This problem is an investigation of Rutherford's model of the atom. Although this model is considered primitive to modern standards, it provides a fairly accurate approximation of the electrostatics of the atom.
Consider an atom with number of protons and number of electrons. The protons are concentrated at the nucleus in the center. The electrons are spread out more or less in a spherical cloud between the nucleus to some atomic radius .
Part 1: Determine the electric field within the atom.
Part 2: Determine the electric potential within the atom.
Part 1: Electric field
We need to include the electric fields of the nucleus and the electron cloud.
If we consider the nucleus as a point charge with , then the electric field is
Consider the charge density of the electron cloud around a sphere
Thus the electric field of the electron cloud is
Part 2: Electric potential
To determine the electric potential, evaluate the integral .