Chapter.3.Appendix1b

Chapter.3.Appendix1b - Appendix 1 Details of the derivation...

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Appendix 1 Electric Field for = 0 in the Radiation zone Details of the derivation (page 78) for the electric field in the radiation zone, kr 1, are given below. E 0 ( r ) = ic ∇ × B 0 ( r ) = = k 2 c ∇×[ r r × p 0 ] f ( r ) where f ( r )= exp ( ikr ) r i 1 kr = k r × p 0 ] 1 r d dr exp ( ikr ) kr r × p 0 ]= ijk x ̂ i j kmn x m p o n = x ̂ i j [ x i p o j x j p o i ] = p 0 3 p 0 + r ( p 0 ∇)− p 0 ( r ∇) E 0 ( r ) = k [− 2 p 0 + r ( p 0 p 0 ( r ∇)] 1 r d dr h ( r ) where h ( r exp ( ikr ) kr = k [ r ( p 0 r ̂ )− p 0 ( r r ̂ )] d dr [ 1 r h ] − 2 k p 0 1 r h = k [ r ̂ ( p 0 r ̂ p 0 ] r d dr [ 1 r h ] − 2 k p 0 1 r h = k [ r ̂ ( p 0 r ̂ p 0 ][− 1 r h + h ′′ ]− 2 k p 0 1 r h = k [ r ̂ ( p 0 r ̂ p 0 ][− 1 r h + h ′′ 2 k p 0 1 r h = k [ r ̂ ( p 0 r ̂ p 0 ][ 2 r h + h ′′ 2 k p 0 1 r h k [ r ̂ ( p 0 r ̂ p 0 ] 3 r h = k [ r ̂ ( p 0 r ̂ p 0 ][− k 2 h ]+ k p 0 1 r h k [ r ̂ ( p 0 r ̂ )] 3 r h Note that h ( r exp ( ikr ) kr = f ( kr ) is a solution to the spherical Bessell equation when = 0 and kr 0: 1 r 2 d dr r 2 d dr f ( kr ) + ( k 2 ( + 1 )/ r 2 ) f ( kr ) = 0 2 r f + f ′′ +[ k 2 ( + 1 )/ r 2 ] f = 0 Finally, E 0 ( r ) = k [ r ̂ ( p 0 r ̂ p 0 ][− k 2 h k [ 3 r ̂ ( p 0 r ̂ p 0 ] 1 r h = k [ r ̂ ( p 0 r ̂ p 0 ][− k 2 h k [ 3 r ̂ ( p 0 r ̂ p 0 ] k r h [ i 1 kr ] =− k 3 [ r ̂ ( p 0 r ̂ p 0 ] exp ( ikr ) kr + k 3 [ 3 r ̂ ( p 0 r ̂ p 0 ] 1 kr exp ( ikr ) kr [ 1 kr i ] = k 3 [( r ̂ × p 0 r ̂ ] exp ( ikr ) kr + k 3 [ 3 r ̂ ( p 0 r ̂ p 0 ] 1 kr [ 1 kr i ] exp ( ikr ) kr A1.1
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The last equation, using ( r ̂ × p 0 r ̂ =− r ̂ ( p 0 r ̂ )+ p 0 , converts the result to that given in Jackson, Eq. 9.18 on page 411. Whereas the magnetic induction, B , is perpendicular to the "radius" vector, r , at all distances, the electric field has components parallel to r , and a component parallel to the dipole moment, p 0 . This derivation gives the electric dipole fields. In the radiation zone B and E are given by; B ( r , t )→ k 3 ( r ̂ × p 0 ) exp i ( kr t ) kr ,f o r kr 1 E ( r , t k 3 [( r ̂ × p 0 r ̂ ] exp i ( kr t ) kr = B ( r , t r ̂ for kr 1 A1.2 A1.3 The units are given by k 3 p 0 q / r 2 .T h e exp i ( kr t ) kr is an outgoing spherical wave, with k = / c .
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This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

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Chapter.3.Appendix1b - Appendix 1 Details of the derivation...

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