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Question #42930

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a uniform volume charge density and carries a total positive charge $Q$ . The electric potential is defined to be zero at the center of the sphere. A particle of mass $m$ and charge $-q$ (where $q > 0$ ) is performing uniform circular motion around the sphere at an orbital radius $r = 2R$ . What is the total mechanical energy of the particle? (Assume the permittivity of free space is $\epsilon_0$ and ignore gravity)

Options

A.
$\frac{5qQ}{16\pi\epsilon_0 R}$
B.
$-\frac{qQ}{16\pi\epsilon_0 R}$
C.
$\frac{3qQ}{16\pi\epsilon_0 R}$
D.
$-\frac{3qQ}{16\pi\epsilon_0 R}$

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

Non-conducting Charged Sphere|Reference Potential Convention|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's fundamental understanding of the electric potential as a line integral of the electric field, specifically challenging the rote memorization of the standard potential formula $V = kQ/r$ .

By shifting the zero-potential reference point to the center of the non-conducting sphere, students cannot simply use the standard formula.

They must correctly integrate the linear interior electric field from $r=0$ to $r=R$ , and the inverse-square exterior field from $r=R$ to $r=2R$ .

Common traps include defaulting to the standard reference at infinity (yielding $-qQ/16\pi\epsilon_0 R$ ), incorrectly setting the surface potential to zero, or mishandling the sign of the test charge when calculating potential energy.

The positive total energy for a bound orbit further challenges students who rely on the heuristic that bound states must have negative total energy, which is only true when $U(\infty) = 0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Non-conducting Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free