Problem 37 - University Physics with Modern Physics

Question

Monochromatic light with wavelength 620 nm passes through a circular aperture with diameter 7.4 \(\mu\)m. The resulting diffraction pattern is observed on a screen that is 4.5 m from the aperture. What is the diameter of the Airy disk on the screen?

Step-by-Step Solution

Answer

The diameter of the Airy disk is approximately 0.9216 m.

Step 1: Understanding the Problem

We need to find the diameter of the Airy disk produced by light diffracting through a circular aperture. The key formula to use is for the angular width of the central maximum in the Airy disk: \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength and \( D \) is the aperture diameter.

Step 2: Calculate Angular Width

Use the formula for the angular width \( \theta = 1.22 \frac{\lambda}{D} \). Here, \( \lambda = 620 \times 10^{-9} \) m and \( D = 7.4 \times 10^{-6} \) m. Substitute these values:\[ \theta = 1.22 \times \frac{620 \times 10^{-9}}{7.4 \times 10^{-6}} \approx 1.024 \times 10^{-1} \text{ radians} \]

Step 3: Determine the Radius on the Screen

The radius \( r \) of the Airy disk on the screen is given by \( r = L \cdot \theta \), where \( L \) is the distance to the screen (4.5 m). Thus:\[ r = 4.5 \cdot 1.024 \times 10^{-1} \approx 0.4608 \text{ m} \]

Step 4: Calculate the Diameter of the Airy Disk

The diameter of the Airy disk is twice the radius:\[ \text{Diameter} = 2 \cdot r = 2 \times 0.4608 = 0.9216 \text{ m} \]

Key Concepts

Airy disk

The Airy disk is a fundamental concept in optics, representing the central bright spot seen in a diffraction pattern. When light waves interfere, the pattern forms due to diffraction through an aperture, like a circular opening. The Airy disk is the region at the center of this pattern, appearing bright because light waves constructively interfere there. It is named after Sir George Biddell Airy, who first described this phenomena.
Here are key points about the Airy disk:

• It forms the central maximum of a diffraction pattern.
• The size of the Airy disk is dependent on the aperture size and the wavelength of light used.
• It plays a crucial role in determining the resolution of optical instruments.

The diameter of this disk is crucial for understanding the resolution limits of devices like telescopes and microscopes, as it sets the smallest detail that can be observed clearly.

circular aperture

A circular aperture is simply a round opening through which light passes. This shape is significant in diffraction because it affects how light waves are distributed after passing through. Thus, when monochromatic light enters a circular aperture, it spreads out and forms a distinct diffraction pattern characterized by concentric rings, with the Airy disk at its center.
Key characteristics of circular apertures include:

• The shape influences diffraction patterns profoundly, producing circular patterns due to symmetry.
• Aperture size ( D ) is inversely related to the size of the diffraction pattern on a screen.

In optical devices like cameras and telescopes, the aperture size can significantly affect image sharpness and clarity due to this diffraction pattern, reinforcing its importance in designing lenses and other components.

monochromatic light

Monochromatic light refers to light of a single wavelength or color. It is crucial in many scientific experiments because it offers consistency and simplicity in studying wave behaviors like diffraction.
Let's highlight a few points about monochromatic light:

• It does not have variations in wavelength, providing a stable pattern of interference and diffraction.
• Common sources include lasers and particular specialized light filters.
• Its wavelength (denoted as \( \lambda \)) significantly affects the visual outcomes, such as the size of the Airy disk.

Using monochromatic light helps us accurately apply equations in optical physics, like calculating diffraction effects, because there are no complications from mixed wavelengths.

wavelength

Wavelength, denoted by \( \lambda \), represents the distance between two consecutive peaks of a wave. In optical physics, it is measured in nanometers (nm) or micrometers (µm) and is crucial in determining how light behaves when interacting with objects.In the context of diffraction through apertures, the wavelength of light directly affects the diffraction pattern's features, such as the size of the Airy disk.
Consider these points about wavelength:

• Longer wavelengths produce larger diffraction patterns, meaning a bigger Airy disk.
• The interaction of light of different wavelengths with apertures can result in colorful patterns if not monochromatic.
• In the problem scenario, the wavelength given is 620 nm, which serves as a key parameter in calculations involving the diffraction pattern.

Thus, understanding wavelength allows predictions on how light will diffract and how patterns will manifest, making it an essential concept in optics.