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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 Frequency-domain: A ( r ) = o 4 I z e- j k r r z Time-domain: 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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