The optimal form for coreless inductors

A fundamental problem of inductor design is to obtain the greatest possible inductance with a piece of round wire. The optimal design is a solenoid with many layers of circular turns, and the maximum inductance scales with the five-thirds power of the total length of wire used. For practical reasons other design types, including solenoids and toroids, also are commonly used. These may have one, few, or many layers, and all have optimal forms with associated scaling rules. By describing the class of coreless inductors in a common notation, a significant unification of design is achieved.>