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Question #42303

Difficulty: 8/10 | 🟢 Free

The Question

A fixed solid non-conducting sphere of radius $R$ with a uniform volume charge density $\rho > 0$ is located beneath a fixed, smooth, non-conducting inclined plane of angle $\theta$ . The inclined plane is tangent to the sphere at a point $P$ . A small block of mass $m$ and charge $q > 0$ is released from rest on the incline at a distance $\sqrt{3}R$ along the incline from $P$ , higher up the plane. Given that the block just loses contact with the plane as it reaches $P$ , what is the speed of the block at $P$ ? (Assume acceleration due to gravity is $g$ and permittivity of free space is $\epsilon_0$ )

Options

A.
$\sqrt{gR (2\sqrt{3}\sin\theta - \cos\theta)}$
B.
$\sqrt{2\sqrt{3}gR \sin\theta}$
C.
$\sqrt{gR (2\sqrt{3}\sin\theta + \cos\theta)}$
D.
$\sqrt{gR (\sqrt{3}\sin\theta - \cos\theta)}$

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

Wedge Constraint Logic|Non-conducting Charged Sphere|Coulomb's Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the electrodynamic coupling of a kinematic constraint (contact loss) with Gauss's Law and the Work-Energy Theorem.

The primary trap lies in the variable normal force $N = mg \cos\theta - qE_{\perp}$ .

Students often incorrectly assume the electric field is constant or neglect its spatial variation when calculating work.

Furthermore, calculating the work done by the electric force requires recognizing the geometry of the tangent plane to correctly determine the radial distances ( $2R$ and $R$ ) from the sphere's center, and applying the potential formula for a solid sphere.

A common mistake is either ignoring the electric work entirely or making a sign error by assuming the electric force aids the motion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Wedge Constraint Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free