The conductance of dipoles of arbitrary size and shape

The real part of either the impedance or the admittance of dipoles of arbitrary size and shape can be computed rigorously without solving a boundary value problem of a partial differential equation. In analogy to a well-known method of potential theory, fields of standing waves can be generated by integrals over current filaments so that for a given frequency there exist dipole shaped surfaces normal to the electric field surrounded by distant surfaces of vanishing electric field strength. Boundaries of perfect conductors may be supposed to coincide with a dipole shaped surface and a distant closed surface. The transients of such fields of standing waves are intimately related to the steady state of the free radiating dipole, since, before the first waves reflected from the distant enclosure have come back, the dipole cannot know whether or not it is enclosed. Corresponding to the type of current filament, either the resistance, or the conductance, of the radiating dipole can be calculated by direct integrations, while the shape of the dipole is determined by an ordinary differential equation of first order. As an example, we compute a family of dipoles that all have the same conductance
G=(254 \Omega)^{-1}
and a length
2h
between limits
\lambda/2\leq 2h\leq 1.36\lambda/2
.