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Question #43679

Difficulty: 8/10 | 🟢 Free

The Question

A heavy nucleus of atomic number $Z$ and radius $R$ is modeled as a non-conducting solid sphere with a uniform volume charge density $\rho = \frac{3Ze}{4\pi R^3}$ , where $e$ is the elementary charge. Suppose a spherical cavity of radius $R/2$ is formed within this nucleus. The cavity touches the outer surface of the nucleus, and its center is located at a distance of $R/2$ from the center of the original nucleus. What is the absolute electrostatic potential at the center of the original nucleus? (Assume the potential at infinity is zero, and $\epsilon_0$ is the vacuum permittivity).

Options

A.
$\frac{5Ze}{16\pi\epsilon_0 R}$
B.
$\frac{3Ze}{16\pi\epsilon_0 R}$
C.
$\frac{9Ze}{32\pi\epsilon_0 R}$
D.
$\frac{7Ze}{16\pi\epsilon_0 R}$

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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 Density Invariance|Non-conducting Charged Sphere|Cavity Field Shielding

Why This Question Is Tricky

Common Mistakes

This problem tests the application of the superposition principle for scalar fields in continuous charge distributions.

A common trap is that students memorize the fact that the electric field inside a spherical cavity is uniform ( $\vec{E} = \frac{\rho \vec{a}}{3\epsilon_0}$ ) and attempt to find the potential by integrating this constant field.

However, they fail to realize that this integration only yields the potential difference between points inside the cavity, not the absolute potential relative to infinity.

Integrating from infinity is mathematically bloated due to the piecewise nature of the field outside the cavity.

The elegant path bypasses the electric field entirely by directly superposing the scalar potentials.

A secondary trap is incorrectly calculating the cavity's potential contribution by using the formula for the center of a sphere instead of its surface, failing to recognize the geometric relationship between the two centers.

Quick Facts About This Question

Question Information

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