Before You Begin

Question #43804

Difficulty: 8/10 | 🟢 Free

The Question

A hydrogen-like atom (atomic number $Z=1$ ) is embedded in a non-homogeneous, isotropic dielectric medium. The relative permittivity of the medium varies with the radial distance $r$ from the nucleus as $\epsilon_r(r) = \frac{2a_0}{r}$ , where $a_0$ is the standard Bohr radius in vacuum. Let $E_0$ be the ionization energy of a hydrogen atom in vacuum. Assuming the Bohr quantization condition holds for the electron's circular orbits, determine the wavelength $\lambda$ of the photon emitted when the electron transitions from the first excited state ( $n=2$ ) to the ground state ( $n=1$ ). (Here $h$ is Planck's constant and $c$ is the speed of light).

Options

A.
$\frac{hc}{E_0 \ln 2}$
B.
$\frac{hc}{2 E_0 \ln 2}$
C.
$\frac{16 hc}{3 E_0}$
D.
$\frac{4 hc}{3 E_0}$

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

Emission vs Absorption|Energy Level Scaling|Resistivity vs Resistance

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of Gauss's Law in variable dielectrics with the Bohr model.

Students commonly fall into the trap of misapplying the standard Coulomb potential formula $V(r) = -q/(4\pi\epsilon r)$ by simply substituting the local $\epsilon(r)$ , which incorrectly yields a constant potential and zero force.

The correct physical principle requires deriving the electric field from the displacement field $\vec{D}$ using Gauss's Law, and then integrating $\vec{E}$ to find the non-Coulombic logarithmic potential.

Furthermore, students often blindly assume the standard Bohr radius scaling ( $r_n \propto n^2$ ), whereas the modified force law fundamentally alters the quantization scaling to $r_n \propto n$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Emission vs Absorption
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free