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Question #43711

Difficulty: 8/10 | 🟢 Free

The Question

A projectile nucleus of mass number $A_1 = 8$ and atomic number $Z_1 = 4$ is fired from infinity with an initial kinetic energy $K$ towards a stationary, free-to-move target nucleus of mass number $A_2 = 27$ and atomic number $Z_2 = 13$ . The strong nuclear force is assumed to have a negligible range beyond the nuclear surface. Assuming the nuclear radius follows the relation $R = R_0 A^{1/3}$ (where $R_0$ is a constant) and the masses of the nuclei are exactly proportional to their mass numbers, what is the minimum value of $K$ required for the nuclei to just touch each other, thereby allowing the strong nuclear force to initiate a reaction? (Let $k = \frac{1}{4\pi\epsilon_0}$ and $e$ be the elementary charge).

Options

A.
$\frac{364}{27} \frac{k e^2}{R_0}$
B.
$\frac{52}{5} \frac{k e^2}{R_0}$
C.
$\frac{1820}{81} \frac{k e^2}{R_0}$
D.
$\frac{91}{2} \frac{k e^2}{R_0}$

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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

Fusion Barriers|Nuclear Density Invariance|Work against Conservative Force|Integration

Why This Question Is Tricky

Common Mistakes

This question tests two critical JEE Advanced concepts simultaneously.

First, it evaluates the application of Nuclear Density Invariance ( $R = R_0 A^{1/3}$ ) to determine the correct spatial onset of the strong nuclear force, trapping students who treat the projectile as a point mass or incorrectly sum the mass numbers before taking the cube root.

Second, it tests two-body collision mechanics.

A common trap is assuming the target nucleus remains stationary (infinite mass approximation) and directly equating the initial kinetic energy to the Coulomb potential energy.

This neglects the recoil energy of the target, violating the conservation of momentum in a free-space interaction, and forces the use of the reduced mass framework to find the true minimum initial kinetic energy.

Quick Facts About This Question

Question Information

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