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?

Verified step by step guidance

1. 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.
2. Step 2: For part (a), use the condition for the first diffraction minima in single-slit diffraction: \( a \sin \theta = m \lambda \), where \( a \) is the slit width, \( \theta \) 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 \( a \): \( a = \frac{m \lambda}{\sin \theta} \). Substitute \( m = 1 \), \( \lambda = 580 \; \text{nm} = 580 \times 10^{-9} \; \text{m} \), and \( \theta = 90.0^\circ \).
3. Step 3: For part (b), use the single-slit diffraction intensity formula: \( I(u) = I_0 \left( \frac{\sin u}{u} \right)^2 \), where \( u = \frac{\pi a \sin \theta}{\lambda} \), \( I_0 \) is the maximum intensity at \( \theta = 0 \), and \( \theta \) is the angle of observation. First, calculate \( u \) for \( \theta = 45.0^\circ \) and \( \theta = 0 \). For \( \theta = 0 \), \( u = 0 \), and for \( \theta = 45.0^\circ \), substitute \( a \) (from part (a)), \( \lambda \), and \( \sin 45.0^\circ \) into the formula for \( u \).
4. Step 4: Compute the ratio of intensities \( \frac{I(45.0^\circ)}{I(0)} \). Since \( I(0) = I_0 \), the ratio simplifies to \( \left( \frac{\sin u}{u} \right)^2 \) for \( u \) at \( \theta = 45.0^\circ \). Substitute the value of \( u \) calculated in Step 3 into this expression to find the ratio.
5. 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.

Key Concepts

Fraunhofer Diffraction

Fraunhofer diffraction occurs when light waves pass through a slit and are observed at a distance where the wavefronts can be considered parallel. This type of diffraction is characterized by the formation of a pattern of bright and dark fringes on a screen, which can be analyzed using mathematical equations. The angle of diffraction and the slit width are crucial in determining the positions of these minima and maxima.

Diffraction Minima

Diffraction minima are points in the diffraction pattern where the intensity of light is zero due to destructive interference. For a single slit, the positions of these minima can be calculated using the formula a sin(θ) = mλ, where 'a' is the slit width, 'θ' is the angle of the minima, 'm' is the order of the minima, and 'λ' is the wavelength of the light. Understanding these minima is essential for determining the slit width and analyzing the intensity distribution.

Intensity Ratio in Diffraction Patterns

The intensity of light in a diffraction pattern varies with angle and can be described by the intensity function derived from the amplitude of the wave. The ratio of intensities at different angles, such as u = 45.0° and u = 0°, can be calculated using the intensity formula I(θ) = I0 (sin(β)/β)², where β is related to the angle and slit width. This ratio provides insight into how the diffraction pattern changes with angle and is important for understanding light behavior in diffraction.