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Question #42704

Difficulty: 8/10 | 🟢 Free

The Question

A particle of rest mass $m_0$ and charge $-q$ is projected tangentially with speed $v_0$ from a radial distance $r = 2R$ in the vacuum space between two fixed concentric non-conducting spherical shells of radii $R$ and $4R$ . The inner shell carries a uniformly distributed charge $+Q$ , while the outer shell carries a uniformly distributed charge $-Q$ . The particle's trajectory just grazes the inner shell. It is given that the characteristic electrostatic energy parameter $U_0 = \frac{qQ}{4 \pi \epsilon_0 R}$ is equal to $10^{-6} m_0 c^2$ , where $c$ is the speed of light in vacuum. Which of the following is the correct expression for the required projection speed $v_0$ ?

Options

A.
$\frac{10^{-3}}{\sqrt{3}} c$
B.
$\sqrt{\frac{2}{3}} 10^{-3} c$
C.
$10^{-3} c$
D.
$\frac{10^{-3}}{2} c$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Potential of Concentric Shells|Quantization of Angular Momentum|Trajectory of Charged Particle

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of electrostatic potential with angular momentum conservation, while deploying a 'masterstroke' trap regarding relativistic mass variation.

By explicitly defining the energy parameter $U_0$ in terms of the rest mass energy $m_0 c^2$ and the speed of light $c$ , students are baited into assuming they must use relativistic momentum ( $L = \gamma m_0 v r$ ) and total energy ( $E = \gamma m_0 c^2 + U$ ).

However, evaluating the energy scale ( $U_0 = 10^{-6} m_0 c^2$ ) reveals the motion is highly non-relativistic ( $v \sim 10^{-3} c$ ).

Students who fall for the trap will suffer severe algebraic bloat attempting to solve transcendental equations, whereas recognizing the classical limit allows a swift solution using standard conservation laws.

Quick Facts About This Question

Question Information

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