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Question #42769

Difficulty: 8/10 | 🟢 Free

The Question

A fixed point charge $+Q$ is located at the origin $(0,0)$ . A small conducting sphere of radius $R$ and mass $m$ is constrained to move on a fixed, frictionless circular track of radius $a$ centered at $(2a, 0)$ . The sphere is connected to ground (potential $V=0$ ) via a long, fine, massless, and frictionless wire. Assume $a \gg R$ so that the charge distribution on the sphere remains approximately uniform. The sphere is given an initial velocity $v_0$ at the point $(3a, 0)$ on the track and undergoes non-uniform circular motion. What is the kinetic energy of the sphere when it reaches the point $(a, 0)$ ?

Options

A.
$\frac{1}{2}mv_0^2 + \frac{R Q^2}{9\pi\epsilon_0 a^2}$
B.
$\frac{1}{2}mv_0^2 - \frac{R Q^2}{9\pi\epsilon_0 a^2}$
C.
$\frac{1}{2}mv_0^2 + \frac{2 R Q^2}{9\pi\epsilon_0 a^2}$
D.
$\frac{1}{2}mv_0^2 + \frac{R Q^2}{18\pi\epsilon_0 a^2}$

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

Capacitance Independence|Work against Conservative Force|Non-Uniform Circular Motion|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to decouple the geometric constancy of capacitance ( $C = 4\pi\epsilon_0 R$ ) from the dynamic state variables ( $q, V$ ) during non-uniform circular motion in an external field.

The 'Lethal Collision' occurs when students miscalculate the work done by the electric field by either ignoring the changing self-energy of the sphere (using only $U = qV_{ext}$ , leading to Option C) or ignoring the interaction energy (using only $U = q^2/2C$ , leading to Option B).

A rigorous application of the Work-Energy Theorem requires recognizing that the total electrostatic potential energy must include both the self-energy of the capacitor and its interaction energy with the external field, and that a grounded source does zero work.

Quick Facts About This Question

Question Information

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