physics problem

No. The charge density doesn't vary with radial distance.
For a pair of charged conductive plates, the electric field set up between the two plates is a uniform field. This indicates that the charge density of the plates is constant throughout (formula: electric feild intensity = charge desnity/premittivity of free space). If not there will be non-uniform electric field set up, which would cause a flow of charges on the plates until a uniform field is established.

Yes, A capacitor is an arrangement of conductors that is used to store electric charge. For a capacitor made of two parallel conductive plates, its charge density vary with radial distance from the center of the plate. Because the potential difference between two plates of area A, separation distance d, and with charges +Q and -Q, is given by
圖片參考：http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter27/Chapter2704.gif