NEET 2026 ALL Q #1926
COMPETENCY BASED
APPLY
4 Marks 2026 NTA-RE-NEET-2026 MCQ SINGLE
Consider a fixed uniformly charged insulating sphere with radius R and total charge +Q. A point charge -q $(q<
(A) $\sqrt{\frac{Qq}{4\pi\epsilon_{0}mR}}$
(B) $\sqrt{\frac{3Qq}{4\pi\epsilon_{0}mR}}$
(C) $\sqrt{\frac{2Qq}{3\pi\epsilon_{0}mR}}$
(D) $\sqrt{\frac{Qq}{3\pi\epsilon_{0}mR}}$
Correct Answer: D

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Detailed Solution

Step 1: Identify the Electric Field inside the sphere

For a uniformly charged insulating sphere of radius R and total charge Q, the electric field E at a distance r from the center (where r < R) is given by Gauss's Law: $$E = \frac{Q r}{4\pi\epsilon_{0} R^3}$$

Step 2: Determine the Restoring Force

The force F on a point charge -q placed at distance r is F = -qE. Substituting the expression for E: $$F = -\left(\frac{Qq}{4\pi\epsilon_{0} R^3}\right)r$$ This force is proportional to the displacement r and directed towards the center, which is the condition for Simple Harmonic Motion (SHM).

Step 3: Calculate the Angular Frequency

Comparing the force equation with the standard SHM equation F = -m\omega^2 r, we get: $$m\omega^2 = \frac{Qq}{4\pi\epsilon_{0} R^3}$$ $$\omega = \sqrt{\frac{Qq}{4\pi\epsilon_{0} m R^3}}$$

Step 4: Calculate Maximum Velocity

The maximum velocity vmax for an SHM is given by vmax = \omega A. Here, the amplitude A is the initial distance R. $$v_{max} = \sqrt{\frac{Qq}{4\pi\epsilon_{0} m R^3}} \times R$$ $$v_{max} = \sqrt{\frac{Qq}{4\pi\epsilon_{0} m R}}$$

Final Answer: (A)

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must synthesize knowledge of Gauss's Law and the dynamics of Simple Harmonic Motion to solve a physical system.
Knowledge Dimension: PROCEDURAL
Justification: The student is required to follow a multi-step derivation process involving electrostatics and mechanics to reach the final velocity expression.
Syllabus Audit: In the context of NEET, this is classified as COMPETENCY. It tests the integration of Electrostatics (Class 12) and Mechanics (Class 11), which is a hallmark of high-level competitive entrance examinations.

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