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Question #43175

Difficulty: 8/10 | 🟢 Free

The Question

An electromagnetic wave with electric field amplitude $E_0$ propagates through a non-magnetic ( $\mu = \mu_0$ ) ideal monatomic gas at a uniform absolute temperature $T$ . The gas acts as a dielectric medium with a temperature-dependent permittivity $\epsilon(T) = \epsilon_0 \left( 1 + \frac{C}{T} \right)$ , where $C$ is a positive constant and $\epsilon_0$ is the vacuum permittivity. According to the thermodynamics of dielectrics, the electric internal energy density of a field $E$ in such a medium is $u_E = \frac{1}{2} \left( \epsilon + T \frac{d\epsilon}{dT} \right) E^2$ . The total time-averaged thermodynamic internal energy density of the electromagnetic wave (sum of electric and magnetic energy densities) is found to be exactly equal to the thermal energy density of the gas. Given that the number density of the gas atoms is $n$ and $k_B$ is the Boltzmann constant, which of the following expressions gives the correct value of $E_0$ ?

Options

A.
$\sqrt{ \frac{6 n k_B T^2}{\epsilon_0 (2T + C)} }$
B.
$\sqrt{ \frac{3 n k_B T^2}{\epsilon_0 (T + C)} }$
C.
$\sqrt{ \frac{3 n k_B T}{\epsilon_0} }$
D.
$\sqrt{ \frac{10 n k_B T^2}{\epsilon_0 (2T + C)} }$

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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

Energy Density Partitioning|Equipartition Theorem|Temperature and Kinetic Energy

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of energy partitioning in electromagnetic waves within a thermodynamic dielectric medium.

The lethal trap is the rote assumption that the time-averaged electric and magnetic energy densities are always equal ( $\langle u_E \rangle = \langle u_B \rangle$ ).

While true in vacuum or simple linear dielectrics, in a medium where permittivity depends on temperature, the thermodynamic internal energy of the electric field includes an entropy-related term ( $T \frac{d\epsilon}{dT}$ ).

Students who ignore this or blindly double either the electric or magnetic energy density will arrive at incorrect pre-factors and temperature dependencies.

The problem forces the synthesis of Maxwell's equations ( $B = E/v$ ) with the Equipartition Theorem and dielectric thermodynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Energy Density Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free