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aklec07 - 7 Hertzian dipole fields In the last lecture we...

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Unformatted text preview: 7 Hertzian dipole fields In the last lecture we found the retarded potential solutions z r kr t z I t z r e z I jkr ˆ ) cos( 4 ) , ( ˆ 4 ) ( ~ r A r A Domain Time Domain Frequency for a z ˆ directed Hertzian dipole. These oscillatory solutions represent spherical waves because of jkr e dependence. Note that in time domain this corresponds to ) cos( kr t . o Compare this to a plane wave traveling in + x direction: jkx e . Corresponding time domain expression is ) cos( kx t . The variable r is radial distance from origin whereas x is the distance from origin along x axis. o Thus, the phasor jkx e represents a plane wave, and the phasor jkr e represents a spherical wave. We will now find the fields radiated by the Hertzian dipole. We will use A B ~ ~ in spherical coordinates. sin ˆ sin ˆ sin ˆ ~ 2 A r A r A r r r r r r A We have earlier found ) ( ~ r A as ˆ ) ( ~ ˆ 4 ) ( ~ z r A z r e z I z jkr r A and we know that cos ˆ ˆ r z sin ˆ ˆ z ˆ ˆ z Therefore we can write in spherical coordinates and components as: ˆ ) ( ~ ~ sin 4 in ) ( ~ ˆ ) ( ~ ~ cos 4 cos ) ( ~ ˆ ) ( ~ ~ r A r A r A A r e z I s r A A r e z I r A r A jkr z jkr z r Substituting these into the curl formula, we can write sin cos ˆ sin ˆ sin ˆ 4 ~ 2 r e r r e r r r r r z I jkr...
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aklec07 - 7 Hertzian dipole fields In the last lecture we...

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