(My last https://www.physicsforums.com/showthread.php?p=4424810#post4424810 post did not get much attention so I try again without all these formulae. Think this will be more clear...) To derive the local field in a non-polar dielectric you assume a very small spherical cavity in which (since there is an applied field to it) you have made up surface charges. Integrating over those made up surface charges you get the Lorentz field from which you can derive the Clausius-Mosotti formula. My question is: when you integrate over those made up surface charges, why dont you also integrate over the real surface charges on the outside of the dielectric? Don't these two fields (made up and real) cancel each other?