Applying the method of cases of equilibrium: From the unprecedented "case of equilibrium of Ampère's motor" to a contemporary deduction of Ampère's electrodynamic Force

Abstract The method of cases of equilibrium is a lesser-known technique that effectively bridges theory and practice to derive fundamental natural laws. In this article, we will explain the method of equilibrium cases: we will introduce the historical motivation behind the development of this technique and outline the differences from the interpolation method. We will describe how to obtain an unprecedented equilibrium case — suitable for easy replication in the classroom — based on Ampère's motor phenomenon, which, along with two already known cases, will allow us to develop a contemporary approach to deducing Ampère's force between current elements. To our knowledge, this method is unknown in courses of experimental or mathematical methods of electrodynamics. The deduction section will require prior knowledge of calculus and unit vector notation.