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Question #42580

Difficulty: 8/10 | 🟢 Free

The Question

Consider a cylindrical conductor of length $L$ and uniform cross-sectional area carrying a steady current. The conductor has a steady-state temperature profile $T(x) = T_0 \left(1 + \frac{3x}{L}\right)$ from $x=0$ to $x=L$ . According to the classical Drude-Lorentz model, the free electrons behave as an ideal gas. Due to temperature-dependent scattering, the electron mean free path $\lambda$ is inversely proportional to the local absolute temperature $T$ . Assuming the free electron density remains uniform, what is the ratio of the local electric field at $x=L$ to the electric field at $x=0$ , $\frac{E(L)}{E(0)}$ ?

Options

A.
$8$
B.
$\frac{1}{8}$
C.
$4$
D.
$2$

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

Mean Free Path|Drift Velocity Mechanism|Kinetic Theory

Why This Question Is Tricky

Common Mistakes

This question tests the microscopic understanding of the Drude-Lorentz model under variable thermodynamic conditions.

A common trap is assuming the mean free path $\lambda$ is constant, which leads to $\tau \propto T^{-1/2}$ and an incorrect field ratio of 2.

Another mistake is ignoring the temperature dependence of the thermal velocity $v_{\text{rms}}$ , leading to a ratio of 4.

By forcing the integration of both $\lambda \propto 1/T$ (due to increased phonon scattering center density) and $v_{\text{rms}} \propto \sqrt{T}$ , the problem rigorously evaluates the student's ability to link microscopic collision frequencies to macroscopic current density and local electric fields.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Mean Free Path
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free