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Question #43155

Difficulty: 8/10 | 🟢 Free

The Question

A series LCR circuit consists of an inductor of inductance $L$ , a capacitor of capacitance $C$ , a metallic resistor of temperature-independent resistance $R_m$ , and a semiconductor resistor of resistance $R_s(T)$ . The semiconductor is intrinsic with a band gap energy $E_g$ , and its resistance varies with absolute temperature $T$ strictly due to the intrinsic carrier concentration $n_i \propto \exp\left(-\frac{E_g}{2k_B T}\right)$ , where $k_B$ is the Boltzmann constant. At a specific temperature $T_0$ , the metallic resistance and the semiconductor resistance are equal, such that $R_m = R_s(T_0) = R_0$ . If $Q_0$ is the quality factor of the circuit at temperature $T_0$ , what is the rate of change of the quality factor with respect to temperature, $\left. \frac{dQ}{dT} \right|_{T=T_0}$ ?

Options

A.
$Q_0 \frac{E_g}{4 k_B T_0^2}$
B.
$Q_0 \frac{E_g}{2 k_B T_0^2}$
C.
$-Q_0 \frac{E_g}{4 k_B T_0^2}$
D.
$-Q_0 \frac{E_g}{2 k_B T_0^2}$

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

Frequency Sensitivity of Z|Energy Band Gap Temperature Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of solid-state physics and AC circuit theory.

The primary trap lies in misapplying the temperature dependence of resistance.

Students often incorrectly assume $\frac{dR}{dT} > 0$ (as in metals) or forget the negative sign from differentiating $1/T$ , leading to a negative $\frac{dQ}{dT}$ .

A secondary trap is ignoring the constant metallic resistance $R_m$ in the derivative chain rule, which incorrectly yields a factor of $1/2$ instead of $1/4$ .

The correct approach requires recognizing that the semiconductor's resistance decreases with temperature, sharpening the resonance peak (increasing Q), and properly accounting for the partial contribution of $R_s$ to the total damping resistance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Frequency Sensitivity of Z
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free