V=    4π R³ /3

(c)The volume charge density of the electron is ρ=-3Ze/4πr³

 (D)In an electrical field E a force F₁ acts on charges given by

              F ₁ =Z.e.E

(E)The positive charge in the nucleus and the centre of the negative charges from the electron "cloud" will thus experience forces in different direction and will become separated. We have idealized situation show.

(F)The separation distanced will have a finite value because the separating force of the external field is exactly balanced by the attractive force between the centres of charges at the distance d. The attractive force F ₂ thus is given by

             F₂=q (Nucleus).q (e in d)/_4πεod2

(G) With q (Nucleus) =Zee and q (e in d=the fraction of the charge of the electrons contained in the sphere with the radios d, which is just the relation of the total volume of the sphere with radios d. Now we calculate the induced dipole moment, which is

                                                                      I_nduced=qd=Ze.d_e=4πε₀R³E

                              P=4πε₀R³E

The polarization P finally is given by multiplying with N, the density of the dipoles, we obtain

                               P=4πε₀R³NE

(H) This is our first basic result concerning the polarization of a material and its resulting susceptibility. There are various interesting points:

We justified the law of a linear relationship between E and P for the number of electronic polarization mechanisms. We can simply extend the result to a mix of different atoms. All we need to do is to sum over the relative densities of every type of atom. Concluding that electronic polarization is completely unimportant would be premature, however. Atoms in crystals or in any e a solids do not generally have spherical symmetry. Consider the Sp3 orbital of Si, Gee or diamond. Without a field, the centre of the negative charge of the electron orbital's will still coincide with the core but an external field breaks that symmetry producing a dipole momentum. The effect can be large compared to spherical S-orbital's: Si has a dielectric constant of 21, which comes exclusively from electronic polarization.