The theory of electrostatics is embedded within Maxwell’s electrodynamics, which is one of the most successful physical theories ever produced. However, a more direct generalisation of electrostatics to make it compatible with relativity would have been to replace it with the sourced massless Klein-Gordon equation. This paper gains insight into these two theories by comparing their structures and consequences in the static limit, finding that the two agree in terms of the field dynamics, but differ substantially in terms of how charged particles interact with the fields. These differences are most prominent for ultra-relativistic particles interacting with strong fields, conditions relevant to the new generation of high power laser facilities. Further connections between electrodynamics and scalar dynamics are explored by unifying the theories in higher dimensional spacetime, analogously to the Kaluza-Klein unification of electrodynamics and gravity.