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Question #43606

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The Question

A uniform plane wave of monochromatic light of wavelength $\lambda$ and intensity $I_0$ is incident normally on an opaque screen containing a circular aperture of diameter $D$ . A planar photocathode is placed parallel to the screen at a distance $L$ (where $L \gg D^2/\lambda$ ). The photocathode is a circular disk of radius $R$ centered on the axis of the aperture. The radius $R$ is chosen to exactly match the radius of the first dark ring of the Fraunhofer diffraction pattern produced by the aperture. The quantum efficiency of the photocathode is $\eta$ (number of photoelectrons emitted per incident photon), and the photon energy exceeds the work function. Given that the central maximum of the Airy pattern contains exactly $84\%$ of the total transmitted power, which of the following expressions gives the saturation photoelectric current $I_{sat}$ ? (Here $h$ is Planck's constant, $c$ is the speed of light, and $e$ is the elementary charge).

Options

A.
$0.21 \frac{\eta e I_0 \pi D^2 \lambda}{h c}$
B.
$0.25 \frac{\eta e I_0 \pi D^2 \lambda}{h c}$
C.
$\eta e I_0 \pi \left( \frac{1.22 \lambda L}{D} \right)^2 \frac{\lambda}{h c}$
D.
$0.84 \eta e I_0 \pi \left( \frac{1.22 \lambda L}{D} \right)^2 \frac{\lambda}{h c}$

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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

Photoelectric Saturation|Wave-Particle Duality Constraint|Resolving Power Limits|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from geometric ray optics to wave optics in the context of the photoelectric effect.

A common trap is assuming that because the photocathode is placed far away ( $L \gg D^2/\lambda$ ), its large radius ( $R \gg D$ ) will geometrically intercept all the light transmitted through the aperture, leading to the incorrect assumption of 100% power interception (Option B).

However, in the far-field regime, light undergoes Fraunhofer diffraction and spreads significantly into higher-order rings.

The cathode only covers the central Airy disk, which contains exactly 84% of the transmitted power.

Students who ignore power conservation entirely might incorrectly multiply the original intensity $I_0$ by the diffraction-limited area of the cathode (Options C and D), failing to recognize that intensity is not conserved over distance in a diffracting beam.

Quick Facts About This Question

Question Information

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