Why Assume Potential Only in the x Direction in Electrostatics BVP?

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Homework Statement

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Homework Equations

The Attempt at a Solution

As you can see, this is an example problem from my text. I'm looking for some clarification in their decision to assume potential exists only in the x direction. Specifically, I don't understand why they assumed the potential only varies in the x direction when the dialectric encloses a volume charge. I know when the charge is a surface charge on parallel plates with surface Area >> than the distance between them, we can assume the plates are of infinite size and so the electric field between them is normal to the surface. But here, the plates are BOTH at a potential of 0 V and the enclosed charged is a volume... So I'm hoping someone can elaborate on this.

I was also hoping someone can help me understand how they are using Gauss's law in exercise 5.1 to solve for the electric field. I tried many ways to solve it and I don't always get the answer they do. I'm thinking that you use the E-field found in the example to find E(x) at x=0 and x=d?
Thanks.

 

[2Tesla]

First, let's say you had a uniformly charged insulator that was arbitrarily large in all 3 directions. Would there be any potential in it? Why (not)? Now think about what happens if it's only arbitrarily large in y and z, but in x has some finite width.

As for the exercise, you don't need anything from the example to find the electric field. Just put your Gaussian "pillbox" so that two of its faces are just outside the plates, note that the system has a symmetry in the x-direction, and use as directed :)