In this work, we present our continuum limit calculations of electrical interactions in ionic crystals and dielectrics. Continuum limit calculations serve two main purposes. First, they give an idea of how the macroscopic behavior of the material is related to the interactions at the atomistic scale. Second, they help in developing a multiscale numerical method, where the goal is to model the material both at the scale of atoms and at the macroscale. We consider two important settings for the continuum limit calculation: nanorod- like materials, where the thickness of a material in the lateral direction is of the order of the atomic spacing, and the materials, where atoms are randomly fluctuating due to the thermal energy. Our calculations, for the nanorod-like materials, show that the electrostatics energy are not long-range in continuum limit. We also consider the discrete system of dipole moments along the straight line and along the helix. We then compute the limit of the energy as the separation between the dipole moments tends to zero. The energy, in the continuum limit, is short-range in nature. This agrees with the calculations of [Gioia and James, 1997] for the magnetic thin films. We consider the system of atoms which are fluctuating due to thermal energy. We model the charge density field as a random field and compute the continuum limit of the electrostatics energy. In second part of the thesis, we present the Quasicontinuum method for the electro- mechanical deformation of the material at a finite temperature. There are two difficulties associated with this: one is the calculation of the phase average, and, second is the long-range interactions of the charged atoms. We use max-ent method presented in [Kulkarni et al., 2008] to formulate the problem as a minimization problem with respect to the atomic position and the atomic momenta. For the electrical interactions in the multi- scale method, we use the continuum limit of the energy for the random charge density field. We have modified the existing Quasicotninuum code, see [Marshall and Dayal, 2013], to implement the multiscale method. The code is an further extension of the code written by Jason Marshall [Marshall and Dayal, 2013].