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Question #43642

Difficulty: 8/10 | 🟢 Free

The Question

A projectile nucleus of mass $m$ and charge $+e$ is fired with an initial kinetic energy $K_0$ towards a free target nucleus of mass $M = 2m$ and charge $+3e$ , which is initially at rest in the laboratory frame. During the head-on collision, the target nucleus is excited to a higher internal energy state, absorbing an excitation energy $\Delta E = \frac{K_0}{6}$ . Assuming the distance of closest approach $d_{min}$ is strictly greater than the sum of their nuclear radii, what is the value of $d_{min}$ ? (Here $\epsilon_0$ is the permittivity of free space)

Options

A.
$\frac{3e^2}{2\pi\epsilon_0 K_0}$
B.
$\frac{9e^2}{10\pi\epsilon_0 K_0}$
C.
$\frac{9e^2}{8\pi\epsilon_0 K_0}$
D.
$\frac{3e^2}{4\pi\epsilon_0 K_0}$

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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

Distance of Closest Approach|Nuclear Density Invariance|Equilibrium Stability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply energy conservation in an inelastic scattering event.

The primary trap is the 'Reduced Mass Effect' combined with internal excitation.

Students often incorrectly assume that the entire initial kinetic energy is converted into electrostatic potential energy (ignoring the recoil kinetic energy of the target) or they forget to subtract the energy lost to the internal excitation of the target nucleus.

By requiring the calculation of the available energy in the center-of-mass frame (or accounting for lab-frame recoil) and then equating it to the sum of the Coulomb barrier and the inelastic excitation energy, the problem forces a rigorous application of the Work-Energy theorem in a two-body dynamic system.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Distance of Closest Approach
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free