The Question is from neet-physics atoms-nuclei , and it was created by the author P.NARASIMHA

Question

Consider a hypothetical universe where the electrostatic force between two charges is proportional to $1/r^3$ instead of $1/r^2$.  According to the Bohr model adapted for this universe, how would the radius of the nth orbit ($r_n$) depend on n?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Bohr Model & Coulomb's Law
	**Related Topics**: Electrostatics, Atomic Physics, Quantization of Angular Momentum
	**Prerequisites**: Coulomb's Law, Centripetal Force, Bohr's Postulates
## SOLUTION PATHWAY
**Given Information**:
	Electrostatic force $F_e ∝ 1/r^3$
	Bohr Model
**Method & Steps**:
	1. **Force Balance**: In a stable orbit, the electrostatic force between the electron and nucleus provides the centripetal force.
		Reasoning:  For an electron to remain in a stable orbit, the inward electrostatic force must equal the outward centrifugal force.
		Formula/Rule: $F_e = F_c$
	2. **Express the forces**: The electrostatic force is given as $F_e = k'/r^3$ (where $k'$ is a constant of proportionality in this universe) and the centripetal force is $F_c = mv^2/r$.
		Working: $k'/r^3 = mv^2/r$
		Key Point:  The constant $k'$ will incorporate the charges of the electron and nucleus.
	3. **Quantization of Angular Momentum**: Bohr's quantization condition states that the angular momentum of the electron is quantized: $mvr = nħ$ (where $n$ is an integer and $ħ$ is the reduced Planck constant).
		Working: $v = nħ/(mr)$
	4. **Substitute and Solve**: Substitute the expression for $v$ from the quantization condition into the force balance equation.
		Working: $k'/r^3 = m(nħ/(mr))^2/r  => k'/r^3 = n^2ħ^2/(mr^3)$
	5. **Solve for r**: Simplify and solve for $r$ in terms of $n$.
		Result: $r = (m/k') (nħ)^2/m = (ħ^2/k')n => r ∝ n$
		Verification: The radius of the orbit is directly proportional to $n$ in this hypothetical universe.
 ## OPTION ANALYSIS
[A] r_n ∝ n² :x: This would be the case if the force were proportional to $1/r$.
[B] r_n ∝ n ✓: This is the correct relationship derived from the modified force law and Bohr's model.
[C] r_n ∝ 1/n :x:  An inverse relationship would imply the force increases with distance, which is not the case.
[D] r_n ∝ 1/n² :x: This is the relationship for energy levels in the standard Bohr model, not the radius.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. Using the standard $1/r^2$ force law: Remember the question specifies a different force law.
		2. Confusing energy levels with radius: Be careful to distinguish between the dependencies of energy and radius on $n$.
	**Quick Tips**:
		1. Remember the force balance is always a key starting point for orbital problems.
		2. Bohr's quantization condition is crucial for relating the radius to the quantum number $n$.

P.NARASIMHA · Answer

r_n ∝ n²

P.NARASIMHA · Answer

The correct Option is "r_n ∝ n"

P.NARASIMHA · Answer

r_n ∝ 1/n

P.NARASIMHA · Answer

r_n ∝ 1/n²

NEET Physics Atoms MCQs

Select Topic

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

NEET Subjects

Our Bots

Company