f = λ v = velocity of wave propagation λ = wavelength of the wave f = frequency of the wave T = period = 1/ f 1-D wave equation: ( ) ( ) 2 2 2 2 2 , 1 , t t x a v x t x a ∂ ∂ = ∂ ∂ ( ) t x a , λ λ f T v = = = = time distance velocity 2 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Electromagnetic Wave Motion - I . . = ∇ = ∇ H E o o r r µ ρ ε t E J H t H E o o ∂ ∂ + = × ∇ ∂ ∂ − = × ∇ r r r r r ε µ Time varying electric and magnetic fields are coupled - this coupling is responsible for the propagation of electromagnetic waves Electromagnetic Wave Equation: Assume free space: = = ⇒ J r ρ ( ) 2 2 t E t H t H E o o o o ∂ ∂ − = ∂ × ∇ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ × −∇ = × ∇ × ∇ r r r r ε µ µ µ ( ) 2 2 t E E o o ∂ ∂ − = × ∇ × ∇ ⇒ r r ε µ ( ) 2 2 2 1 t E c E ∂ ∂ − = × ∇ × ∇ ⇒ r r m/s 10 3 1 8 × ≈ = o o c µ ε ECE 303 – Fall 2007 – Farhan Rana – Cornell University ( ) 2 2 2 1 t E c E ∂ ∂ − = × ∇ × ∇ ⇒ r r Equation for a wave traveling at speed c in free space Electromagnetic Wave Motion - II ( ) 2 2 2 2 1 . t E c E E ∂ ∂ − = ∇ − ∇ ∇ ⇒ r r r 2 2 2 2 1 t E c E ∂ ∂ = ∇ ⇒ r r . = = ∇ ρ ε E o r ( ) ( ) ( ) 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 ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ Wave equation is essentially three equations stacked together – one for each component of the E-field Wave must also satisfy: . = ∂ ∂ + ∂ ∂ + ∂ ∂ ⇒ = ∇ z E y E x E E z y x o r ε Use the vector Identity: ( ) ( ) F F F r r r 2 . ∇ − ∇ ∇ = × ∇ × ∇ 3 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Electromagnetic Wave Motion - III The H-field also satisfies a similar wave equation t E J H t H E o o ∂ ∂ + = × ∇ ∂ ∂ − = × ∇ r r r r r ε µ Start from Maxwell’s equations: Assume free space: = = ⇒ J r ρ ( ) 2 2 t H t E t E H o o o o ∂ ∂ − = ∂ × ∇ ∂ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ × ∇ = × ∇ × ∇ r r r r ε µ ε ε ( ) 2 2 t H H o o ∂ ∂ − = × ∇ × ∇ ⇒ r r ε µ ( ) 2 2 2 1 t H c H ∂ ∂ − = × ∇ × ∇ ⇒ r r ( ) 2 2...
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This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell University (Engineering School).

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Lecture13 - 1 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lecture 13 Electromagnetic Waves in Free Space In this lecture you will

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