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Unformatted text preview: 6 Spherical waves In this lecture we will find out that short-filaments of oscillatory currents produce uniform spherical waves of vector potential propagating away from the filament. The relationship between spherical waves of vector potential and the corresponding electromagnetic wave fields will be examined in the next lecture. We recall that time-varying solutions of Maxwell’s equations can be ob- tained via B = ∇ × A , where the vector potential A ( r , t ) is related to time-varying current density J ( r , t ) via Time-domain: A ( r , t ) = μ o J ( r , t- | r- r | c ) 4 π | r- r | d 3 r . Frequency-domain: ˜ A ( r ) = μ o ˜ J ( r ) e- jk | r- r | 4 π | r- r | d 3 r , where k = ω √ μ o o . Δ z x y z θ φ r I ( z, t ) = I rect( z Δ z ) cos( ωt ) Hertzian dipole • We will next examine the implications of the above results from Lecture 4 for an ˆ z directed infinitesimal current filament defined as I ( r , t ) = I cos( ωt ) , for x = 0 , y = 0 ,- Δ z 2 < z < Δ z 2 , otherwise. 1 where constant I is specified in units of amperes (A). We can associate with this infinitesimal current the following current density function J ( r , t ) = Iδ ( x ) δ ( y ) cos( ωt )ˆ z, for- Δ z 2 < z < Δ z 2 , otherwise. = Iδ ( x ) δ ( y ) rect ( z Δ z ) cos( ωt )ˆ z A m 2 recalling that the dimension of an impulse δ ( x ) is m- 1 . Δ z x y z θ φ r J ( r , t ) = Iδ ( x ) δ ( y )rect( z Δ z ) cos( ωt )ˆ z • The oscillatory and ˆ z directed infinitesimal current filament of a length Δ z can in turn can be represented in terms of a phasor ˜ J ( r ) = Iδ ( x ) δ ( y ) rect ( z Δ z )ˆ z A m 2 . We can also re-write this as ˜ J ( r ) = I Δ z δ ( x ) δ ( y ) rect ( z Δ z ) Δ z ˆ z A m 2 in which the ratio with the rectangle in the numerator can be treated as “ δ ( z ) ” provided that – the width, Δ z , of the rectangle is considered an infinitesimal so that the ratio rect ( z Δ z ) Δ z represents...
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This note was uploaded on 09/27/2011 for the course ECE 450 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08