Electric Field and Potential of Three Charges on Circle

To determine the correct option, we analyze the given system of charges using geometric and electrostatic principles.

1. Distance Between Charges at $C$ and $B$

The three points $A$, $B$, and $C$ lie on a circle of radius $R$ centered at $O$.
The angle subtended by the chord $CB$ at the point $A$ on the circumference is given as:
$\angle \text{CAB} = 60^\circ$

By the inscribed angle theorem, the angle subtended by the arc $CB$ at the center $O$ is twice the angle subtended at the circumference:
$\angle \text{COB} = 2 \times \angle \text{CAB} = 2 \times 60^\circ = 120^\circ$

Now, considering triangle $\triangle COB$, where $OC = OB = R$ and $\angle \text{COB} = 120^\circ$, we apply the Law of Cosines to find the distance $CB$:
$CB^2 = OC^2 + OB^2 - 2(OC)(OB)\cos(\angle \text{COB})$
$CB^2 = R^2 + R^2 - 2R^2 \cos(120^\circ)$
$CB^2 = 2R^2 - 2R^2 \left(-\frac{1}{2}\right) = 3R^2$
$CB = \sqrt{3}R$

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2. Force Between Charges at $C$ and $B$

The charges placed at points $B$ and $C$ are:
$q_B = \frac{q}{3}, \quad q_C = -\frac{2q}{3}$

Using Coulomb's Law, the magnitude of the electrostatic force $F_{CB}$ between the charges at $C$ and $B$ is:
$F_{CB} = \frac{1}{4\pi\varepsilon_0} \frac{|q_C \cdot q_B|}{CB^2}$

Substitute the values of $q_B$, $q_C$, and $CB^2$:
$F_{CB} = \frac{1}{4\pi\varepsilon_0} \frac{\left|-\frac{2q}{3}\right| \cdot \left|\frac{q}{3}\right|}{3R^2}$
$F_{CB} = \frac{1}{4\pi\varepsilon_0} \frac{\frac{2q^2}{9}}{3R^2}$
$F_{CB} = \frac{2q^2}{108\pi\varepsilon_0 R^2} = \frac{q^2}{54\pi\varepsilon_0 R^2}$

This matches statement (C).

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3. Verification of Other Options

• Electric Potential at Point $O$ (Option D):
  $V_O = \frac{1}{4\pi\varepsilon_0 R} (q_A + q_B + q_C) = \frac{1}{4\pi\varepsilon_0 R} \left(\frac{q}{3} + \frac{q}{3} - \frac{2q}{3}\right) = 0$
  Hence, Option D is incorrect.

• Potential Energy of the System (Option B):
  The potential energy depends on the pairwise distances $r_{AB}, r_{BC}, r_{CA}$. Since the net potential energy is generally non-zero, Option B is incorrect.

• Electric Field at Point $O$ (Option A):
  The vector sum of electric fields depends on the explicit position vectors of $A, B, C$, which are not given to yield a net magnitude of $\frac{q}{8\pi\varepsilon_0 R^2}$ along the negative x-axis in general.

Thus, the correct statement is C.