Abstract

An electric dipole of finite size in the form of a spherical current sheet with a surface current density in the θ direction varying as sinθ is introduced. External to the spherical surface, the electromagnetic fields are of the same form as that of a point electric dipole situated at the origin and oriented in the z direction. The power and the power flux density are separated into the radiative and the reactive parts. This type of separation of power is related to the usual separation made on the basis of propagating and evanescent waves. There is a natural relationship between the radiative and the reactive parts with the far-field and the near-field effects, respectively.

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References

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  1. E. Wolf and J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16-18 (1998).
    [CrossRef]
  2. E. Wolf and J. T. Foley, “Do evanescent waves contribute to the far field? Errata,” Opt. Lett. 23, 1142 (1998).
    [CrossRef]
  3. T. Setala, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200-1206 (1999).
    [CrossRef]
  4. A. Lakhtakia and W. S. Weiglhofer, “Evanescent plane waves and the far field; resolution of a controversy,” J. Mod. Opt. 47, 759-763 (2000).
    [CrossRef]
  5. C. J. R. Sheppard and F. Aguilar, “Comment: Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177-180 (2001).
    [CrossRef]
  6. S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971), pp. 461-470.
  7. S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971), pp. 223-225.
  8. T. Padhi and S. R. Seshadri, “Radiated and reactive powers in a magnetoionic medium,” Proc. IEEE 55, 1123-1125 (1967).
  9. S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971), pp. 67-71.

2001

C. J. R. Sheppard and F. Aguilar, “Comment: Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177-180 (2001).
[CrossRef]

2000

A. Lakhtakia and W. S. Weiglhofer, “Evanescent plane waves and the far field; resolution of a controversy,” J. Mod. Opt. 47, 759-763 (2000).
[CrossRef]

1999

T. Setala, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200-1206 (1999).
[CrossRef]

1998

1967

T. Padhi and S. R. Seshadri, “Radiated and reactive powers in a magnetoionic medium,” Proc. IEEE 55, 1123-1125 (1967).

J. Mod. Opt.

A. Lakhtakia and W. S. Weiglhofer, “Evanescent plane waves and the far field; resolution of a controversy,” J. Mod. Opt. 47, 759-763 (2000).
[CrossRef]

C. J. R. Sheppard and F. Aguilar, “Comment: Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177-180 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. E

T. Setala, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200-1206 (1999).
[CrossRef]

Proc. IEEE

T. Padhi and S. R. Seshadri, “Radiated and reactive powers in a magnetoionic medium,” Proc. IEEE 55, 1123-1125 (1967).

Other

S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971), pp. 67-71.

S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971), pp. 461-470.

S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, 1971), pp. 223-225.

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Equations (46)

