Before You Begin

Question #43806

Difficulty: 8/10 | 🟢 Free

The Question

A dilute gas of $Li^{2+}$ ions (assume an infinitely massive nucleus) is placed in a uniform magnetic field along the $z$ -axis. The ions are excited to the $n=2$ state. Due to the Zeeman effect, the photons emitted along the $z$ -axis from the $n=2 \to n=1$ transition are right-circularly polarized. A collimated beam of these photons, with a uniform photon number density $n_p$ , propagates in vacuum and strikes a specialized flat surface at normal incidence. The surface perfectly reflects the electric field component of the wave oscillating along the $x$ -axis and perfectly absorbs the component oscillating along the $y$ -axis. Let $u_B$ be the time-averaged magnetic energy density of the incident beam, $h$ be Planck's constant, $c$ be the speed of light, and $R$ be the Rydberg constant. Which of the following correctly gives the radiation pressure $P$ exerted on the surface in terms of $u_B$ , and the value of $u_B$ ?

Options

A.
$P = 3 u_B \; ; \; u_B = \frac{27}{8} n_p h c R$
B.
$P = \frac{3}{2} u_B \; ; \; u_B = \frac{27}{4} n_p h c R$
C.
$P = 2 u_B \; ; \; u_B = \frac{27}{8} n_p h c R$
D.
$P = 3 u_B \; ; \; u_B = \frac{9}{8} n_p h c R$

Check Your Answer

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

Photon Polarization and Selection|Radiation Pressure|Energy Density Partitioning

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of quantum transitions, electromagnetic wave properties, and radiation pressure.

The primary trap lies in 'Energy Density Partitioning'.

Students often forget that for a plane wave in vacuum, the total energy density $u$ is the sum of the electric and magnetic energy densities ( $u = u_E + u_B$ ), and that they are equal ( $u_E = u_B$ ).

A student falling for this trap might assume $u_B$ represents the total energy density, leading to an incorrect pressure relation $P = \frac{3}{2} u_B$ and an incorrect $u_B$ value.

Additionally, the problem tests the understanding of circular polarization; students must recognize that the intensity is split equally between orthogonal axes, resulting in different momentum transfer efficiencies for the reflected and absorbed components.

Finally, forgetting the $Z^2$ dependence in the Bohr model for $Li^{2+}$ serves as a secondary trap.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Photon Polarization and Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free