Before You Begin

Question #43701

Difficulty: 8/10 | 🟢 Free

The Question

In a phenomenological model of a heavy nucleus of radius $R$ , the effective potential energy of a single nucleon at a distance $r$ from the center ( $0 < r \le R$ ) is given by: $U(r) = -U_0 \left[ 1 - \left(\frac{r}{R}\right)^2 \right] + \frac{k}{r^2}$ where $U_0$ and $k$ are positive constants. The first term represents the attractive mean field of the saturated nuclear matter, and the second term represents a hard-core repulsion. A nucleon is initially at its stable equilibrium position $r_0$ inside the nucleus. What is the minimum external work required to move this nucleon slowly to the nuclear surface ( $r = R$ )? Assume the nucleus remains otherwise unperturbed.

Options

A.
$\left( \sqrt{U_0} - \frac{\sqrt{k}}{R} \right)^2$
B.
$U_0 + \frac{k}{R^2}$
C.
$U_0 - \frac{k}{R^2}$
D.
$\left( \sqrt{U_0} + \frac{\sqrt{k}}{R} \right)^2$

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

Nuclear Force Range and Saturation|Conservative Force Field|Calculus (Potential Energy Gradient)

Why This Question Is Tricky

Common Mistakes

This question tests the application of conservative force fields within a phenomenological nuclear model.

A common misconception is confusing the 'saturation' of nuclear forces (constant binding energy per nucleon across different nuclei) with a constant spatial potential inside a single nucleus.

Students often incorrectly assume the interior force is zero everywhere or ignore the hard-core repulsion's role in establishing a local equilibrium.

By applying the Work-Energy Theorem to a conservative field, the work is strictly the potential difference $\Delta U$ .

The trap is failing to find the true equilibrium position $r_0$ where $dU/dr = 0$ , and instead assuming $r_0 = 0$ or misinterpreting the saturation term, which leads to incorrect work values like $U_0 + k/R^2$ .

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