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Question #43720

Difficulty: 8/10 | 🟢 Free

The Question

A dilute colloidal suspension of spherical semiconductor particles, each of radius $r$ and positive charge $q$ , is confined in a fluid medium between $x = 0$ and $x = L$ . A uniform electric field $E$ is applied in the $+x$ direction. The fluid is subjected to a steady temperature gradient such that the local absolute temperature is $T(x) = T_0(1 + \alpha x)$ , where $T_0$ and $\alpha$ are positive constants. The dynamic viscosity of the fluid varies with temperature as $\eta(T) = \eta_0 \frac{T_0}{T}$ .

The particle flux is governed by the Nernst-Planck equation $J = n(x)v_d(x) - D(x)\frac{dn}{dx}$ , where $n(x)$ is the local particle concentration, $v_d(x)$ is the local drift velocity determined by Stokes' drag, and $D(x)$ is the local diffusion coefficient given by the Stokes-Einstein relation $D(x) = \frac{k_B T(x)}{6\pi\eta(x) r}$ .

Assuming steady-state conditions with zero net flux ( $J = 0$ ), which of the following expressions correctly represents the concentration profile $n(x)$ ?

Options

A.
$n(x) = n(0) (1 + \alpha x)^{\frac{qE}{\alpha k_B T_0}}$
B.
$n(x) = n(0) \exp\left[ \frac{qE}{k_B T_0} \left( x + \frac{\alpha x^2}{2} \right) \right]$
C.
$n(x) = n(0) \exp\left[ \frac{qE}{\alpha k_B T_0} \left( 1 - \frac{1}{1+\alpha x} \right) \right]$
D.
$n(x) = n(0) \exp\left[ \frac{qE}{k_B T_0} x \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Drift vs Diffusion Current|Trajectory of Charged Particle|Viscosity and Stokes' Law

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental origins of the drift-diffusion model under non-equilibrium thermodynamic conditions.

The primary trap lies in the temperature-dependent viscosity $\eta(T)$ .

Students often mistakenly calculate an effective constant diffusion coefficient $D_0$ or assume a constant drift velocity $v_{d0}$ , leading to incorrect exponential profiles (Options B and C).

By forcing the use of the local force balance (Stokes' drag) and the local Stokes-Einstein relation, the problem reveals a beautiful physical cancellation: the local viscosity $\eta(x)$ drops out entirely when net flux is zero.

The concentration profile is dictated purely by the ratio of the conservative electric work to the local thermal energy, yielding a power-law distribution rather than the standard isothermal exponential decay.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Drift vs Diffusion Current
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free