The current distribution on a very thin wire dipole antenna of overall length is given by à,lo sin 2 i, - ai sin where 1 is a constant. representing the distance r of (6-112) by the far-field approximations of (6-112a) through (6-112b), derive the far-zone electric and magnetic fields radiated by the dipole using (6-97a) and the far-field formulations of section 6.7. and (6-96b), or (6-97a) and (6-97b). however, the integrations, as will be shown next, can be simplified if the observations are made in the far field. if the observations are made in the far field (br 1), it can be shown [1] that the radial distance r of figure 6-3 a from any point on the source or scatterer to the observation point can be assumed to be parallel to the radial distance r from the origin to the observation point, as shown in figure 6-36. in such cases the relation between the magnitudes of r and r of figure 6-3a, given by r-+(2r cos (6-112) can be approximated, according to figure 6-3b, most commonly by [1] - r'cos for phase variations for amplitude variations (6-112a) (6-112b) r- where is the angle between r and r'. these approximations yield a maximum phase error of /8 (22.5°) provided the observations are made at distances 2d (6-113) r2a where d is the largest dimension of the radiator or scatterer. the distance (6-113) inf 6 113) for electric and magnetic currents i, and im, they in turn reduce to line integrals of the form -br e a , z') r (6-97a)