Hertzian dipole

The Hertzian dipole is the textbook idealisation of an antenna: an infinitesimally short conducting segment of length L ≪ λ carrying a uniform sinusoidal current I(t) = I₀ cos(ωt). The model treats the dipole as a point source at the origin with oscillating current moment p_m = I₀ L, from which the full electromagnetic field at every point in space is computed by the retarded-potential formalism. The name honours Heinrich Hertz, whose 1887 spark-gap antenna was effectively one at the wavelengths he worked with, though the mathematical idealisation is typically attributed to later theoretical developments by Pocklington and others.

The far-field radiation from a Hertzian dipole is the canonical sin²θ doughnut pattern, with maximum broadside (θ = 90°) and perfect null along the axis (θ = 0° and 180°). The radiated power is P = (μ₀ ω² I₀² L²)/(12πc), the radiation resistance is R_rad = (2π/3)(μ₀c)(L/λ)² ≈ 80π²(L/λ)² Ω, and the directive gain is G = 1.5 (1.76 dBi). The Hertzian dipole is the building block for every more complex antenna: a half-wave dipole is a collection of Hertzian elements with a sinusoidal current envelope, an array is a collection of such dipoles with phase relationships, and a reflector or aperture antenna is the continuous limit. Its shortcomings as a real antenna are also instructive — the radiation resistance scales as (L/λ)² and becomes vanishingly small for L ≪ λ, requiring impractically large matching networks, which is why practical broadcast antennas are always at least a quarter-wavelength long.