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A r ( r , θ ) = μ I m exp ( i k r ) 4 π r cos θ , A θ ( r , θ ) = μ I m exp ( i k r ) 4 π r sin θ ,
A ϕ ( r , θ ) = 0 .
H r ( r , θ ) = H θ ( r , θ ) = 0 ,
H ϕ ( r , θ ) = I m k 2 sin θ exp ( i k r ) 4 π ( h ϕ 1 i k r + h ϕ 2 k 2 r 2 ) .
E r ( r , θ ) = I m k 2 η 0 cos θ exp ( i k r ) 2 π ( e r 2 k 2 r 2 e r 3 i k 3 r 3 ) ,
E θ ( r , θ ) = I m k 2 η 0 sin θ exp ( i k r ) 4 π ( e θ 1 i k r + e θ 2 k 2 r 2 e θ 3 i k 3 r 3 ) ,
E ϕ ( r , θ ) = 0 .
J ( r ) = θ ̂ J θ sin θ δ ( r a ) = J * ( r ) ,
A r ( r , θ ) = cos θ μ M sin k r r ; A θ ( r , θ ) = sin θ μ M sin k r r ,
A ϕ ( r , θ ) = 0 .
H r ( r , θ ) = H θ ( r , θ ) = 0 ,
H ϕ ( r , θ ) = M k 2 sin θ ( cos k r k r sin k r k 2 r 2 ) ,
E r ( r , θ ) = i M k 2 η 0 2 cos θ ( cos k r k 2 r 2 sin k r k 3 r 3 ) ,
E θ ( r , θ ) = i M k 2 η 0 sin θ ( sin k r k r + cos k r k 2 r 2 sin k r k 3 r 3 ) ,
E ϕ ( r , θ ) = 0 .
M = M N I m M D 4 π ,
M N = exp ( i k a ) ( 1 1 i k a 1 k 2 a 2 ) ,
M D = sin k a + cos k a k a sin k a k 2 a 2 .
J θ sin θ = H ϕ ( r = a , θ ) H ϕ ( r = a + , θ ) .
J θ = I m k 4 π a M D .
P ( t ) = E ( r , t ) J ( r , t ) d r ,
E ( r , t ) = 1 2 [ E ( r ) exp ( i ω t ) + E * ( r ) exp ( i ω t ) ] ,
J ( r , t ) = 1 2 [ J ( r ) exp ( i ω t ) + J * ( r ) exp ( i ω t ) ] ,
P ( t ) = P R 2 cos 2 ω t + P I sin 2 ω t ,
P C = P R + i P I = 1 2 E ( r ) J * ( r ) d r .
P C = π a 2 J θ 0 π d θ E θ ( a , θ ) sin 2 θ .
P C = J θ 3 I m k 2 a 2 η 0 exp ( i k a ) ( 1 k 2 a 2 i k a + i k 3 a 3 ) .
P C = ( I m 2 η 0 k 2 12 π ) S ( k a ) ,
S ( k a ) = exp ( i k a ) M D ( 1 k a + i k 2 a 2 i ) .
S ( r , θ ) = 1 2 Re { [ r ̂ E r ( r , θ ) + θ ̂ E θ ( r , θ ) ] × ϕ ̂ H ϕ * ( r , θ ) } = r ̂ I m 2 η 0 k 2 sin 2 θ 32 π 2 r 2 .
P 0 = 0 2 π d ϕ 0 π d θ sin θ r 2 I m 2 η 0 k 2 32 π 2 r 2 sin 2 θ = I m 2 η 0 k 2 12 π .
S ( k a ) = 1 + i 3 2 k 3 a 3 for k a 1 .
S ( r , θ ) = S R ( r , θ ) + i S I ( r , θ ) ,
S R ( r , θ ) = r ̂ S R r ( r , θ ) + θ ̂ S R θ ( r , θ ) ,
S R r ( r , θ ) = η 0 I m 2 k 4 sin 2 θ 32 π 2 [ e θ 1 h ϕ 1 k 2 r 2 + N k 4 r 4 ( e θ 2 h ϕ 2 e θ 3 h ϕ 1 ) ] ,
S R θ ( r , θ ) = η 0 I m 2 k 4 sin 2 θ 32 π 2 N k 4 r 4 ( e r 2 h ϕ 2 e r 3 h ϕ 1 ) ,
S I ( r , θ ) = r ̂ S I r ( r , θ ) + θ ̂ S I θ ( r , θ ) ,
S I r ( r , θ ) = η 0 I m 2 k 4 sin 2 θ 32 π 2 [ N k 3 r 3 ( e θ 2 h ϕ 1 e θ 1 h ϕ 2 ) + e θ 3 h ϕ 2 k 5 r 5 ] ,
S I θ ( r , θ ) = η 0 I m 2 k 4 sin 2 θ 32 π 2 ( e r 2 h ϕ 1 k 3 r 3 + e r 3 h ϕ 2 k 5 r 5 ) .
J ( r ) = z ̂ I m δ ( r ) .
A z ( ρ , z ) = μ i I m 4 π 0 d η η J 0 ( η ρ ) ζ 1 exp ( i ζ z ) ,
ζ = ( k 2 η 2 ) 1 2 for η 2 < k 2 , = i ( η 2 k 2 ) 1 2 for η 2 > k 2 .
A z ( ρ , z ) = μ i I m 8 π d ζ exp ( i ζ z ) H 0 ( 1 ) ( η ρ ) ,
η = ( k 2 ζ 2 ) 1 2 for ζ 2 < k 2 , = i ( ζ 2 k 2 ) 1 2 for ζ 2 > k 2 .
Re S ( k a ) = 1 ,
Im S ( k a ) = [ sin k a k a + ( 1 k 2 a 2 1 ) cos k a ] M D ,

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