Resistance of thin disks and rings

Abstract

In this paper the following, classic problem [1] is taken into consideration: calculate the resistance between two, perfectly conducting contacts on the rim of a thin disk (t << a, where a indicates disk’s radius and t is its thickness) of conductivity σ. An example of a simple, approximate solution to above problem can be found in [2], however, it requires contacts to be sufficiently small (i.e. d >> δ, where δ is the contacts’ size and d indicates the distance between them). Our purpose was to find more general solution, valid for contacts of finite size. Moreover, we propose the extension of above problem to thin rings. The exact analytical method of solving the Laplace equation to find the scalar potential was used. Afterwards, the Ohm law to obtain the formulas for disks and rings resistance was utilized. Using the dilogarithm function we obtained simplified, approximate formulas for the disks and rings resistance.