Before You Begin

Question #42656

Difficulty: 8/10 | 🟢 Free

The Question

A small bead of mass $m$ (in $\text{kg}$ ) and charge $+q$ (in $\text{C}$ ) is threaded on a smooth, rigid, insulating wire shaped as a parabola $y = \alpha x^2$ (where $\alpha$ is a positive constant in $\text{m}^{-1}$ ) in the $x$ - $y$ plane. A uniform electric field $\vec{E} = E_0 \hat{i}$ (in $\text{V/m}$ ) exists throughout the region. The bead is released from rest at the origin $(0,0)$ . Assuming the entire setup is in a gravity-free space, find the magnitude of the normal force (in $\text{N}$ ) exerted by the wire on the bead when it reaches the point where the tangent to the wire makes an angle of $60^\circ$ with the positive $x$ -axis.

Options

A.
$\frac{3\sqrt{3}}{4} q E_0$
B.
$\frac{3\sqrt{3}}{2} q E_0$
C.
$\sqrt{3} q E_0$
D.
$\frac{\sqrt{3}}{4} q E_0$

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

Neutral Equilibrium Points|The Velocity-Position Identity|Relative Velocity in 2D|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply Newton's Second Law and the Work-Energy theorem in a constrained 2D path.

A common trap (The Velocity-Position Identity trap) is assuming that because the external force is purely in the x-direction, the x-acceleration is simply $a_x = qE_0/m$ .

Students then incorrectly use $v_x^2 = 2a_x x$ and $v^2 = v_x^2 + v_y^2$ to find the velocity.

This is fundamentally flawed because the normal force from the wire has an x-component, meaning $a_x \neq qE_0/m$ .

The correct approach requires using the Work-Energy theorem to find the true speed.

A secondary trap is failing to project the electric force onto the instantaneous normal axis when calculating the required centripetal force, leading students to equate the normal force solely to $m v^2 / \rho$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Neutral Equilibrium Points
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free