Question

Monochromatic light of wavelength 580 nm passes through a single slit and the diffraction pattern is observed on a screen. Both the source and screen are far enough from the slit for Fraunhofer diffraction to apply. (a) If the first diffraction minima are at ±90.0°, so the central maximum completely fills the screen, what is the width of the slit? (b) For the width of the slit as calculated in part (a), what is the ratio of the intensity at θ = 45.0° to the intensity at θ = 0?

Accepted Answer

Step 1: Understand the problem and identify the key concepts. This problem involves Fraunhofer diffraction through a single slit. The first part (a) requires finding the slit width using the condition for the first diffraction minima, and the second part (b) involves calculating the intensity ratio at specific angles using the single-slit diffraction intensity formula. Step 2: For part (a), use the condition for the first diffraction minima in single-slit diffraction: $$ a \sin 	heta = m \lambda $$, where $$ a  is $$the slit width, $$ 	heta  is $$the angle of the minima, $$ m  is $$the order of the minima ($$ m = \pm 1 $$ for the first minima), and $$ \lambda  is $$the wavelength of the light. Rearrange the formula to solve for $$ a $$: $$ a = \frac{m \lambda}{\sin 	heta} . $$Substitute $$ m = 1 $$, $$ \lambda = 580 \; 	ext{nm} = 580 	imes 10^{-9} \; 	ext{m} $$, and $$ 	heta = 90.0^\circ . $$Step 3: For part (b), use the single-slit diffraction intensity formula: $$ I(u) = I_0 \left( \frac{\sin u}{u} ight)^2 $$, where $$ u = \frac{\pi a \sin 	heta}{\lambda} $$, $$ I_0  is $$the maximum intensity at $$ 	heta = 0 $$, and $$ 	heta  is $$the angle of observation. First, calculate $$ u $$ for $$ 	heta = 45.0^\circ $$ and $$ 	heta = 0 . $$For $$ 	heta = 0 $$, $$ u = 0 $$, and for $$ 	heta = 45.0^\circ $$, substitute $$ a  ($$from part (a)), $$ \lambda $$, and $$ \sin 45.0^\circ $$ into the formula for $$ u . $$Step 4: Compute the ratio of intensities $$ \frac{I(45.0^\circ)}{I(0)} . $$Since $$ I(0) = I_0 $$, the ratio simplifies to $$ \left( \frac{\sin u}{u} ight)^2 $$ for $$ u  at$$ $$ 	heta = 45.0^\circ . $$Substitute the value of $$ u $$ calculated in Step 3 into this expression to find the ratio. Step 5: Summarize the results. The slit width $$ a  is $$determined from part (a) using the diffraction minima condition, and the intensity ratio is calculated in part (b) using the single-slit diffraction intensity formula. Ensure all units are consistent and verify the calculations for accuracy.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Fraunhofer Diffraction

Diffraction Minima

Intensity Ratio in Diffraction Patterns