An accessible approach to understanding the role of fields in mediating interactions

Abstract This article presents a comprehensive exposition on deriving the interaction potential between point-like particles within the framework of a scalar field. The Yukawa interaction energy, as well as the Coulomb interaction as a particular case, is obtained purely through classical arguments, without invoking field quantization, and directly from the field sources, with no need for artifacts such as the work-on-sources concept. Furthermore, we adopt an approach grounded in the language of classical field theory and based solely on dynamical equations, avoiding the use of analytic tools such as the Lagrangian and Hamiltonian density formalisms, in the hope that this exposition will help beginners appreciate the role of fields in mediating interactions and spark their interest in classical field theory. We also hope this approach is accessible, comprehensive, and self-contained for undergraduate physics students in the final year of a typical program. We recover a general formula that enables the calculation of the interaction energy between arbitrary stationary source distributions, not necessarily point-like, whose interaction is mediated by the scalar field. As an exercise for interested readers, we suggest investigating additional physical systems using the same framework employed in this work.