Before You Begin

Question #42729

Difficulty: 8/10 | 🟢 Free

The Question

A thermistor is made of an intrinsic semiconductor whose resistance varies with absolute temperature $T$ as $R(T) = A \exp\left(\frac{E_g}{2k_B T}\right)$ , where $E_g$ is the energy band gap, $k_B$ is the Boltzmann constant, and $A$ is a constant. The thermistor is connected in series with a temperature-independent resistor of resistance $R_0$ and an ideal battery of electromotive force $V$ . At a specific temperature $T_0$ , the potential difference across the thermistor is observed to be $\frac{V}{2}$ . What is the rate of change of the potential difference across the thermistor with respect to temperature, $\frac{dV_{th}}{dT}$ , at $T = T_0$ ?

Options

A.
$-\frac{V E_g}{8 k_B T_0^2}$
B.
$-\frac{V E_g}{4 k_B T_0^2}$
C.
$\frac{V E_g}{8 k_B T_0^2}$
D.
$-\frac{V E_g}{2 k_B T_0^2}$

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

Resistivity vs Resistance|Energy Band Gap Temperature Dependency|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to abandon the standard linear approximation for resistance ( $R = R_0(1 + \alpha \Delta T)$ ) and work directly with the intrinsic semiconductor model.

The primary trap is treating the temperature coefficient $\alpha$ as a constant or assuming metallic behavior (positive $\alpha$ ).

By forcing the derivation of $\alpha = \frac{1}{R}\frac{dR}{dT} = -\frac{E_g}{2k_B T^2}$ , the problem exposes students who fail to recognize that semiconductor resistivity drops exponentially with temperature.

Additionally, it integrates circuit analysis, where students often make algebraic errors in differentiating the voltage divider equation, leading to incorrect prefactors like $1/4$ or $1/2$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Resistivity vs Resistance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free