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lecture28 - 1 ECE 303 – Fall 2005 – Farhan Rana –...

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Unformatted text preview: 1 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Lecture 28 Antennas and Radiation and the Hertzian Dipole In this lecture you will learn: • Generation of radiation by oscillating charges and currents • Hertzian dipole antenna ECE 303 – Fall 2005 – Farhan Rana – Cornell University Maxwell’s Equations and Radiation Maxwell’s equation predict outgoing radiation from sinusoidally time-varying currents (and charges - recall that current and charge densities are related through the continuity equation: ( ) , . = ∇ t r H o r r µ ( ) ( ) t t r H t r E o ∂ ∂ − = × ∇ , , r r r r µ ( ) ( ) t r t r E o , , . r r r ρ ε = ∇ ( ) ( ) ( ) t t r E t r J t r H o ∂ ∂ + = × ∇ , , , r r r r r r ε Time-varying currents as the “source” or the “driving term” for the wave equation: ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t r J t t r E c t r E t t r E c t t r J t t r H t r E o o o ∂ ∂ − = ∂ ∂ + × ∇ × ∇ ⇒ ∂ ∂ − ∂ ∂ − = ∂ × ∇ ∂ − = × ∇ × ∇ , , 1 , , 1 , , , 2 2 2 2 2 2 r r r r r r r r r r r r r r µ µ µ ( ) ( ) , , . = ∂ ∂ + ∇ t t r t r J r r r ρ 911 2 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Electro- and Magneto-quasistatics and Potentials ( ) ( ) t r t r E o , , . r r r ρ ε = ∇ ( ) ( ) t r J t r H , , r r r r = × ∇ ( ) , . = ∇ t r H o r r µ ( ) , = × ∇ t r E r r Electroquasistatics Magnetoquasistatics ( ) , = × ∇ t r E r r Since: One could write: ( ) ( ) t r t r E , , r r r φ −∇ = Since: One could write: ( ) ( ) t r A t r H , , o r r r r × ∇ = µ ( ) , . = ∇ t r H o r r µ Scalar potential Vector potential ECE 303 – Fall 2005 – Farhan Rana – Cornell University Electrodynamics and Potentials - I ( ) , . = ∇ t r H o r r µ ( ) ( ) t t r H t r E o ∂ ∂ − = × ∇ , , r r r r µ ( ) ( ) t r t r E o , , . r r r ρ ε = ∇ ( ) ( ) ( ) t t r E t r J t r H o ∂ ∂ + = × ∇ , , , r r r r r r ε One still has: ( ) , . = ∇ t r H o r r µ Therefore, one can still introduce a vector potential: ( ) ( ) t r A t r H , , o r r r r × ∇ = µ Faraday’s law then becomes: ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + × ∇ ⇒ ∂ × ∇ ∂ − = × ∇ ⇒ ∂ ∂ − = × ∇ t t r A t r E t t r A t r E t t r H t r E o r r r r r r r r r r r r µ Vector Potential 3 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Electrodynamics and Potentials - II Using the vector and scalar potentials, the expressions for E-field and H-field become: ( ) ( ) t r A t r H , , o r r r r × ∇ = µ ( ) ( ) ( ) t r t t r A t r E , , , r r r r r φ ∇ − ∂ ∂ − = Scalar Potential Since: ( ) ( ) , , = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + × ∇ t t r A t r E r r r r One may introduce a scalar potential as follows: ( ) ( ) ( ) ( ) ( ) ( ) t r t t r A t r E t r t t r A t r E , , , , , , r r r r r r r r r r φ φ ∇ − ∂ ∂ − = ⇒ −∇ = ∂ ∂ + ECE 303 – Fall 2005 – Farhan Rana – Cornell University...
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