cant understand certain part of a solution..

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

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

but i dont 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??

but i dont 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??

Click to expand...

If you are talking about eq 7, that's just the definition of potential, the line integral of the electric field. Actually it should be the negative line integral, but they were sloppy and didn't write limits on the integral.

[latex]\vec{E} = -\vec{\nabla U}[/latex]
this means that the field is the divergent of the potential

[latex]V =-\int\vec{E} \cdot \vec{d \ell}[/latex]
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

??

I don't know what the question is. All I know about is the solutions that you posted. Eq 8 and 9 express the electric field in the two regions in terms of the potential, V.

this means that the field is the divergent of the potential

Click to expand...

BTW. The electric field is the negative gradient of the potential. You cannot take the divergence of a scalar.

how to know by the question that i need to present two expressions
for the filed??

i to know that i need to express it using the potential?