Final answer:
The expressions for the electric field at a very far away point and at the center of the ring are E_far = 0 and E_center = kQ/R^2, respectively.
Step-by-step explanation:
In the given question, we are asked to determine the expressions for the electric field at point P, which is a very far away point (z >> R) and at the center of the ring (z = 0).
For a very far away point, the correct expression for the electric field is E_far = 0, and for the center of the ring, the correct expression is E_center = kQ/R^2.
Therefore, option c) E_far = kQ/R^2, E_center = 0 is the correct answer.