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Question #43868

Difficulty: 8/10 | 🟢 Free

The Question

A rigid massless rod of length $L$ has a point charge $+q$ of mass $m$ attached to one end and a point charge $-q$ attached to the other end. The rod is pivoted at the $-q$ end (which is fixed at the origin) and is free to rotate in the $xy$ -plane. A torsional spring of stiffness $C$ is connected to the pivot, exerting a restoring torque $\tau_s = -C\theta$ when the rod is at an angle $\theta$ with the positive $x$ -axis. The entire system is immersed in a non-uniform electric field given by $\vec{E} = E_0(1 + \alpha x)\hat{i}$ , where $E_0$ and $\alpha$ are positive constants. If the rod is slightly displaced from its stable equilibrium position at $\theta = 0$ , what is the angular frequency $\omega$ of the resulting small oscillations?

Options

A.
$\sqrt{\frac{C + qE_0L(1 + \alpha L)}{mL^2}}$
B.
$\sqrt{\frac{C + qE_0L\left(1 + \frac{\alpha L}{2}\right)}{mL^2}}$
C.
$\sqrt{\frac{C + qE_0L}{mL^2}}$
D.
$\sqrt{\frac{qE_0L(1 + \alpha L)}{mL^2}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Dipole Potential Angular Dependence|SHM Definition via Differential Equation|Effective Spring Constant

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the effective spring constant $k_{eff} = d^2U/d\theta^2$ in non-uniform fields.

A common trap is to assume the electrostatic restoring torque is simply $\tau = -pE_{eff}\sin\theta$ , and incorrectly evaluate the electric field at the dipole's center of mass ( $x = L/2$ ), leading to an erroneous term $(1 + \alpha L/2)$ .

Another trap is ignoring the field gradient entirely for the torque calculation.

By forcing students to Taylor expand the exact potential energy function (or calculate the exact torque via $\vec{r} \times \vec{F}$ ), the problem differentiates those who rely on memorized uniform-field formulas ( $U = -pE\cos\theta$ ) from those who build the differential equation from first principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Dipole Potential Angular Dependence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free