Question

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?
(\(\epsilon\) is the permittivity of the medium)

• A. \(\dfrac{Qq}{4\pi\epsilon R}\)
• B. \(\dfrac{Qq}{4\pi\epsilon R^2}\)
• C. \(0\)
• D. \(\dfrac{q}{4\pi\epsilon R}\times\dfrac{Q}{2\pi R}\)

Hint:

By symmetry, every charge element on a uniform ring has a diametrically opposite twin whose field at the center exactly cancels it.

Answer 1

The Correct Option is C

Step 1: Set up the ring using symmetry.
Consider the uniform ring of radius \(R\) with total charge \(Q\) spread evenly around its circumference, and the point charge \(q\) placed exactly at the geometric center \(O\) of the ring.
Step 2: Consider one small charge element on the ring.
Pick a small element of charge \(dq\) on the ring at some angle \(\theta\). By Coulomb's law, this element produces a field at the center of magnitude \(dE=dq/(4\pi\epsilon R^2)\), directed along the line joining that element to the center.
Step 3: Find the diametrically opposite element.
Directly across the ring at angle \(\theta+180^\circ\), there is another element carrying the same charge \(dq\), since the ring is uniform. It produces a field at the center of exactly the same magnitude \(dq/(4\pi\epsilon R^2)\), but pointing exactly opposite to the field from the first element, since the two elements sit on opposite ends of a diameter through the center.
Step 4: Cancel every such pair.
Since the ring is uniformly charged, every element has a diametrically opposite twin producing an equal and opposite field at the center. Adding up all such pairs around the full ring, every contribution cancels exactly, so the net electric field at the center of a uniformly charged ring is zero.
Step 5: Apply this to the force on \(q\).
The force on the point charge is \(F=qE_{center}\). Since \(E_{center}=0\), the force is \(F=0\), regardless of the values of \(Q\), \(q\), \(R\), or \(\epsilon\).
Step 6: Final Answer.
Options (A) and (B) look like they come from the point-charge formula \(Qq/(4\pi\epsilon R)\) or \(Qq/(4\pi\epsilon R^2)\), as if the ring's charge \(Q\) were concentrated at a single point at distance \(R\), which is not valid here since the ring's charge is spread out and the contributions cancel by symmetry rather than add up. Option (D) combines mismatched terms that do not correspond to any correct formula for this configuration.
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