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Question #42927

Difficulty: 8/10 | 🟢 Free

The Question

A system consists of three concentric conducting spherical shells of radii $R$ , $2R$ , and $3R$ . The innermost shell is isolated and carries zero net charge. The middle shell is grounded, and the outermost shell is isolated and carries a charge $Q$ . A small particle of mass $m$ and positive charge $q$ is attached to one end of an ideal non-conducting spring of spring constant $k$ and zero natural length. The other end of the spring is fixed at the common center of the shells. Initially, the particle is at equilibrium at the center. The particle is then displaced by a very small distance $x$ ( $x \ll R$ ) and released. Assuming $k > \frac{q^2}{4\pi\epsilon_0 R^3}$ , what is the time period of small oscillations of the particle?

Options

A.
$2\pi \sqrt{\frac{m}{k}}$
B.
$2\pi \sqrt{\frac{m}{k - \frac{q^2}{4\pi\epsilon_0 R^3}}}$
C.
$2\pi \sqrt{\frac{m}{k + \frac{q^2}{4\pi\epsilon_0 R^3}}}$
D.
$2\pi \sqrt{\frac{m}{k - \frac{qQ}{4\pi\epsilon_0 (3R)^3}}}$

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Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Charged Conducting Sphere|Potential of Concentric Shells|Neutral Equilibrium Points|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of electrostatic shielding, induced charges, and the method of image charges within a cavity.

A common trap is the misapplication of the Shell Theorem: students often assume that the electric field inside any conducting shell is strictly zero, ignoring the field produced by the non-uniform induced charge on the inner surface when the source charge is off-center.

This leads them to incorrectly choose the unperturbed time period.

Another trap is assuming the induced charge provides a stabilizing (restoring) force rather than a destabilizing one, leading to a sign error.

The presence of the outer shells acts as a distractor to test if students truly understand that the cavity is completely shielded from external charge configurations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Charged Conducting Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free