On bounding the equivalent radius

The concept of the equivalent radius, first introduced by Hallén, has been useful in the solution of antenna and scattering problems that involve electrically thin wires of arbitrary cross section. Generally, the solution is first calculated for circular wires of an arbitrary radius, and this radius is then replaced by the equivalent radius of the noncircular wire. In this way the original problem is reduced to that of finding the equivalent radius. Here we bound the equivalent radius through the use of certain isoperimetric inequalities so that the bounds are completely described by knowledge of the wire´s cross-sectional area and perimeter. For a given cross-sectional area it is shown that the circular wire possesses the smallest equivalent radius. Applications of these results to problems in shielding and scattering are suggested.