Before You Begin

Question #43809

Difficulty: 8/10 | 🟢 Free

The Question

Consider a hypothetical heavy nucleus of mass number $A$ and atomic number $Z$ . Instead of a uniform charge distribution, assume the proton charge density within the nucleus varies radially as $\rho(r) = k r^2$ for $r \le R$ , and $\rho(r) = 0$ for $r > R$ , where $k$ is a positive constant and $r$ is the radial distance from the center. The nuclear radius scales with mass number as $R = R_0 A^{1/3}$ , where $R_0$ is a constant. Calculate the electrostatic self-energy $U$ of this nucleus. (Let $e$ be the elementary charge and $\epsilon_0$ be the vacuum permittivity).

Options

A.
$\frac{5 Z^2 e^2}{36 \pi \epsilon_0 R_0 A^{1/3}}$
B.
$\frac{3 Z^2 e^2}{20 \pi \epsilon_0 R_0 A^{1/3}}$
C.
$\frac{Z^2 e^2}{72 \pi \epsilon_0 R_0 A^{1/3}}$
D.
$\frac{Z^2 e^2}{7 \pi \epsilon_0 R_0 A^{1/3}}$

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

Nuclear Radius Scaling|Self-Energy of Charged Sphere|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to calculate the electrostatic self-energy of a continuous, non-uniform charge distribution.

A common trap is assuming a uniform charge density (leading to the standard 3/5 coefficient) or calculating only the energy stored in the electric field strictly inside the nucleus (ignoring the field from R to infinity, which yields the 1/72 coefficient).

Students must correctly normalize the variable charge density using Gauss's law in integral form and apply the self-energy assembly integral dU = V dq before substituting the nuclear radius scaling law.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Radius Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free