Step-by-step Derivations:

1. Determine the Geometry of the System

• The charges are placed at points A, B, and C on a circle of radius R with center O.
• The problem states that the angle ∠CAB = 60°. This is an angle subtended by the arc CB at point A on the circumference.
• The angle subtended by the same arc at the center, ∠COB, is twice the angle at the circumference. Therefore, ∠COB = 2 × ∠CAB = 2 × 60° = 120°.
• Let's determine the distance between points B and C. In triangle OBC, OB = OC = R (radii). We can use the law of cosines to find the length of the chord BC: $BC^2 = OB^2 + OC^2 - 2(OB)(OC) \cos(∠COB)$ $BC^2 = R^2 + R^2 - 2(R)(R) \cos(120°)$ Since $\cos(120°) = -1/2$: $BC^2 = 2R^2 - 2R^2(-1/2) = 2R^2 + R^2 = 3R^2$ $BC = R\sqrt{3}$
• The figure shows a symmetric arrangement where C is on the negative y-axis and the chord AB is horizontal. This implies symmetry about the y-axis. Due to this symmetry, AC = BC. This also means ∠CBA = ∠CAB = 60°. Consequently, ∠ACB = 180° - 60° - 60° = 60°. So, triangle ABC is an equilateral triangle with side length $L = R\sqrt{3}$.
• We can assign coordinates based on the figure: C at (0, -R). With ∠COB = 120° and symmetry, B is at $(R\cos(30°), R\sin(30°)) = (R\sqrt{3}/2, R/2)$ and A is at $(-R\sqrt{3}/2, R/2)$.

2. Evaluate Each Option

A: The electric field at point O

• The electric field is a vector quantity. $\vec{E}_O = \vec{E}_A + \vec{E}_B + \vec{E}_C$.
• Charges: $q_A = q/3$, $q_B = q/3$, $q_C = -2q/3$.
• Distances: OA = OB = OC = R.
• Electric field due to $q_A$ at O: $\vec{E}_A$ points from A to O. Its magnitude is $E_A = k |q_A| / R^2$ where $k=1/(4\pi\epsilon_0)$.
• Electric field due to $q_B$ at O: $\vec{E}_B$ points from B to O.
• The horizontal components of $\vec{E}_A$ and $\vec{E}_B$ are equal and opposite due to symmetry and $q_A=q_B$, so they cancel out. The vertical components add up.
• The angle that OA and OB make with the y-axis is 60°. The vertical component of $\vec{E}_A$ is $-E_A \cos(60°)$. The vertical component of $\vec{E}_B$ is $-E_B \cos(60°)$. $E_{A,y} + E_{B,y} = -2 (k(q/3)/R^2) \cos(60°) = -2(kq/(3R^2))(1/2) = -kq/(3R^2)$.
• Electric field due to $q_C$ at O: $q_C$ is negative, so $\vec{E}_C$ points towards C, which is in the negative y-direction. $E_{C,y} = -E_C = -k|q_C|/R^2 = -k|-2q/3|/R^2 = -2kq/(3R^2)$.
• Total electric field at O: $E_{O,y} = E_{A,y} + E_{B,y} + E_{C,y} = -kq/(3R^2) - 2kq/(3R^2) = -3kq/(3R^2) = -kq/R^2$.
• $\vec{E}_O = -{q \over {4\pi\epsilon_0 R^2}} \hat{j}$. The field has magnitude ${q \over {4\pi\epsilon_0 R^2}}$ and is directed along the negative y-axis. Option A is incorrect.

B: The potential energy of the system

• Potential energy $U = k(q_A q_B / r_{AB} + q_B q_C / r_{BC} + q_C q_A / r_{CA})$.
• As established, triangle ABC is equilateral with side length $L = R\sqrt{3}$. So $r_{AB} = r_{BC} = r_{CA} = R\sqrt{3}$.
• $U = {k \over {R\sqrt{3}}} [ (q/3)(q/3) + (q/3)(-2q/3) + (-2q/3)(q/3) ]$
• $U = {k \over {R\sqrt{3}}} [ q^2/9 - 2q^2/9 - 2q^2/9 ] = {k \over {R\sqrt{3}}} [-3q^2/9] = -{k q^2 \over {3R\sqrt{3}}}$
• $U = -{q^2 \over {12\pi\epsilon_0 R\sqrt{3}}}$. The potential energy is not zero. Option B is incorrect.

C: The magnitude of the force between the charges at C and B

• Using Coulomb's law: $F_{CB} = k {|q_C q_B| \over r_{BC}^2}$.
• Charges: $q_C = -2q/3$, $q_B = q/3$.
• Distance: $r_{BC} = R\sqrt{3}$, so $r_{BC}^2 = 3R^2$.
• $F_{CB} = {1 \over {4\pi\epsilon_0}} { |(-2q/3)(q/3)| \over (R\sqrt{3})^2 } = {1 \over {4\pi\epsilon_0}} { (2q^2/9) \over 3R^2 }$
• $F_{CB} = {1 \over {4\pi\epsilon_0}} {2q^2 \over 27R^2} = {2q^2 \over 108\pi\epsilon_0 R^2} = {q^2 \over 54\pi\epsilon_0 R^2}$.
• This matches the expression in the option. Option C is correct.

D: The potential at point O

• Potential is a scalar quantity: $V_O = V_A + V_B + V_C$.
• $V_O = k{q_A \over r_A} + k{q_B \over r_B} + k{q_C \over r_C}$.
• $r_A = r_B = r_C = R$.
• $V_O = {k \over R} (q_A + q_B + q_C) = {k \over R} (q/3 + q/3 - 2q/3)$
• $V_O = {k \over R} (0) = 0$.
• The potential at point O is zero. Option D is incorrect.

Conclusion

Based on the step-by-step analysis, only option C is correct.