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Question #43757

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic light beam of wavelength $\lambda$ passes through a long single slit of width $a$ and length $L$ ( $L \gg a$ ). The resulting diffraction pattern falls on a flat photoelectric plate placed at a distance $D$ ( $D \gg a$ ) from the slit. The work function of the plate is $\phi$ (where $\phi < hc/\lambda$ ), and its quantum efficiency is $\eta$ (defined as the number of photoelectrons emitted per incident photon). The peak intensity of the central diffraction maximum on the plate is $I_0$ . Assuming the small-angle approximation ( $\sin \theta \approx \theta$ ), what is the total number of photoelectrons emitted per second from the region of the plate corresponding strictly to the central maximum?

(Given: $\int_{-\pi}^{\pi} \left(\frac{\sin x}{x}\right)^2 dx = \alpha$ , where $h$ is Planck's constant and $c$ is the speed of light.)

Options

A.
$\frac{2 \eta I_0 L \lambda^2 D}{h c a}$
B.
$\frac{\eta I_0 L \lambda^2 D \alpha}{h c a}$
C.
$\frac{\eta I_0 L \lambda^2 D \alpha}{\pi h c a}$
D.
$\frac{\eta I_0 L \lambda^2 D}{h c a}$

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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

Einstein's Photoelectric Equation|Single Slit Diffraction Minima|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of wave optics and quantum mechanics by challenging the assumption of uniform intensity in the photoelectric effect.

Students often memorize the macroscopic photoelectric equation $N = \eta I A / (hf)$ and incorrectly apply it by assuming the peak intensity $I_0$ is constant across the entire geometric width of the central diffraction maximum ( $2\lambda D/a$ ), which leads to Trap A.

To solve this correctly, they must recognize that intensity is a spatial distribution $I(y)$ and integrate the energy flux over the specific area of the central maximum.

Furthermore, it tests mathematical rigor in variable substitution during integration, specifically extracting the $\frac{\lambda D}{\pi a}$ factor when converting $dy$ to $d\beta$ , the omission of which leads to Trap B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Einstein's Photoelectric Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free