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Question #76965

Difficulty: 5/10 | 🟢 Free

The Question

An infinitely long straight wire with a uniform linear charge density $\lambda > 0$ is fixed along the $z$ -axis. A small dipole of mass $m$ and dipole moment $\vec{p}$ is attached to one end of a massless, inextensible string of length $L$ . The other end of the string is fixed at the origin $(0,0,0)$ . The dipole is constrained to move in the $x$ - $y$ plane. The orientation of the dipole is maintained constant such that $\vec{p} = p \hat{j}$ at all times. Gravity acts in the $-\hat{j}$ direction with acceleration $g$ . The magnitude of the electric field produced by the wire at a distance $L$ is $E_0$ . It is given that $\frac{p E_0}{L} = mg$ . If the dipole is given a horizontal velocity $v_0$ at the lowest point of its circular path, what is the minimum value of $v_0$ required for the dipole to complete a full vertical circle?

Options

A.
$\sqrt{2gL}$
B.
$\sqrt{6gL}$
C.
$\sqrt{10gL}$
D.
$\sqrt{5gL}$

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

Minimum Speed for Looping|Dipole in Non-Uniform Field|Electric Field Symmetry|Vector Cross-Product

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the superposition of gravitational and electrostatic potential energies for a dipole in a non-uniform field.

Students commonly fall into two traps: (1) They evaluate the electric force at the top and bottom (where it points downwards with magnitude $mg$ ) and incorrectly assume it acts as a constant effective gravity $g_{eff} = 2g$ everywhere, leading to $v_0 = \sqrt{10gL}$ . (2) They ignore the work done by the electric field, assuming that moving on a circle around a line charge means moving on an equipotential, forgetting that the dipole's fixed orientation changes its alignment with the radial field.

This leads to standard energy conservation and $v_0 = \sqrt{6gL}$ .

The correct application of the work-energy theorem reveals that the electric field does exactly enough positive work to cancel the negative work of gravity, resulting in a constant looping speed.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Minimum Speed for Looping
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free