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Question #42946

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The Question

A semiconductor bar of cross-sectional area $A$ is maintained at a constant temperature $T$ . The electron concentration along the $x$ -axis is given by $n(x) = n_0 e^{-x/\lambda}$ , where $n_0$ and $\lambda$ are positive constants. A steady total electric current $I$ flows through the bar in the $+x$ direction. Assuming the current is entirely due to electrons and the electron mobility is $\mu$ , which of the following expressions gives the required electric field $E(x)$ in the bar? (Here $e$ is the elementary charge and $k_B$ is the Boltzmann constant).

Options

A.
$\frac{I}{A n_0 e \mu} e^{x/\lambda} + \frac{k_B T}{e \lambda}$
B.
$\frac{I}{A n_0 e \mu} e^{x/\lambda} - \frac{k_B T}{e \lambda}$
C.
$\frac{I}{A n_0 e \mu} e^{-x/\lambda} + \frac{k_B T}{e \lambda}$
D.
$\frac{I}{A n_0 e \mu} e^{x/\lambda}$

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

Potential Gradient Direction|Drift Velocity Mechanism|Drift vs Diffusion Current

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between drift and diffusion mechanisms in a semiconductor.

A common trap is treating the electric field as solely responsible for the total current (ignoring diffusion), leading to the omission of the thermal voltage term.

Another lethal trap is misassigning the sign of the electron diffusion current.

Since electrons diffuse towards $+x$ (down the concentration gradient), they carry a negative charge, meaning the diffusion current is in the $-x$ direction.

To maintain a net positive current $I$ , the drift current must overcome this opposing diffusion current, requiring a larger positive electric field.

Students who use $J_{\text{diff}} = -e D \frac{dn}{dx}$ or fail to account for the negative charge of electrons will incorrectly subtract the diffusion term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Potential Gradient Direction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free