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Question #42795

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ is fixed in space and has a spherically symmetric volume charge density $\rho(r) = kr$ , where $k$ is a positive constant and $r$ is the distance from the center of the sphere. A particle of mass $m$ and charge $-q$ (where $q > 0$ ) is released from rest at a distance $2R$ from the center of the sphere. Assuming no other forces act on the particle, what is its speed when it reaches the center of the sphere? (Let $\epsilon_0$ be the permittivity of free space)

Options

A.
$\sqrt{\frac{5 q k R^3}{12 m \epsilon_0}}$
B.
$\sqrt{\frac{q k R^3}{2 m \epsilon_0}}$
C.
$\sqrt{\frac{q k R^3}{6 m \epsilon_0}}$
D.
$\sqrt{\frac{17 q k R^3}{36 m \epsilon_0}}$

Check Your Answer

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

Potential of Concentric Shells|Gauss Law Applicability|Integration

Why This Question Is Tricky

Common Mistakes

Students often memorize the potential formula for a uniformly charged sphere and attempt to adapt it by simply substituting a total charge $Q$ , ignoring that the non-uniform density $\rho(r) = kr$ fundamentally alters the electric field profile inside the sphere.

Furthermore, a common trap is failing to properly establish the boundary condition at the surface ( $r=R$ ) when integrating the internal electric field to find the central potential.

They either integrate $E_{\text{in}}$ from $R$ to $0$ and forget to add $V(R)$ , or they incorrectly assume the potential at the surface is zero.

This question forces a rigorous application of Gauss's Law to derive piecewise electric fields, followed by a nested integration from infinity to the center to correctly apply the Work-Energy Theorem.

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