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Question #43902

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical semiconductor resistor has an initial resistance $R_0$ at ambient temperature $T_0$ . It is connected to a variable current source $I$ . Due to the asymmetry of the interatomic potential well, the crystal lattice constant $a$ expands with temperature as $a(T) = a_0[1 + \alpha(T - T_0)]$ , where $\alpha$ is the linear thermal expansion coefficient. The resistor expands isotropically. The total number of free charge carriers in the resistor remains constant. The carrier mobility $\mu$ depends on the lattice constant as $\mu \propto a^{-3}$ . The resistor dissipates heat to the surroundings according to Newton's law of cooling with a cooling constant $K$ (in $\text{W/K}$ ). Assuming $\alpha(T - T_0) \ll 1$ and $K > 5\alpha R_0 I^2$ , which of the following expressions correctly represents the dynamic resistance $r_d = \frac{dV}{dI}$ of the device in the steady state?

Options

A.
$R_0 \frac{K^2 + 5\alpha R_0 K I^2}{(K - 5\alpha R_0 I^2)^2}$
B.
$\frac{K R_0}{K - 5\alpha R_0 I^2}$
C.
$R_0 \frac{K^2 + 2\alpha R_0 K I^2}{(K - 2\alpha R_0 I^2)^2}$
D.
$R_0 \frac{K^2 + 3\alpha R_0 K I^2}{(K - 3\alpha R_0 I^2)^2}$

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

Load Line Analysis|Dynamic Resistance|Interatomic Force Curve|Thermal Expansion of Area and Volume

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of solid-state physics, thermodynamics, and circuit analysis.

The primary trap lies in the definition of dynamic resistance ( $r_d = dV/dI$ ) versus static resistance ( $V/I$ ).

Students often calculate the temperature-modified static resistance and incorrectly select it as the answer (Option B), ignoring the electro-thermal feedback loop where an incremental change in current induces a further change in temperature and resistance.

Secondary traps involve miscalculating the geometric scaling of resistance: failing to recognize that constant total carriers $N$ in an expanding volume decreases carrier concentration $n$ , which combined with $L \propto a$ and $\mu \propto a^{-3}$ yields a steep $R \propto a^5$ dependence.

Options C and D represent these partial geometric oversights.

Quick Facts About This Question

Question Information

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