Problem 35 - Physics

Question

At the surface of the moon, which is \(3.77 \times 10^{8} \mathrm{~m}\) away, the light strikes a reflector left there by astronauts. The reflected light returns to the earth, where it is detected. When it leaves the spotlight, the circular beam of light has a diameter of about \(0.20 \mathrm{~m}\), and diffraction causes the beam to spread as the light travels to the moon. In effect, the first circular dark fringe in the diffraction pattern defines the size of the central bright spot on the moon. Determine the diameter (not the radius) of the central bright spot on the moon.

Step-by-Step Solution

Answer

The diameter of the central bright spot on the moon is approximately 5.1 km.

Step 1: Understand the Problem

When light passes through an opening, it spreads out or diffracts. For a circular aperture, this diffraction pattern is defined by the Airy disk, where the central maximum is surrounded by dark and bright rings.

Step 2: Use Diffraction Formula

The angle to the first minimum (dark fringe) in a diffraction pattern for a circular aperture is given by \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of the light and \( D \) is the diameter of the aperture.

Step 3: Calculate the Linear Size of the Central Spot

The spread of the central bright spot on the moon can be calculated using the formula for small angles \( x = L \times \theta \), where \( L \) is the distance to the moon and \( \theta \) is the angular separation to the first dark fringe.

Step 4: Solve for Diameter of Central Bright Spot

Combine \( \theta = 1.22 \frac{\lambda}{D} \) with \( x = L \times \theta \) to find \( x = L \times 1.22 \frac{\lambda}{D} \). Solving for the diameter of the central bright spot \( (\Delta x) \), we get \( \Delta x = 2x = 2L \times 1.22 \frac{\lambda}{D} \).

Step 5: Substitute Known Values

Here, \( L = 3.77 \times 10^8 \) m, \( D = 0.20 \) m, and we need the wavelength \( \lambda \). Assuming visible light with \( \lambda \approx 550 \times 10^{-9} \) m, substitute these into the diameter equation to find the diameter of the central bright spot.

Step 6: Compute the Answer

Plug the values into the formula \( \Delta x = 2 \times 3.77 \times 10^8 \times 1.22 \frac{550 \times 10^{-9}}{0.20} \). Calculate the result to get the diameter of the central bright spot.

Key Concepts

Circular Aperture

When light passes through an aperture, or opening, it doesn't just go straight through. Instead, it tends to bend and spread out. This phenomenon is known as diffraction. When this opening is circular, like the beam of light mentioned in this exercise, we call it a circular aperture.
This is common in telescopes, microscopes, and cameras, essentially anything with a round lens or opening. The effect of the diffraction is more prominent when the aperture is smaller relative to the wavelength of the light.
It's like trying to fit a wide water stream through a small hole. The water spreads out on the other side. This is similar to how light spreads, forming a pattern as it passes through the aperture. This diffraction pattern is something we can predict and calculate using specific formulas.

Airy Disk

The pattern light forms when it diffracts through a circular aperture is known as the Airy disk. It is named after the British scientist Sir George Biddell Airy who first described it.
The Airy disk pattern consists of a central bright spot surrounded by alternating dark and bright rings. It's like dropping a pebble in a pond and watching the ripples spread out. The central ripple is the brightest and the most distinct.

• The central bright spot is called the Airy disk.
• The subsequent rings surrounding it become gradually dimmer with increasing distance from the center.

This concept is crucial for understanding many optical systems, as it defines how light will behave as it focuses through lenses. The Airy disk limits the resolution of imaging systems such as cameras and telescopes.

Central Bright Spot

The Airy disk's most noticeable part is its central bright spot. This is the main bright area of the diffraction pattern on the other side of the circular aperture. It's the focal point and the brightest spot in the pattern.
The diameter of this central bright spot, measured from edge to edge across the center, is used to define resolution limits of optical instruments. This is primarily because smaller central bright spots allow for sharper images, key for activities like fine microscopy or precise astronomical observation.
In this exercise, the question seeks to determine the diameter of this bright spot as it appears on the moon as light diffracts through the circular aperture and spreads over the vast distance to the moon's surface.

Wavelength

Wavelength, denoted by the symbol \( \lambda \), is the distance between two consecutive peaks or troughs in a wave. Light, being a wave, has different wavelengths that determine its color.
In this exercise, the wavelength is part of the formula needed to calculate how the light spreads out when passing through the circular aperture. A typical choice might be visible light with a wavelength of around \(550 \times 10^{-9}\) m, which corresponds to a greenish-yellow light.

• Shorter wavelengths spread out less and form smaller Airy disks.
• Longer wavelengths spread more, resulting in larger Airy disks.

Understanding wavelength is crucial because it directly affects the diffraction and, consequently, the size of the central bright spot on the moon.

Angular Separation

Angular separation refers to the angle at which two points in a diffraction pattern (like the central bright spot and the first dark fringe) appear apart when observed from a distance.
In diffraction through a circular aperture, the angular separation (\( \theta \)) to the first dark fringe is calculated using the formula \( \theta = 1.22 \frac{\lambda}{D} \), where \( D \) is the diameter of the aperture and \( \lambda \) is the wavelength of the light.
This angle is critical for predicting how the diffraction pattern will spread. A smaller aperture or longer wavelength results in a larger angular separation, causing the light to spread out more significantly over a distance. In this exercise, knowing the angular separation allows us to calculate how big the central bright spot is as the light travels to the moon.

Dark Fringe

In a diffraction pattern created by a circular aperture, the dark fringes are the rings of darkness that alternate with bright rings around the central bright spot.
The first dark fringe marks the boundary between the central bright spot and the surrounding bright rings. Essentially, the angular point to this first dark fringe is used to define the angular extent of the central bright spot.
Because these dark fringes alternate with bright regions, they help to map out the pattern of diffraction. Knowing where these dark fringes fall helps scientists measure and use diffraction principles to apply them accurately in various technologies and scientific explorations.