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Question #43177

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a non-uniform volume charge density given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)$ for $r \le R$ , and $\rho(r) = 0$ for $r > R$ , where $\rho_0$ is a positive constant and $r$ is the radial distance from the center of the sphere. Let $U_{in}$ be the total electrostatic potential energy stored in the electric field within the volume of the sphere ( $r \le R$ ), and $U_{out}$ be the total electrostatic potential energy stored in the electric field outside the sphere ( $r > R$ ). What is the ratio $\frac{U_{in}}{U_{out}}$ ?

Options

A.
$\frac{1}{7}$
B.
$\frac{1}{5}$
C.
$\frac{1}{8}$
D.
$\frac{3}{5}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Energy Density Partitioning|Self-Energy of Charged Sphere|Shell Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to partition electrostatic self-energy by integrating the energy density of the electric field ( $u_E = \frac{1}{2}\epsilon_0 E^2$ ) over distinct spatial regions.

A common trap is to blindly apply the standard self-energy formula for a uniformly charged sphere ( $U = \frac{3kQ^2}{5R}$ ) or to assume the internal energy fraction is identical to the uniform case (which yields a ratio of $1/5$ ).

Furthermore, students often fail to correctly set up the volumetric integration element ( $dV = 4\pi r^2 dr$ ), mistakenly integrating the 1D energy density profile, or they incorrectly evaluate the piecewise discontinuity of the electric field at the boundary $r=R$ .

By forcing the derivation from microscopic energy density, the problem exposes superficial reliance on macroscopic formulas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Energy Density Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free