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Chapter 10: Problem 33

A 2.4 -cm-diameter positive lens with a focal length of \(100 \mathrm{cm}\) forms an image of a small far-away red \((656 \mathrm{nm})\) hydrogen lamp. Determine the linear size of the central circular spot appearing on the focal plane.

Short Answer

Expert verified
The central spot size is approximately 0.067 mm.

Step by step solution

01

Calculate the Aperture Radius

The diameter of the lens is given as 2.4 cm. To find the radius \( r \), divide the diameter by 2. Thus, \( r = \frac{2.4}{2} = 1.2 \text{ cm} = 0.012 \text{ m} \) (after converting to meters).
02

Determine the Wavelength in Meters

The wavelength of the light \( \lambda \) is given as 656 nm. Convert this to meters: \( 656 \text{ nm} = 656 \times 10^{-9} \text{ m} \).
03

Use the Formula for Linear Size of Airy Disk

The linear size of the central bright spot (Airy disk) on the focal plane can be calculated using the formula for the diameter of the Airy disk: \( d = \frac{2.44 \cdot \lambda \cdot f}{D} \). Here, \( f = 1 \text{ m} \) (focal length), \( \lambda = 656 \times 10^{-9} \text{ m} \), and \( D = 0.024 \text{ m} \) (diameter of the lens).
04

Calculate the Diameter of the Airy Disk

Substitute the values into the formula: \( d = \frac{2.44 \cdot 656 \times 10^{-9} \cdot 1}{0.024} \). Calculate the result to find \( d \approx 6.67 \times 10^{-5} \text{ m} \).
05

Convert to More Accessible Units

Convert the result into millimeters for easier interpretation: \( 6.67 \times 10^{-5} \text{ m} = 0.067 \text{ mm} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Aperture radius
The aperture radius is a crucial concept when discussing lenses and the creation of images. The aperture radius is essentially half of the diameter of a lens or optical system.
In this exercise, the lens is 2.4 cm in diameter, so to find the aperture radius, simply divide by 2. This converts to an aperture radius of 1.2 cm or 0.012 meters.
Understanding this measurement is critical because it directly affects how the lens captures light and ultimately focuses it to form an image.
  • The larger the aperture, the more light the lens can gather.
  • Larger apertures typically allow for better resolutions in images.
In the context of Airy disks, the aperture radius helps determine the size of the central bright spot created on a focal plane. This relationship is crucial in optical calculations.
Wavelength in meters
Wavelength is the distance between consecutive peaks of a wave. This measurement is important in optics because it influences phenomena like diffraction and image resolution.
The wavelength of light used in this exercise is given as 656 nm, which refers to red light emitted by a hydrogen lamp. It's also crucial to convert the wavelength from nanometers (nm) to meters (m) for standardization in calculations.
To convert 656 nm to meters, multiply by the factor of : .
  • This conversion ensures that all values in an equation share the same units for accuracy.
  • Using the consistent metric unit of meters simplifies complex calculations and reduces errors.
Correct conversion allows for accurate computation of phenomena such as the size of the Airy disk, essential in understanding image formation.
Focal length
The focal length is a fundamental property of lenses and optical systems. It refers to the distance from the lens to the focal point, where light rays converge to form a clear image.
In this problem, the focal length is given as 100 cm, which is equivalent to 1 meter. This distance is ultimately where the image will be sharpest and is pivotal in determining the size of the Airy disk.
The size of the Airy disk is proportional to the focal length, alongside the aperture diameter and the wavelength of light:
  • It influences the scale of the image produced by the lens, impacting clarity and detail.
  • In optical systems, understanding and correctly applying focal length is essential for maneuvers like focusing, zooming, and scaling.
As the focal length increases, the lens can capture images farther away, but it can also impact the resolution, typically enlarging the Airy disk.

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Most popular questions from this chapter

Collimated light from a krypton ion laser at \(568.19 \mathrm{nm}\) impinges normally on a circular aperture. When viewed axially from a distance of \(1.00 \mathrm{m}\), the hole uncovers the first half-period Fresnel zone. Determine its diameter.

The angular distance between the center and the first minimum of a single-slit Fraunhofer diffraction pattern is called the half-angular breadth; write an expression for it. Find the corresponding half-linear width when no focusing lens is present and the distance from the slit to the viewing screen is \(L\). Notice that the half-linear width is also the distance between the successive minima.

Monochromatic plane waves perpendicularly illuminate a small circular hole in a screen. From point- \(P\), beyond the hole on the central axis, exactly 3 Fresnel zones appear to fill the hole. If the incident irradiance on the aperture screen is \(I_{u}\), prove that the irradiance at \(P\) is very nearly \(4 I_{u^{-}}\) [Hint: Because these are plane waves, the unobstructed irradiance at \(\left.P \text { would equal } I_{u} .\right]\)

Light having a frequency of \(4.0 \times 10^{14} \mathrm{Hz}\) is incident on a grating formed with 10000 lines per centimeter. What is the highestorder spectrum that can be seen with this device? Explain.

A collimated beam from a ruby laser \((694.3 \mathrm{nm})\) having an irradiance of \(10 \mathrm{W} / \mathrm{m}^{2}\) is incident perpendicularly on an opaque screen containing a square hole \(5.0 \mathrm{mm}\) on a side. Compute the irradiance at a point on the central axis \(250 \mathrm{cm}\) from the aperture. Check that this is near-field diffraction.
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