# Why it can be asserted that the bottom plate takes the highest potential and the top plates takes the lowest of the capacitor? The ideal parallel capacitor has been filled with 2 dielectrics as shown in the above diagram.

$$V_{1} :=\text{potential at the top plate}$$

$$V_{2} :=\text{potential at the bottom plate}$$

$$V' :=\text{ potential at the border between the dielectrics }$$

$$V :=\text{ potential at any point inside the any colored domain }$$

We want to deduce the formula of $$\V\$$

$$D :=\text{ electric flux density } ~~ \leftarrow~~ \text{this value is constant at any point between the plates.}$$

$$E_{1}=\frac{ D }{ \epsilon_{1} }$$

$$E_{2}=\frac{ D }{ \epsilon_{2} }$$

The below 2 equations are the problems for me currently.

$$V'-V_{1} = E_{1} d_{1}$$

$$V_{2} -V'= E_{2} d_{2}$$

Concretely I can't get relations of magnitudes of the each potential.

Since

$$E_{1} d_{1} ~~,~~ E_{2} d_{2} >0$$

The below inequations must be held.

$$V_{2} \geq V' \geq V_{1} \geq 0$$

How can it be determined?

Why $$\V_{2}\$$ can be asserted that $$\V_{2}\$$ takes a highest value?

I thought the below.

$$Q:=\text{charge at the top plate}$$

$$\sigma :=\text{surface charge density of the top plate}$$

Since the electric flux density $$\D\$$ is constant at any point between the plates,

the surface charge density of the bottom plate can be said $$\ -\sigma \$$

By the way and needless to say , $$\V'\$$ is constant at any point inside the surface of border of the 2 dielectrics.

Of course electric lines of force mostly exist between the plates(few of electric lines of force exist outside between the plates).

What I've been missing or mistaking?

When you assumed $$E_{1} d_{1} ~~,~~ E_{2} d_{2} >0$$ you assumed the electric field points upwards. Therefore you assumed the bottom plate is at a higher potential than the top plate.
It's probably better if you assume $$\V_1\$$ and $$\V_2\$$ are known values and use them to calculate $$\E_1\$$ and $$\E_2\$$. You might even decide that either $$\V_1\$$ or $$\V_2\$$ is taken to be 0 (i.e. is the reference potential) to simplify the calculations.