Introduction:
In this problem, we have n identical point charges distributed symmetrically on the periphery of a circle in the xy plane. We are required to find the ratio of the resultant electric field at two different points - (0, 0, R) and (0, 0, 2R).

Given:
- The charges are distributed symmetrically on the periphery of the circle x^2 + y^2 = R^2.
- The number of charges is n.
- The electric field at (0, 0, R) is denoted as E1.
- The electric field at (0, 0, 2R) is denoted as E2.

Analysis:
To find the electric field at a point due to multiple charges, we can use the principle of superposition. The total electric field at a point is the vector sum of the electric fields due to each individual charge.

Electric Field at (0, 0, R):
To calculate the electric field at (0, 0, R), we consider the electric field due to each individual charge and then sum them up.

Electric Field due to a Single Charge:
The electric field due to a point charge q at a distance r from the charge is given by Coulomb's Law:

E = k * q / r^2

Where E is the electric field, k is the Coulomb's constant, q is the charge, and r is the distance.

Electric Field due to n Identical Charges:
Since the charges are distributed symmetrically on the periphery of the circle, the electric field due to each charge will have the same magnitude. However, the direction of the electric field vectors will cancel out each other in pairs.

Electric Field at (0, 0, R) (contd.):
Considering the symmetry of the charges, we can conclude that the electric field vectors due to charges on opposite sides of the circle will cancel out each other, resulting in zero net electric field in the x and y directions.

The only component that will contribute to the electric field at (0, 0, R) is the z-component. The z-component of the electric field due to each charge will be the same and will add up.

Hence, the resultant electric field at (0, 0, R) is given by:

E1 = n * Ez

Where Ez is the z-component of the electric field due to a single charge.

Electric Field at (0, 0, 2R):
To find the electric field at (0, 0, 2R), we can use the same approach as before. However, the distance between the charges and the point of interest has changed.

Considering the symmetry of the charges, we can conclude that the electric field vectors due to charges on opposite sides of the circle will cancel out each other, resulting in zero net electric field in the x and y directions.

The only component that will contribute to the electric field at (0, 0, 2R) is the z-component. The z-component of the electric field due to each charge will be the same and will add up.

Hence, the resultant electric field at (0, 0, 2R) is given by:

E2 = n *