Frequency-domain: µo J(r )e jk r r. where

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1 6Spheicalwaves In this lectue we will find out that shot-filaments of oscillato cuents poduceunifom spheical waves of vecto potential popagating awa fom the filament. The elationship between spheical waves of vecto potential and the coesponding electomagnetic wave fields will be eamined in the net lectue. We ecall that time-vaing solutions of Mawell s equations can be obtained via B = A, whee the vecto potential A(,t) is elated to time-vaing cuent densit J(,t) via Time-domain: µo J(,t ) c A(,t)= d 3. 4π Fequenc-domain: µo J( )e jk Ã() = d 3, 4π whee k = ω µ o ɛ o. We will net eamine the implications of the above esults fom Lectue 4foanẑ diected infinitesimal cuent filament defined as { I cos(ωt), fo =0,=0, I(,t) = << 0, othewise. 1 φ I(, t) =Iect( )cos(ωt) Hetian dipole

2 whee constant I is specified in units of ampees (A). We can associate with this infinitesimal cuent the following cuent densit function { Iδ()δ()cos(ωt)ẑ, fo J(,t) = << 0, othewise. = Iδ()δ()ect( )cos(ωt)ẑ A m ecalling that the dimension of an impulse δ() is m 1. φ J(,t)=Iδ()δ()ect( )cos(ωt)ẑ The oscillato and ẑ diected infinitesimal cuent filament of a length can in tun can be epesented in tems of a phaso We can also e-wite this as J() =Iδ()δ()ect( )ẑ A m. J() =I δ()δ() ect( ) ẑ A m in which the atio with the ectangle in the numeato can be teated as δ() povidedthat the width,, oftheectangleisconsideedaninfinitesimal so that the atio ect( ) epesents in effect an infinitel thin and tall function centeed about =0having a unit aea undeneath.

3 Net we substitute this cuent densit phaso J() (with consideed an infinitesimal) into the phaso fomula fo the etaded vecto potential to obtain µo J( )e jk Ã() = d 3 4π = J( ) {}}{ µo I δ( )δ( )δ( )ẑe jk d d d 4π whee the integations ae to be caied ove,,and in the ange to +. φ J() =I δ()δ()δ()ẑ These ae ve eas integals to take because of δ( ), δ( ),andδ( ) factos in the integand, and lead to (afte eplacing all,,and elsewhee in the integand b 0) Ã() = µ o I e jk 4π whee = as usual. Conveting this esult into time domain b multipling it with e jωt and taking the eal pat of the poduct we obtain A(,t)= µ o 4π ẑ, k) I cos(ωt ẑ. We have just finished deiving the etaded vecto potential solution of an oscillato infinitesimal cuent filament known as the Hetian dipole. 3

4 Ou esults indicate that fo a Hetian dipole oiented in ẑ diection, the vecto potential solution Fequenc-domain: Ã() = µ o I e jk 4π ẑ Time-domain: A(,t)= µ o k) I cos(ωt ẑ 4π is also oiented in the ẑ diection and oscillate in time at the fequenc ω of the oscillating dipole. Note that: 1. These vecto potential solutions descibe a spheical wave (as opposed to a plane wave) chaacteied b spheical sufaces of constant phase associated with e jk and cos(ωt k) vaiations in fequenc and time domains. λ = π k J φ. Spheical wave solution is unifom in the sense that the vecto potential phaso à is constant (in diection and magnitude) on spheical sufaces of constant phase (in analog to unifom TEM plane waves of electic and magnetic fields studies in ECE 39). 3. Cleal, the popagation speed of the spheical wave is v p = ω k = ω ω = c. µ o ɛ o 4

5 4. The spheical wave is also chaacteied b an oscillation amplitude that vaies as 1 awa fom the adiating souce In the net lectue we will take the cul of this esult (using spheical coodinates opeatos) to obtain spheical (but non-unifom) waves of B that accompan the A-waves, and then deive the accompaning spheical (but non-unifom) E-waves using Ampee s law. We will find out E- andb-waves deived fom A-waves ae in geneal non-unifom and fom beams of diections along which field magnitudes Ẽ and B maimie ove spheical planes of constant phase. The mathematical desciption of these beams is povided b the gain function and the solid angle of the adiating sstem to be defined and eploed in Lectue 10. In deiving E- andb-waves fom A we will not eplicitl wo about V (,t) and ρ(,t) that accompanies the Hetian dipole behavio (since J contains all infomation included in ρ vaiations). Fo completeness sake, howeve, let us eamine what kind of ρ(,t) vaiation should be epected fo the Hetian dipole. The Hetian dipole is a hpothetical adiation element defined and intoduced above. Its main utilit is that it has the simplest adiation popeties 5

6 that one could imagine and use as a building block to epesent moe complicated (and pactical athe than hpothetical) adiation elements. AHetiandipolewasdefinedasafilamentofaninfinitesimallength which is caing a constant (-independent) cuent at each instant of time t. Since outside the filament the cuent vanishes, chage consevation and the continuit equation ρ + J =0 t demand that thee has to be a time-vaing chage accumulation at the two ends of the filament. Since fo a ẑ diected Hetian dipole, J =ẑj,wecanwitethe phaso domain fom of the continuit equation as Thus, with and we get J jω ρ + J =0. J = Iδ()δ() ect( ) = Iδ()δ()[δ( + ) δ( )], ρ = 1 J jω = j I δ()δ()[δ( + ) δ( ω )]. 6 Cuent J = Iδ()δ()ect( ) Chage δ( /) ++δ( + /) ρ = j I ω δ()δ() [δ( + /) δ( /)] Depicted chage densit (ed)leads the depicted cuent densit (blue) pofile b a quate peiod because of j tem in chage densit. Positive esevoi of chage at <0 end of the dipole dischages into the negative esevoi at the othe end causing half a ccle of -diected cuent acoss the filament. B the end of half-ccle the top end is positivel chaged and the bottom end negativel, so a new half-ccle with motions in the opposite diection stats.

7 In time-domain this coesponds to ρ(,t)= I ω δ()δ()[δ( ) δ( + )] sin(ωt) C m 3 accompaning the cuent densit vaiation Cuent J = Iδ()δ()ect( ) J(,t) = Iδ()δ()ect( )cos(ωt)ẑ A m. Cleal, the esult above shows that the ends of a Hetian dipole element located at = ± seve as point-chage esevois (of opposite polaities) sustaining the cuent vaiations of the element. Radiated fields of the Hetian dipole should be attibuted to both the time-vaing ρ and the time-vaing J even though consideations of J will be sufficient to detemine the adiated fields owing to the dependence of ρ on J that is built-in within Mawell s equations. Chage δ( /) ++δ( + /) ρ = j I ω δ()δ() [δ( + /) δ( /)] Depicted chage densit (ed)leads the depicted cuent densit (blue) pofile b a quate peiod because of j tem in chage densit. Positive esevoi of chage at <0 end of the dipole dischages into the negative esevoi at the othe end causing half a ccle of -diected cuent acoss the filament. B the end of half-ccle the top end is positivel chaged and the bottom end negativel, so a new half-ccle with motions in the opposite diection stats. 7