A uniform ring charge of radius R carries a total charge Q. Which one of the following options correctly quantifies the magnitude of the force on a point charge of strength q kept at the center of the ring? (ϵ is the permittivity of the medium)

The correct option is:

$$0$$

Step-by-Step Explanation:

Consider a uniform ring of radius R carrying a total charge Q distributed uniformly along its circumference. We want to find the force acting on a point charge q placed exactly at the geometric center of this ring.

By definition, a uniform charge distribution means that the charge is spread evenly. Thus, for any small element of charge on one side of the ring, there exists an identical, diametrically opposite charge element on the other side of the ring.

Let us analyze the electrostatic force exerted by these opposite elements on the point charge q at the center:
1. A small charge element $dQ$ on the ring exerts a electrostatic force on the charge q directed along the line joining them.
2. A diametrically opposite charge element $dQ$ of equal magnitude is located at the exact same distance R from the center.
3. This opposite element exerts an equal force on q, but in the exactly opposite direction.

Consequently, the electrostatic forces due to every pair of diametrically opposite elements on the ring cancel each other out vectorially. Since this cancellation occurs for all charge elements around the entire ring, the net electric field at the center of the uniformly charged ring is exactly zero:
$$E_{\text{center}}=0$$

Since the electric force on a point charge q in an electric field E is given by $F=qE$, the net force on the charge at the center of the ring is:
$$F=q\times 0=0$$

Thus, due to the symmetric distribution of charge around the center of the ring, the magnitude of the net force on the point charge q is indeed 0.