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Question #42929

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a radially varying volume charge density given by $\rho(r) = k r^2$ , where $k$ is a positive constant and $r$ is the radial distance from the center. The total charge of the sphere is $Q$ . A point charge $q_0$ is initially placed at a distance $r = R/2$ from the center. What is the work done by the conservative electric field of the sphere when the point charge $q_0$ is moved from $r = R/2$ to the surface of the sphere ( $r = R$ )? Assume the presence of $q_0$ does not alter the charge distribution of the sphere, and let $\epsilon_0$ be the vacuum permittivity.

Options

A.
$\frac{15 q_0 Q}{256 \pi \epsilon_0 R}$
B.
$\frac{3 q_0 Q}{32 \pi \epsilon_0 R}$
C.
$\frac{q_0 Q}{4 \pi \epsilon_0 R}$
D.
$\frac{q_0 Q}{8 \pi \epsilon_0 R}$

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

Non-conducting Charged Sphere|Conservative Force Field|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the integral form of Gauss's Law in a non-uniform charge distribution.

A common trap is reflexively treating the sphere as a point charge for internal points (yielding $E \propto 1/r^2$ ) or assuming a uniform charge density (yielding $E \propto r$ ).

Both misconceptions bypass the mandatory volume integration of $\rho(r) = kr^2$ required to find the true enclosed charge $q_{enc}(r) \propto r^5$ .

Consequently, students miscalculate the internal electric field as $E \propto r$ instead of $E \propto r^3$ , leading to an incorrect potential difference and work done.

This highlights the critical distinction between the algebraic shortcuts of Gauss's Law and its fundamental integral definition.

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