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Question #42563

Difficulty: 8/10 | 🟢 Free

The Question

Let $E_{avg}^{(d)}$ be the actual average kinetic energy per electron for a completely degenerate ( $T=0$ ) free electron gas in $d$ dimensions, having a Fermi energy $E_F$ . A student incorrectly assumes the gas behaves classically and applies the equipartition theorem to find a temperature $T_d$ such that the classical average energy of a $d$ -dimensional ideal gas equals $E_{avg}^{(d)}$ . Let $R_d = \frac{T_d}{T_F}$ , where $T_F = \frac{E_F}{k_B}$ is the Fermi temperature. What is the value of the ratio $\frac{R_3}{R_2}$ ?

Options

A.
$\frac{4}{5}$
B.
$\frac{5}{4}$
C.
$\frac{3}{5}$
D.
$\frac{2}{3}$

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

Equipartition Theorem|Fermi Level Position|Approximations (Binomial)

Why This Question Is Tricky

Common Mistakes

This question targets the lethal collision between classical Maxwell-Boltzmann statistics (Equipartition Theorem) and quantum Fermi-Dirac statistics in the degenerate regime ( $T \ll T_F$ ).

Students often blindly apply $\frac{3}{2} k_B T$ for energy, ignoring that the Pauli Exclusion Principle forces electrons into higher energy states even at absolute zero, creating a massive zero-point energy.

By comparing the classical temperature required to match this quantum ground-state energy across different dimensions, the trap exposes the failure to adjust the density of states $g(E) \propto E^{d/2-1}$ and highlights the fundamental breakdown of equipartition in high-density, low-temperature fermion systems.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Equipartition Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free