Coulomb's Law from Ambient Electrical Energy Axioms

Coulomb's law is traditionally introduced as an empirical foundation of electrostatics, established experimentally in the eighteenth century. In standard formulations, this law is typically recovered from Maxwell's equations or Gauss's law in the static limit. In this work, we present an alternative derivation based on a single physical axiom: the Ambient Electrical Energy E_0 fills all space with a ground state energy density u_0 , charges create smooth deformations of E_0 , and E_0 settles into the configuration that minimizes its total deformation energy. Assuming that E 0 responds linearly to deformation and that physical configurations minimize the total deformation energy, we derive the energy density u = 1/2 ε_0 |∇V |^ 2 instead of postulating it. Minimization of the associated energy functional leads directly to Laplace's equation as the equilibrium condition of the ambient energy. The unique spherically symmetric solution that vanishes at infinity yields the potential V (r) = q 4πε_0/r. The interaction energy between charges follows from the deformation of E_0, and the electrostatic force emerges as the gradient of this energy, giving Coulomb's inverse square law. Within this framework, the 1/r 2 dependence is not introduced empirically, but arises from the geometry of three-dimensional isotropic space together with the variational equilibrium of the ambient energy field.