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Question #42836

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density $\rho(r) = kr$ , where $k$ is a positive constant and $r$ is the radial distance from the center. A very narrow, smooth tunnel is drilled through the center of the sphere along a diameter. A point particle of mass $m$ and charge $-q$ ( $q > 0$ ) is released from rest inside the tunnel at a distance $A$ ( $A < R$ ) from the center. Let $T$ be the time period of the resulting oscillatory motion. Given that $\int_0^1 \frac{du}{\sqrt{1 - u^3}} = \beta$ , which of the following expressions correctly represents $T$ ?

Options

A.
$4\beta \sqrt{\frac{6m\epsilon_0}{qkA}}$
B.
$2\pi \sqrt{\frac{4m\epsilon_0}{qkR}}$
C.
$4\beta \sqrt{\frac{3m\epsilon_0}{qkA}}$
D.
$2\pi \sqrt{\frac{3m\epsilon_0}{qkA}}$

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

Continuous Charge Distribution|Non-conducting Charged Sphere|SHM Definition via Differential Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between general oscillatory motion and Simple Harmonic Motion (SHM).

A common trap is assuming that any restoring force inside a charged sphere leads to SHM.

Students often memorize that the internal field of a uniformly charged sphere is linear ( $E \propto r$ ), leading to a constant time period independent of amplitude.

Here, the non-uniform charge density $\rho = kr$ results in a non-linear electric field ( $E \propto r^2$ ).

Consequently, the restoring force is proportional to $x^2$ , making the differential equation non-linear.

Students who blindly apply the SHM time period formula by assuming an effective uniform density (e.g., evaluating $\rho$ at $r=A$ ) or linearizing incorrectly will choose the wrong option.

The correct approach requires integrating the equation of motion from first principles, revealing that the time period depends on the amplitude as $A^{-1/2}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Continuous Charge Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free