# Fully filled capacitors with parallel dielectrics problem

consider that we have two dielectrics inside a capacitor as shown in the picture, let0s consider also that Q is the charge of the capacitor and d the distance between the two plates , the first dielectric occupy a surface of S/3 with a dielectric constant of er1 and the second a surface of 2S/3 with a dielectric constant of er2, the question is calculate the electric field inside the capacitor and the surface density of the induced charge

During the calculation I faced the following doubts: I applied gauss theorem to the flux density in both areas of the dielectrics and I got:

$$σ$$ is the surface density of the free charges $$=Q/S$$,

doing the same with the second surface I got that

and because my two dielectric are in parallel so $$E1=E2$$

this gives me

BUT this not true! so where did my logic fails?