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Unformatted text preview: 7 Hertzian dipole fields We concluded last lecture with the retarded potential solutions x y z r J = 2 k Frequencydomain: A ( r ) = o 4 I z e j k r r z Timedomain: A ( r , t ) = o 4 I z cos( t k r ) r z of a z directed Hertzian dipole. We noted that these oscillatory solutions describe spherical waves by virtue of the e j k r dependence of the potential phasor on r : the variable r measures distance in all directions away from the origin, as opposed to, say, x measuring distance only along one coordinate axis labelled as x . Thus, while the phasor variation e j kx describes a plane wave, the pha sor e j k r describes a spherical wave (see margin). We will next determine the magnetic and electric fields produced by a Hertzian dipole. 1 To calculate the magnetic field phasor B we will make use of B = A and A = r r 2 sin r sin r r A r r A r sin A in spherical coordinates. Given that A ( r ) = o 4 I z e j k r r z A z ( r ) and z r = cos , z = sin , z = 0 , it follows that A r = A ( r ) r = A z ( r ) cos , A = A ( r ) = A z ( r ) sin , A = A ( r ) = 0 . Substituting A r , A , A into the curl formula, we proceed as x y z r J A ( r ) = z A z ( r ) r A z cos  A z s in A ( r ) = A z ( r ) s in A r ( r ) = A z ( r ) cos A = o 4 I z r r 2 sin r sin r r e jkr r...
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 Fall '08
 Staff
 Electromagnet, Frequency, Hertz

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