Cant understand some step in a solution

[nabliat]

here is the problem and solution:
http://i26.tinypic.com/2vvm8vm.jpg
http://i31.tinypic.com/nqwq4n.jpg

i solved the 1st subquestion.
my problem starts with the second subquestion:
i understood how they got this expression
$$E=\frac{\epsilon_0}{\epsilon}E_0$$

but i don't know what they do next
what law principle are they doing in the next 3 lines??

why the integral equals the expression that they show??

 

[nabliat]

replied in a wrong place

 

[kuruman]

The principle is that the electric field can be derived from the electric potential

$$\vec{E} = -\vec{\nabla U}$$

from which it follows that

$$V =-\int\vec{E} \cdot \vec{d \ell}$$

The line integral is taken along a straight line from one plate to the other and needs to be split into two terms because the E-field changes when the dielectric-vacuum boundary is crossed. That's why you get two terms with two different E values.

Does this make sense?

 

[nabliat]

$$\vec{E} = -\vec{\nabla U}$$
this means that the field is the divergent of the potential

$$V =-\int\vec{E} \cdot \vec{d \ell}$$
why are we starting to look for the potenitial
we are asked to find the total field
??

in the result i was not presented with the total electric field
but with sepated fields that i knew them already before the integral part

??

 

[kuruman]

My explanation was how they got equations (7) - (9). They did the line integral in two steps because there are two different electric fields between the plates. After doing the integral, they solve the equation to find E₀ in terms of the voltage, which we assume is a given quantity. Once you have E₀, you can find E.

 

[nabliat]

ok thanks :)