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Question #43835

Difficulty: 8/10 | 🟢 Free

The Question

In a specific semiconductor photocathode, the photoelectric threshold energy is exactly equal to its temperature-dependent band gap energy $E_g(T)$ . The band gap varies with absolute temperature $T$ according to the Varshni empirical relation: $E_g(T) = E_0 - \frac{\alpha T^2}{T+\beta}$ where $E_0$ is the band gap at $0\text{ K}$ , and $\alpha, \beta$ are positive material constants. If $\lambda_0$ is the threshold wavelength for photoelectric emission at temperature $T$ , which of the following expressions correctly represents the temperature sensitivity of the threshold wavelength, $\frac{d\lambda_0}{dT}$ ? (Here $h$ is Planck's constant and $c$ is the speed of light in vacuum).

Options

A.
$\frac{\alpha \lambda_0^2 T(T + 2\beta)}{hc(T+\beta)^2}$
B.
$-\frac{\alpha \lambda_0^2 T(T + 2\beta)}{hc(T+\beta)^2}$
C.
$\frac{\alpha \lambda_0^2 T^2}{hc(T+\beta)^2}$
D.
$\frac{\alpha \lambda_0^2 T(T + \beta)}{hc(T+2\beta)^2}$

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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

Energy Band Gap Temperature Dependency|Einstein's Photoelectric Equation|Calculus/Rate of Change

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple solid-state physics (temperature dependence of the band gap via the Varshni relation) with the Einstein photoelectric equation.

The primary trap is the misconception that the photoelectric work function (and thus threshold wavelength) is a strict constant for a given material.

By forcing the derivation of $d\lambda_0/dT$ , it evaluates whether students can correctly apply the chain rule to an inverse relationship (missing the negative sign leads to Option B) and accurately differentiate a rational function using the quotient rule (errors here lead to Options C or D).

It highlights that as temperature increases, the band gap shrinks, causing the threshold wavelength to shift to longer (redder) values.

Quick Facts About This Question

Question Information

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