Is the Uniformity of Electric Field Between Grounded Metallic Plates Possible?

[pcm]

Homework Statement

Suppose there are 2 mettalic plates 1(left)&2(right). Left plate is grounded and right plate is given surface charge density sigma. Now since potential of left plate is 0, so there is no potential difference difference between infinity and plate and since field has to be uniform(?) it has to be zero otherwise there will be potentil diff between plate 1 and infinity . Now suppose the field between the plates is E.
Therefore potential of right plate will be Ed(d is distance between plates )..since field to the right of plate 2 is uniform it has to be zero or otherwise pd between plate 2 and infinity will be infinite..hence I conclude plate 2 is also at 0 potential .so E=0 everywhere which it can't be since I have charge on plate 2. What is wrong with it?

Homework Equations

The Attempt at a Solution

 

[pcm]

I am assuming plates to be infinite and that on all four faces charge is
uniformly distributed

 

[pcm]

"possibly" the charges distribute itself in such a way (non uniformly) that field is not zero everywhere, and at the same time potential of grounded plate is zero.
any comments on that...