Math Problem Statement

What is the brightness of the light at a distance of 7.65 cm from the central maxima, relative to the brightness of the central maxima, given monochromatic light (wavelength λ = 956 nm) incident on a metal plate with slits, where the diffraction pattern is observed 51 cm away from the plate?

Solution

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Problem Summary:
We are working with monochromatic light (wavelength λ=956nm\lambda = 956 \, \text{nm}) incident on a metal plate with either single or double slits, where the diffraction pattern is observed on a screen 51 cm away. The first two minima are observed at distances of 2.73 cm and 8.29 cm from the central maximum.

The goal is to determine the brightness of light at a distance of 7.65 cm from the central maxima, relative to the brightness of the central maxima.

Given Information:

  • Wavelength λ=956nm=9.56×107m\lambda = 956 \, \text{nm} = 9.56 \times 10^{-7} \, \text{m}
  • Screen distance: L=51cm=0.51mL = 51 \, \text{cm} = 0.51 \, \text{m}
  • First two minima positions: y1=2.73cm,y2=8.29cmy_1 = 2.73 \, \text{cm}, \, y_2 = 8.29 \, \text{cm}
  • Distance of interest: y=7.65cmy = 7.65 \, \text{cm}
  • Intensity formulas for single and double slits are provided:
    • Single slit:
      I=I0(sin(πdsinθλ)πdsinθλ)2I = I_0 \left( \frac{\sin \left( \frac{\pi d \sin \theta}{\lambda} \right)}{\frac{\pi d \sin \theta}{\lambda}} \right)^2
    • Double slit:
      I=I0cos2(πdsinθλ)I = I_0 \cos^2 \left( \frac{\pi d \sin \theta}{\lambda} \right)

Steps to Solve:

  1. Determine the type of slit configuration: Using the positions of the minima (y1y_1 and y2y_2), we will analyze the spacing dd and check for consistency with either a single-slit or double-slit pattern.

  2. Calculate angle θ\theta for the given position y=7.65cmy = 7.65 \, \text{cm}:
    Using the small-angle approximation (sinθtanθy/L\sin \theta \approx \tan \theta \approx y / L).

  3. Plug into the appropriate intensity formula: Once the configuration is determined (single or double slit), substitute sinθ\sin \theta into the respective intensity formula to find I/I0I/I_0.


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Math Problem Analysis

Mathematical Concepts

Wave Optics
Diffraction
Interference
Small Angle Approximation

Formulas

I = I_0 (sin(πd sin(θ) / λ) / (πd sin(θ) / λ))^2 (single slit)
I = I_0 cos^2(πd sin(θ) / λ) (double slit)
sin(θ) ≈ tan(θ) ≈ y / L

Theorems

Diffraction and Interference Principles

Suitable Grade Level

Undergraduate Physics or Advanced High School