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Question #42907

Difficulty: 8/10 | 🟢 Free

The Question

A small bead of mass $m$ and electric dipole moment of magnitude $p_0$ is threaded on a smooth, rigid quarter-circular wire of radius $R$ . The wire is fixed in the vertical $x$ - $y$ plane, spanning from point $A(0, R)$ to point $B(R, 0)$ . Gravity acts in the $-y$ direction. An external electrostatic field $\vec{E} = \alpha (y \hat{i} + x \hat{j})$ exists in the region, where $\alpha$ is a positive constant. The bead is released from rest at point $A$ and slides down to point $B$ . A mechanical constraint continuously aligns the bead's dipole moment vector $\vec{p}$ so that it is always tangent to the wire, pointing in the direction of the bead's velocity. Assuming the bead can be treated as a point mass with negligible rotational kinetic energy, what is its translational kinetic energy when it reaches point $B$ ?

Options

A.
$mgR - \alpha p_0 R$
B.
$mgR - 2\alpha p_0 R$
C.
$mgR + \alpha p_0 R$
D.
$mgR + 2\alpha p_0 R$

Check Your Answer

Correct Answer

Step-by-Step Solution

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

Dipole in Non-Uniform Field|Gravitational Potential Energy (System)|Work-Energy Theorem Validity

Why This Question Is Tricky

Common Mistakes

This question tests a profound misconception regarding the Work-Energy Theorem applied to extended objects (dipoles) with constrained orientations.

Students will typically attempt to use Conservation of Energy by calculating the change in the dipole's potential energy, $U_{elec} = -\vec{p} \cdot \vec{E}$ .

Evaluating this from $A$ to $B$ yields $\Delta U_{elec} = 2\alpha p_0 R$ , leading to the trap answer $mgR - 2\alpha p_0 R$ .

However, $U_{elec}$ accounts for BOTH the translational work and the rotational work done by the electric field.

Because the dipole is mechanically forced to remain tangent to the wire, the wire exerts a constraint torque.

This constraint torque does non-zero rotational work that absorbs exactly half of the electric potential energy change.

To find the translational kinetic energy, one must strictly integrate the translational force $\vec{F} = (\vec{p} \cdot \nabla)\vec{E}$ along the path.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dipole in Non-Uniform Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free