How does electric field at a point change with distance r from an infinite thin sheet of charge?

How does electric field at a point change with distance r from an infinitely long charged wire?

A point charge q is placed at the origin. How does the electric field due to the charge vary with distance r from the origin?

State Gauss' theorem in electrostatics. Derive an expression for the electric field intensity at any point of to an infinite plane sheet of charge.

A thin circular ring of radius r is charged uniformaly so that its linear charge density becomes lambda . Derive an expression for the electric for the electric field at a point P at a distance x from it along the axis of the ring. Hence, prove that at large distances (x>>r), the ring behaves as a point charge.

How is electric field at a point related to potential gradient?