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Question #42794

Difficulty: 8/10 | 🟢 Free

The Question

A small bead of mass $m$ and positive charge $q$ is threaded on a rigid, rough, straight, non-conducting wire. The wire is fixed in the $xy$ -plane along the line $y = x$ , extending from the origin $(0,0)$ to the point $(L,L)$ . The coefficient of kinetic friction between the bead and the wire is $\mu$ . The entire setup is in a gravity-free space. An external electrostatic field exists in the region, with its electric potential given by $V(x,y) = \frac{E_0}{L} (x^2 - y^2)$ , where $E_0$ is a positive constant. The bead is projected from the origin along the wire with an initial speed $v_0$ . What is the minimum value of $v_0$ required for the bead to just reach the end of the wire at $(L,L)$ ?

Options

A.
$2 \sqrt{\frac{q E_0 L}{m}}$
B.
$\sqrt{\frac{2 \mu q E_0 L}{m}}$
C.
$2 \sqrt{\frac{\mu q E_0 L}{m}}$
D.
$2 \sqrt{\frac{q E_0 L (1 + \mu)}{m}}$

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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

Symmetry and Virtual Work|Kinetic Friction Transition|Equipotential Surface Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the magnitude of a force and its work-producing component in a spatially varying field.

A common trap is to reflexively calculate work by integrating the magnitude of the electric force ( $\int q|\vec{E}| ds$ ) when presented with a complex potential.

By placing the constrained path exactly on an equipotential line ( $V=0$ ), the electric force is strictly normal to the displacement, doing zero work.

The electric field's only mechanical role is to generate the normal constraint force, which in turn dictates the non-conservative frictional dissipation.

This evaluates the transition from conservative potential gradients to non-conservative work-energy applications.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Symmetry and Virtual Work
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free