R1 Figure 6.4: Dipole antenna in a lossless unbounded medium. (f×E*) E. ER R Substituting from (6.21), we can also write Sau in terms of the directional weighting function as Sau=fi(Ace+ad Aio)-(4r aeon co) k2r We note that Sau describes the variation of the power density with 8, and is thus sometimes used as a descriptor of the power pattern of the sources Example of a current source radiating into an unbounded medium: the dipole antenna. A common type of antenna consists of a thin wire of length 2/ and radius a fed at the center by a voltage generator as shown in Figure 6.4. The generator induces an impressed current on the surface of the wire which in turn radiates an electromagnetic wave. For very thin wires(a <i, a <1 embedded in a lossless medium, the current may be accurately approximated using a standing-wave distribution J(r, o)=il(o)sin [k(-zD]8(x)8(y) We may compute the field produced by the dipole antenna by first finding the vector potential from(6.15)and then calculating the magnetic field from(6.16). The electric field may then be found by the use of Ampere's ②2001 by CRC Press LLCFigure 6.4: Dipole antenna in a lossless unbounded medium. fields we have, by (6.22), Sav = 1 2 Re Eˇ × (rˆ × Eˇ ∗) η  = rˆ Eˇ · Eˇ ∗ 2η . Substituting from (6.21), we can also write Sav in terms of the directional weighting function as Sav = rˆ ωˇ 2 2η  Aˇ eθ Aˇ ∗ eθ + Aˇ eφ Aˇ ∗ eφ  = rˆ k2η (4πr)2 1 2 aˇeθaˇ ∗ eθ + 1 2 aˇeφaˇ ∗ eφ  . (6.25) We note that Sav describes the variation of the power density with θ,φ, and is thus sometimes used as a descriptor of the power pattern of the sources. Example of a current source radiating into an unbounded medium: the dipole antenna. A common type of antenna consists of a thin wire of length 2l and radius a, fed at the center by a voltage generator as shown in Figure 6.4. The generator induces an impressed current on the surface of the wire which in turn radiates an electromagnetic wave. For very thin wires (a  λ, a  l) embedded in a lossless medium, the current may be accurately approximated using a standing-wave distribution: J˜i (r,ω) = zˆ˜I(ω)sin [k(l − |z|)] δ(x)δ(y). (6.26) We may compute the field produced by the dipole antenna by first finding the vector potential from (6.15) and then calculating the magnetic field from (6.16). The electric field may then be found by the use of Ampere’s law