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Chapter 9: Problem 12

A \(3 \times 5\) card containing two pinholes, \(0.08 \mathrm{mm}\) in diameter and separated center to center by \(0.10 \mathrm{mm}\), is illuminated by parallel rays of blue light from an argon ion laser \(\left(\lambda_{0}=487.99 \mathrm{nm}\right) .\) If the fringes on an observing screen are to be \(10 \mathrm{mm}\) apart, how far away should the screen be?

Short Answer

Expert verified
The screen should be approximately 2.05 meters away.

Step by step solution

01

Understand the Conditions for Interference

When light passes through two pinholes, it creates an interference pattern due to the superposition of light waves. The distance between these pinholes acts as the slit separation in a double-slit experiment, which is given as \(d = 0.10 \text{ mm}\). The wavelength \(\lambda\) of the light is \(487.99 \text{ nm}\), which is \(487.99 \times 10^{-9} \text{ m}\). The fringe separation \(y\) on the screen is \(10 \text{ mm} = 0.010 \text{ m}\). We need to find the distance \(L\) from the pinholes to the screen.
02

Recall the Fringe Separation Formula

For double-slit interference, the fringe separation \(y\) between adjacent fringes is given by the formula:\[y = \frac{\lambda L}{d}\]where \(y\) is the fringe separation, \(\lambda\) is the wavelength of light, \(d\) is the distance between the centers of the pinholes, and \(L\) is the distance from the pinholes to the screen.
03

Rearrange the Formula to Solve for L

We want to find \(L\), so we rearrange the formula for fringe separation to solve for \(L\):\[L = \frac{y \, d}{\lambda}\]We need to ensure all units are consistent when plugging in the values.
04

Plug in the Values

Substitute the known values into the equation:\[L = \frac{(0.010 \text{ m}) \times (0.10 \times 10^{-3} \text{ m})}{487.99 \times 10^{-9} \text{ m}}\]Calculate \(L\) to find how far away the screen should be.
05

Calculate L

Compute the value:\[L = \frac{0.010 \times 0.0001}{487.99 \times 10^{-9}} = \frac{0.000001}{487.99 \times 10^{-9}} \approx 2.05 \text{ meters}\]Therefore, to achieve the desired fringe separation, the screen should be approximately \(2.05\) meters away from the card.

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Key Concepts

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

Interference Pattern
When light travels through two closely spaced pinholes, it creates an interference pattern, a key phenomenon in physics known as the double-slit experiment. This experiment beautifully demonstrates how light exhibits wave-like behaviors. Each pinhole acts as a source of waves, and when these waves overlap, they can interfere with each other.

  • If the peaks (crests) of two waves align, they add together in a process called constructive interference. This results in bright spots on the observing screen, or bright fringes.
  • If a peak of one wave aligns with the trough of another, they cancel each other out in destructive interference, creating dark areas or dark fringes.
This alternating pattern of bright and dark bands is called the interference pattern. The setup of the double-slit experiment allows these patterns to emerge, visually illustrating the wave nature of light.
Fringe Separation
Fringe separation refers to the distance between successive bright (or dark) interference fringes on the observing screen. In double-slit interference, fringe separation is directly influenced by several factors. The key formula used to calculate this separation involves the wavelength of the light (\( \lambda \)) and the distance (\( L \)) from the slits to the screen, as well as the distance (\( d \)) between the slits.

When expressed as \[ y = \frac{\lambda L}{d} \], this formula shows that the fringe separation (\( y \)) increases when:
  • The wavelength of the light is longer.
  • The distance from the pinholes to the observing screen is greater.
  • The distance between the pinholes is smaller.
This tells us how adjusting these variables can alter the pattern seen during the experiment.
Argon Ion Laser
An argon ion laser is a common light source used in creating interference patterns due to its ability to emit light of a single wavelength, often in the blue part of the light spectrum. This consistency is critical for the repeatability and accuracy of experiments like the double-slit experiment.

  • They are widely used in laboratory settings due to their reliability and precision.
  • The specific wavelength mentioned in this exercise, \( 487.99 \text{ nm} \), allows precise measurements and calculations.
  • An argon ion laser is highly efficient in producing monochromatic light, crucial for observing clear interference patterns without overlapping spectra of different colors.
This makes these lasers a preferred choice for experimental applications in fields like physics and engineering.
Wavelength of Light
The wavelength of light is a fundamental concept in understanding optical phenomena such as the interference pattern observed in the double-slit experiment. Wavelength (\( \lambda \)) is the distance between successive crests of a wave, and it plays a crucial role in determining how light interacts with different materials or structures.

  • Shorter wavelengths, like the \( 487.99 \text{ nm} \) in this exercise, are associated with higher energies and blue light.
  • The wavelength impacts the spacing of the interference fringes; longer wavelengths result in wider fringe separations.
  • Knowing the precise wavelength is necessary for calculating other important variables, such as the distance to the observing screen or fringe separation.
The careful application and measurement of wavelengths enable scientists and engineers to perform accurate experiments and develop technologies that depend on precise light behavior.

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Most popular questions from this chapter

Given that the mirrors of a Fabry-Perot Interferometer have an amplitude reflection coefficient of \(r=0.8944,\) find (a) the coefficient of finesse, (b) the half-width, (c) the finesse, and, (d) the contrast factor defined by $$C \equiv \frac{\left(I_{t} / I_{i}\right)_{\max }}{\left(I_{t} / I_{i}\right)_{\min }}$$

With regard to Young's Experiment, derive a general expression for the shift in the vertical position of the \(m\) th maximum as a result of placing a thin parallel sheet of glass of index \(n\) and thickness \(d\) directly over one of the slits. Identify your assumptions.

A glass camera lens with an index of 1.55 is to be coated with a cryolite film \((n \approx 1.30)\) to decrease the reflection of normally incident green light \(\left(\lambda_{0}=500 \mathrm{nm}\right) .\) What thickness should be deposited on the lens?

A line source of sodium light \(\left(\lambda_{0}=589.3 \mathrm{nm}\right)\) illuminates a Lloyd's mirror \(10.0 \mathrm{mm}\) above its surface. A viewing screen is \(5.00 \mathrm{m}\) from the source and the whole apparatus is in air. How far apart are the first and third maxima?

Suppose we place a chamber \(10.0 \mathrm{cm}\) long with flat parallel windows in one arm of a Michelson Interferometer that is being illuminated by 600 -nm light. If the refractive index of air is 1.00029 and all the air is pumped out of the cell, how many fringe-pairs will shift by in the process?
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