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Question #42848

Difficulty: 8/10 | 🟢 Free

The Question

A small conducting sphere of mass $m$ , radius $R$ , and net charge $Q$ is attached to one end of a rigid, massless rod of length $L$ . The other end of the rod is pivoted at the origin $(0,0)$ and connected to a torsional spring of restoring constant $C$ . The system is constrained to rotate in a smooth horizontal $xy$ -plane. An external electric field is present in the region, given by $\vec{E} = \left(E_0 + E_1 \frac{x}{L}\right) \hat{i} - E_1 \frac{y}{L} \hat{j}$ , where $E_0$ and $E_1$ are positive constants. The rod is initially in equilibrium along the positive $x$ -axis. If the rod is given a small angular displacement $\theta$ and released, what is the angular frequency $\omega$ of the resulting simple harmonic motion? (Assume $R \ll L$ and neglect any radiation or damping effects. The permittivity of free space is $\epsilon_0$ .)

Options

A.
$\sqrt{ \frac{C + Q L (E_0 + 2 E_1) + 4\pi\epsilon_0 R^3 E_0 E_1}{m L^2} }$
B.
$\sqrt{ \frac{C + Q L (E_0 + 2 E_1)}{m L^2} }$
C.
$\sqrt{ \frac{C + Q E_0 L + 4\pi\epsilon_0 R^3 E_0 E_1}{m L^2} }$
D.
$\sqrt{ \frac{C + Q L (E_0 + 2 E_1) + 8\pi\epsilon_0 R^3 E_0 E_1}{m L^2} }$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Charged Conducting Sphere|Torsional Oscillations|Mechanical Energy Conservation|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the subtle interplay between mechanical restoring forces and electrostatic energy in non-uniform fields.

The primary trap lies in treating the conducting sphere merely as a point charge $Q$ and ignoring its finite size and polarizability.

In a non-uniform electric field, a conducting sphere redistributes its charge, creating an induced dipole moment $\vec{p} \propto \vec{E}$ .

The interaction of this dynamic dipole with the field gradient results in a polarization potential energy $U_{dipole} = -\frac{1}{2} \alpha_e E^2$ .

Students commonly misapply the standard SHM logic by only calculating the torque on the net charge $Q$ (Option B), failing to account for the additional restoring torque generated by the spatial variation of the field magnitude which couples with the induced dipole.

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