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Question #42528

Difficulty: 8/10 | 🟢 Free

The Question

Consider a blackbody cavity of volume $V$ at an equilibrium temperature $T$ . Due to photon neutrality, the number of photons is not conserved, and their chemical potential is zero. Let $E_w$ be the energy of a single photon corresponding to the wavelength $\lambda_{max}$ at which the spectral energy density per unit wavelength is maximum. Let $\langle E \rangle$ be the average energy of a photon in the cavity. If $x$ is the non-zero root of the equation $x = 5(1 - e^{-x})$ , which of the following expressions correctly represents the ratio $\frac{E_w}{\langle E \rangle}$ ? \ (Given: $\int_0^\infty \frac{y^{s-1}}{e^y - 1} dy = \Gamma(s)\zeta(s)$ , where $\zeta(s)$ is the Riemann zeta function, $\Gamma(4)=6$ , $\Gamma(3)=2$ , and $\zeta(4)=\frac{\pi^4}{90}$ )

Options

A.
$\frac{30 x \zeta(3)}{\pi^4}$
B.
$\frac{15 x \zeta(3)}{\pi^4}$
C.
$\frac{x \pi^4}{30 \zeta(3)}$
D.
$\frac{x \pi^4}{15 \zeta(3)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Wien’s Displacement Law|Photon Neutrality|Wave-Particle Duality Constraint

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge classical thermodynamics (Wien's Displacement Law) with quantum statistical mechanics (Bose-Einstein distribution for a photon gas).

A common trap is assuming that the peak of the energy density distribution corresponds directly to the average photon energy, or incorrectly applying classical equipartition theorems.

By explicitly stating that photon number is not conserved (zero chemical potential), it forces the evaluation of the average energy via the ratio of total energy density to total number density using the Riemann zeta function.

The wave-particle duality constraint is highlighted as students must navigate between the continuous wave spectrum (wavelength distribution for Wien's peak) and the discrete particle nature (average energy per photon).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Wien’s Displacement Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free