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Question #43643

Difficulty: 8/10 | 🟢 Free

The Question

A smooth horizontal table has a small hole at the origin $O$ . A smooth vertical rod passes through $O$ along the $z$ -axis. A nucleus of mass $M$ and charge $+Q$ is constrained to slide on the vertical rod above the table. An electron of mass $m$ and charge $-q$ is constrained to move on the table. The system is in a gravity-free space.

A light inextensible string is attached to the nucleus, passes upwards along the $z$ -axis, and wraps around a uniform solid disk of radius $R$ which is free to rotate about a fixed horizontal axis located high above the table.

Initially, the nucleus is at a height $z_0$ above the table, and the electron is in a steady circular orbit of radius $r_0$ around $O$ . An external torque is applied to the disk, quasi-statically unwinding the string to lower the nucleus until it reaches the hole $O$ . During this process, the electron continuously adjusts its orbit to remain in steady circular motion.

Find the total work done by the external torque. (Let $\alpha = \frac{1}{4\pi\epsilon_0} Qq$ )

Options

A.
$\frac{\alpha}{2} \left[ \frac{r_0^2 + 2z_0^2}{(r_0^2 + z_0^2)^{3/2}} - \frac{(r_0^2 + z_0^2)^{3/2}}{r_0^4} \right]$
B.
$\alpha \left[ \frac{1}{(r_0^2 + z_0^2)^{1/2}} - \frac{(r_0^2 + z_0^2)^{3/2}}{r_0^4} \right]$
C.
$\alpha \left[ \frac{1}{(r_0^2 + z_0^2)^{1/2}} - \frac{1}{r_0} \right]$
D.
$\frac{\alpha}{2} \left[ \frac{(r_0^2 + z_0^2)^{3/2}}{r_0^4} - \frac{r_0^2}{(r_0^2 + z_0^2)^{3/2}} \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Coulombic Central Force|Work-Energy Theorem for Rotation|Constraint Relations via String Length

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the Work-Energy Theorem to a coupled electro-mechanical system.

The primary 'lethal trap' is treating the Coulomb force purely as a static potential energy term (Option B) or assuming the electron's orbital radius remains constant (Option C).

Students often fail to recognize that as the nucleus moves, the changing geometric constraint alters the radial component of the Coulomb force acting on the electron.

Because angular momentum is conserved, this forces the electron's orbital radius and its kinetic energy to change continuously.

A correct application of the Work-Energy Theorem must account for the work done by the changing tension constraint, which manifests as a change in the electron's macroscopic orbital kinetic energy alongside the system's electrostatic potential energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Coulombic Central Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free