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Question #42891

Difficulty: 8/10 | 🟢 Free

The Question

A rough horizontal insulating rod is aligned along the $x$ -axis. A small bead B of mass $m$ and charge $+q$ is initially at rest at the origin $(0,0)$ on the rod. Another small bead A of charge $+Q$ is slowly moved along the line $y = h$ in the vertical $xy$ -plane (where the $y$ -axis points vertically upwards). The coefficient of static friction between bead B and the rod is $\mu_s = \frac{1}{3}$ . What is the maximum value of $Q$ such that bead B never slips on the rod during the entire motion of bead A from $x = -\infty$ to $x = +\infty$ ? (Assume acceleration due to gravity is $g$ and $k = \frac{1}{4\pi\epsilon_0}$ )

Options

A.
$\frac{\sqrt{2} m g h^2}{k q}$
B.
$\frac{\sqrt{3} m g h^2}{2 k q}$
C.
$\frac{\sqrt{5} m g h^2}{4 k q}$
D.
$\frac{2\sqrt{2} m g h^2}{3 k q}$

Check Your Answer

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

Coulomb's Law Vector Form|Static Friction Threshold|The Directional Sign Convention

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly model a position-dependent normal force and apply the static friction inequality across a continuous range of positions.

The primary trap is assuming the normal force remains constant ( $N = mg$ ), which leads to maximizing only the horizontal Coulomb force component (yielding Option B).

A secondary trap involves a sign error in the vertical force component, assuming the repulsive force lifts the bead and reduces the normal force (yielding Option C).

The problem requires recognizing that the critical slipping point is not where the driving force $F_x$ is maximum, but where the ratio of the driving force to the normal force is maximized.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Coulomb's Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free