In this paper, a model that describes the transient heating of a thin wire causing the tip to melt, roll-up of the molten mass into a ball due to surface tension forces, and the subsequent solidification of the molten material due to conduction up the wire and convection and radiation from the surface, has been provided. The wire is assumed to be heated at its lower tip to a temperature beyond the melting temperature of the wire material by heat Aux from an electrical discharge. The shape of the melt is analytically/numerically determined by solving equations based on minimum energy principles. The departure from sphericity of the melt that is formed is examined by perturbation schemes, based on expansions for small ratio of gravity to surface tension forces and small ratio of surface tension gradient to surface tension forces, both of which are true for the problems considered. Temperature fields in the melt have been obtained by solving the energy equation using a body-fitted coordinate system. Temperature fields in the wire above the melt were calculated as well. Comparisons of those temperatures with experimental measurements described in Part II of this study are excellent.