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Question #42671

Difficulty: 8/10 | 🟢 Free

The Question

A bead of mass $m$ and charge $+q$ is constrained to slide without friction along a vertical non-conducting wire aligned with the $y$ -axis. Two fixed point charges, each of charge $+Q$ , are located at coordinates $(d, 0)$ and $(-d, 0)$ in the $xy$ -plane. The bead is released from rest at $y = h$ . A programmable external mechanism applies a vertical force to the bead such that it undergoes exact vertical free fall kinematics, maintaining a constant downward acceleration $g$ . Simultaneously, a time-varying uniform electric field $\vec{E}(t) = E_0 t \hat{j}$ is active in the region. What is the total work done by the external mechanism from the moment of release until the bead reaches the origin ( $y=0$ )? (Assume $k = \frac{1}{4\pi\epsilon_0}$ )

Options

A.
$2kQq \left( \frac{1}{d} - \frac{1}{\sqrt{d^2+h^2}} \right) + \frac{2}{3} q E_0 h \sqrt{\frac{2h}{g}}$
B.
$2kQq \left( \frac{1}{d} - \frac{1}{\sqrt{d^2+h^2}} \right)$
C.
$2kQq \left( \frac{1}{d} - \frac{1}{\sqrt{d^2+h^2}} \right) - \frac{2}{3} q E_0 h \sqrt{\frac{2h}{g}}$
D.
$2kQq \left( \frac{1}{d} - \frac{1}{\sqrt{d^2+h^2}} \right) + \frac{1}{2} q E_0 h \sqrt{\frac{2h}{g}}$

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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

Superposition of Potentials|Mechanical Energy Conservation|Variable Force Integration|Kinematics of Vertical Free Fall

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple conservative and non-conservative work components in a constrained kinematic system.

The primary trap is blindly applying the Mechanical Energy Conservation theorem while either ignoring the time-varying electric field or incorrectly treating it as a static potential.

Furthermore, students often fail to express the path integral of a time-dependent force ( $\int \vec{F}(t) \cdot d\vec{r}$ ) by parameterizing the displacement $d\vec{r}$ in terms of the known velocity profile $\vec{v}(t)dt$ , leading to incorrect spatial averaging (Option D) or sign errors (Option C).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Superposition of Potentials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free