1. The problem statement, all variables and given/known data Consider an x1:x2 mix of two linear isotropic dielectrics (by volume). x1+x2=1. On large length scales the mixture is homogenous and can be characterized with a dielectric constant ε_mix but on short length scales the material is granular so the dielectric constant is either ε_1 or ε_2. Neglecting spatial variations completely we have ε_mix = x1*ε_1 + x2*ε_2. Calculate the first correction to this result, if spatial variations δε = ε_2 - ε_1 are small (that is |δε| << ε_1,2. 2. Relevant equations Are you kidding? it's a dielectrics problem even if it wasn't 4:30 AM I probably wouldn't write them all. Basically equations for linear isotropic dielectrics and maybe Claussius-Mossotti. 3. The attempt at a solution I couldn't tell what form to assume for ρ the charge density, so unable to directly calculate the average dipole moment. So I tried to algebraically solve for ε_mix in terms of the dielectric constants and volume ratios...to no avail (probably). Then I re-read an email, the Prof. seemed to be implying that he wanted us to derive (in general) the displacement when the media is a dielectric mixture. However he does not specify we must use certain methods to solve a problem, so I think I can do it by directly calculating the average dipole moment. So I need to know a reasonable assumption to make for the charge density. Can you advise? Or point me to a reference? Or can you point me to a reference with the general derivation done? It's due tomorrow. Yes I'd like to start on hw early. The reality is, it's nearly impossible.