Before You Begin

Question #43825

Difficulty: 8/10 | 🟢 Free

The Question

A thin spherical shell of radius $R$ contains $N$ uniformly distributed nucleons. The strong nuclear potential energy between two nucleons separated by a distance $r$ is modeled by the Yukawa potential $U(r) = -k \frac{e^{-\alpha r}}{r}$ , where $k$ (in $\text{J}\cdot\text{m}$ ) and $\alpha$ (in $\text{m}^{-1}$ ) are positive constants. A test nucleon is placed at a distance $x$ ( $0 < x < R$ ) from the center of the shell. What is the magnitude of the net nuclear force experienced by the test nucleon?

Options

A.
0 \text{ N}
B.
$\frac{k N e^{-\alpha R}}{\alpha R x^2} [\alpha x \cosh(\alpha x) - \sinh(\alpha x)] \text{ N}$
C.
$\frac{k N e^{-\alpha R}}{\alpha R x^2} [\alpha x \sinh(\alpha x) - \cosh(\alpha x)] \text{ N}$
D.
$\frac{k N e^{-\alpha x}}{\alpha R x^2} [\alpha R \cosh(\alpha R) - \sinh(\alpha R)] \text{ N}$

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 Force Range and Saturation|Shell Theorem (Internal vs External)|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the 'inverse-square bias' where students reflexively apply the Classical Shell Theorem to conclude the internal force is zero (Option A).

The Shell Theorem is a direct mathematical consequence of Gauss's Law, which strictly requires a $1/r^2$ force (or $1/r$ potential).

The strong nuclear force follows a Yukawa potential ( $e^{-\alpha r}/r$ ), which decays exponentially.

Because the force from the nearer side of the shell is exponentially stronger than the force from the farther side, the geometric cancellations of a $1/r^2$ field do not occur.

The test nucleon is pulled towards the nearest wall, requiring explicit integration to find the non-zero force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Force Range and Saturation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free