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Question #42253

Difficulty: 8/10 | 🟢 Free

The Question

A fixed solid non-conducting sphere of radius $R$ has a non-uniform mass density $\rho_m = \rho_0 r$ and a uniform volume charge density $\rho_e$ , where $r$ is the radial distance from the center. A point particle of mass $m$ and charge $-q$ ( $q > 0$ ) is released from rest at the surface of the sphere. Assuming the particle can move freely through the sphere without any resistance, which of the following expressions represents the speed of the particle when it reaches the center of the sphere? (Given $G$ is the universal gravitational constant and $\epsilon_0$ is the permittivity of free space).

Options

A.
$\sqrt{ \frac{2}{3} \pi G \rho_0 R^3 + \frac{q \rho_e R^2}{3 m \epsilon_0} }$
B.
$\sqrt{ \pi G \rho_0 R^3 + \frac{q \rho_e R^2}{3 m \epsilon_0} }$
C.
$\sqrt{ 2 \pi G \rho_0 R^3 + \frac{2 q \rho_e R^2}{3 m \epsilon_0} }$
D.
$\sqrt{ \frac{1}{3} \pi G \rho_0 R^3 + \frac{q \rho_e R^2}{6 m \epsilon_0} }$

Check Your Answer

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

Principle of Superposition|Electric Field Symmetry|Non-Uniform Acceleration Constraint

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-uniform acceleration fields generated by continuous, non-uniform mass and charge distributions.

A common trap is applying standard kinematic equations ( $v^2 = u^2 + 2as$ ) by evaluating the acceleration at the surface and assuming it remains constant, which leads to Option C.

Another trap is assuming the gravitational field inside a non-uniform sphere scales linearly with $r$ as it does in a uniform sphere, leading to Option B.

The correct approach requires deriving the position-dependent force fields using Gauss's Law for both gravity and electrostatics, and then integrating $v(dv/dr) = a(r)$ or applying the Work-Energy theorem to find the final velocity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Principle of Superposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free