Far-field zone

The far-field zone is the region at distances much greater than a wavelength from an oscillating source (r ≫ λ, kr ≫ 1). In this region the field takes the form of an outgoing spherical wave: the electric and magnetic components are mutually perpendicular, transverse to the radial direction, in-phase with each other, with |B| = |E|/c, and with amplitudes that fall only as 1/r. This slow falloff is what allows radiation to escape: integrating the Poynting flux S = (1/μ₀)E×B over a sphere of radius r gives a total power independent of r, because the 1/r² decrease of S is exactly compensated by the 4πr² increase of the sphere's surface area. The same power crosses every concentric shell, forever — this is literally light, or radio, or X-rays, depending on ω.

The angular distribution of the far-field intensity is the radiation pattern, which for a simple oscillating dipole is the sin²θ doughnut of §10.2. For more complex sources (half-wave dipoles, arrays, reflector antennas) the pattern is obtained as the magnitude-squared Fourier transform of the current distribution on the source evaluated at the radiation wavevector k. The boundary with the near-field zone is conventionally placed at the Fraunhofer distance 2D²/λ, where D is the largest linear dimension of the source — inside this radius the wavefronts have not yet flattened into the quasi-planar form assumed by far-field antenna-pattern formulae; outside it, they have. In astrophysical contexts every observer is automatically in the far field of every source, and the radiation pattern directly determines what fraction of a star's, pulsar's, or synchrotron source's power is pointed at the Earth.