A slab of insulating material (not metal) with infinite area (infinite y-z plane

Question

A slab of insulating material (not metal) with infinite area (infinite y-z plane) has a uniform positive charge density . An edge view of the slab is shown in the above figure. You will find the electric field E inside the slab at distance x from the center (O).
a) Show that E = t/o.
b) Imagine a rectangular solid as a Gaussian surface of area A and thickness t (in x direction) inside the slab. Assume the left surface of the Gaussian surface is at x=0 (on the y-z plane). It is clear that electric field E, if any, is perpendicular to the y-axis and in the positive/negative x direction. Find the net flux that penetrates the Gaussian surface in terms of E and A.
c) Find the amount of charge inside the Gaussian surface in terms of A, t, and .

Please show all calculations, work, steps and answer question completely for full credit. Thank you!

Explanation / Answer

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Given : Slab is in Y - Z plane with infinite area .

Thickness is d in +ve x direction from 0 to d.

Volume charge density =

Gaussian Surface : surface area A in Y - Z plane .

Thickness t in the x direction inside the slab.

b) The net flux is = charge enclosed /0 = At/0 .

c) The amount of charge = (volume)() = (At)().

a) Flux is also given by = EA.

Equating two different representations of gives

EA = At/0 Implies

E = A/0 .

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