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Question #42940

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ is fixed at the origin. The volume charge density within the sphere varies radially as $\rho(r) = \rho_0 \left( 1 - \frac{3r}{2R} \right)$ for $r \le R$ , where $\rho_0$ is a positive constant. For $r > R$ , the charge density is zero. A point charge $+q$ is moved from infinity to a point $P$ located at a distance $r = R/2$ from the center. The charge is moved along a complex, non-radial spiral path. What is the work done by the electrostatic field on the point charge during this entire process?

Options

A.
$\frac{5 q \rho_0 R^2}{192 \epsilon_0}$
B.
$-\frac{5 q \rho_0 R^2}{192 \epsilon_0}$
C.
$\frac{11 q \rho_0 R^2}{192 \epsilon_0}$
D.
$\frac{q \rho_0 R^2}{64 \epsilon_0}$

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

Conservative Field Path Independence|Non-conducting Charged Sphere|Electric Charges and Fields - Class 12

Why This Question Is Tricky

Common Mistakes

This problem tests the principle of path independence in conservative fields and the rigorous application of Gauss's Law across boundaries.

The primary trap lies in calculating the absolute potential at an interior point.

Students often erroneously integrate the interior electric field expression directly from infinity, ignoring the piecewise nature of the field and violating boundary continuity at $r=R$ .

Alternatively, they calculate the potential difference $\int_{R/2}^R E_{in} dr$ but incorrectly assume $V(R)=0$ , leading to Option D.

Another common mistake is misinterpreting 'work done by the electrostatic field' as work done by an external agent, resulting in a sign error (Option B).

The complex spiral path acts as a conceptual filter to ensure students rely on potential energy rather than attempting an impossible line integral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Conservative Field Path Independence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free