Fully filled capacitors with parallel dielectrics problem

[Amaelle]

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 (see the uploaded image)
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??

Many thanks in advance!

 

[cnh1995]

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

Charge density on the plates will not be uniform.

 

[Amaelle]

Thanks for your prompt reply, it makes sens but can you elborate more?

 

[Amaelle]

I just go why the charge density is different from dielectric to dielectric
many thanks !