C

Step 1: Consider the forces acting on the particle at the highest point and after being slightly displaced.
Step 2: The particle will fall under the influence of gravity. At the point of leaving the circle, the gravitational force will equal the centripetal force needed to maintain circular motion.
Step 3: Set the centripetal force equal to the component of the weight of the particle at the point of leaving. By applying energy conservation or centripetal force equations, we find that the critical height at which the particle loses contact is related to the radius of the circle.
Step 4: The relationship derived gives us ${h = \frac{R}{3}}$, which matches option C.
Therefore, the correct answer is C.