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Chapter 27: Problem 6

Two parallel slits are illuminated by light composed of two wavelengths. One wavelength is \(\lambda_{\mathrm{A}}=645 \mathrm{nm} .\) The other wavelength is \(\lambda_{\mathrm{B}}\) and is unknown. On a viewing screen, the light with wavelength \(\lambda_{\mathrm{A}}=645 \mathrm{nm}\) produces its third-order bright fringe at the same place where the light with wavelength \(\lambda_{\mathrm{B}}\) produces its fourth dark fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?

Short Answer

Expert verified
The unknown wavelength is 430 nm.

Step by step solution

01

Understand the Fringe Conditions

For the bright fringes produced by light with wavelength \(\lambda_A\), the condition is \(d \sin \theta = m \lambda_A\), where \(m\) is the order of the bright fringe. For the dark fringes produced by light with wavelength \(\lambda_B\), the condition is \(d \sin \theta = (n+1/2)\lambda_B\), where \(n\) is the order of the dark fringe. We know that the third bright fringe for wavelength \(\lambda_A\) coincides with the fourth dark fringe for wavelength \(\lambda_B\).
02

Set Up the Equations

For the third-order bright fringe of \(\lambda_A\), we have:\[d \sin \theta = 3 \lambda_A\]For the fourth-order dark fringe of \(\lambda_B\), we have:\[d \sin \theta = 4.5 \lambda_B\]Since these two distances are equal, set the equations equal to each other:\[3 \lambda_A = 4.5 \lambda_B\]
03

Solve for the Unknown Wavelength

Use the equation from Step 2 to find the unknown wavelength \(\lambda_B\):\[3 \lambda_A = 4.5 \lambda_B\]\[\lambda_B = \frac{3 \lambda_A}{4.5}\]Substitute \(\lambda_A = 645 \mathrm{nm}\):\[\lambda_B = \frac{3 \times 645}{4.5}\]\[\lambda_B = 430 \mathrm{nm}\]

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Key Concepts

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

Interference Patterns
Interference patterns are a fascinating phenomenon that arise when two or more waves overlap and combine with each other. This concept is foundational in understanding the double slit experiment, a classic demonstration in physics. The patterns arise because of the constructive and destructive interference of light waves.
  • Constructive Interference: This occurs when the waves align perfectly, meaning their peaks and troughs match up. This results in a brighter, more intense light spot, known as a bright fringe.
  • Destructive Interference: On the other hand, when the waves are out of phase 鈥� peaks align with troughs 鈥� the waves effectively cancel each other out, creating a dark fringe.
By illuminating two slits with light and examining the pattern on a screen, we can observe the alternating series of bright and dark fringes. These patterns depend on the wavelength of the light used and the geometry of the setup, such as the distance between the slits and the screen. Understanding these interference patterns helps in exploring wave properties of light and similar phenomena in other wave types.
Order of Fringes
The order of fringes relates to the sequence of bright and dark spots seen in an interference pattern. Each bright or dark spot is labeled with an order number, typically starting with zero at the central maximum.
  • In the scenario of a double slit experiment, **bright fringes** are numbered by an integer (m), starting from the central maximum (zeroth order).
  • **Dark fringes**, representing positions of destructive interference, generally have non-integer indices like half-integers ((n + 1/2){\lambda}_B).
These order numbers help specify which bright or dark fringe we are talking about. For instance, in the given exercise, the third bright fringe for a wavelength \(\lambda_A\) and the fourth dark fringe for \(\lambda_B\) coincide, which is significant for wavelength calculations. Using the order of fringes, one can better understand how different wavelengths produce specific interference patterns, which can be crucial for calculating unknown wavelengths.
Wavelength Calculation
Wavelength calculation is an essential aspect of analyzing interference patterns. In many cases, as with the textbook problem, you might need to determine an unknown wavelength based on the interference pattern it produces relative to another known wavelength. To calculate the unknown wavelength, one must use the relationships established for bright and dark fringes.
  • The formula for **bright fringes** is \(d \sin \theta = m \lambda\), where "\(m\)" is the fringe order.
  • For **dark fringes**, the formula is \(d \sin \theta = (n + 1/2) \lambda\).
By equating these when fringes coincide, as was done in the problem, one can solve for the unknown wavelength. This involves setting the equations for two coinciding fringes equal, rearranging the formula to isolate the unknown wavelength, and substituting known values. In the example, substituting the known \(\lambda_A = 645 \mathrm{nm}\) in the formula allowed calculating \(\lambda_B = 430 \mathrm{nm}\). Calculations like these illustrate the precise and mathematical nature of understanding light behavior in wave optics.

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Most popular questions from this chapter

Light of wavelength \(410 \mathrm{nm}\) (in vacuum) is incident on a diffraction grating that has a slit separation of \(1.2 \times 10^{-5} \mathrm{m} .\) The distance between the grating and the viewing screen is \(0.15 \mathrm{m} .\) A diffraction pattern is produced on the screen that consists of a central bright fringe and higher-order bright fringes (see the drawing). (a) Determine the distance \(y\) from the central bright fringe to the second-order bright fringe. (Hint: The diffraction angles are small enough that the approximation \(\tan \theta \approx \sin \theta\) can be used.) (b) If the entire apparatus is submerged in water \(\left(n_{\text {water }}=1.33\right),\) what is the distance \(y ?\)

A dark fringe in the diffraction pattern of a single slit is located at an angle of \(\theta_{\mathrm{A}}=34^{\circ} .\) With the same light, the same dark fringe formed with another single slit is at an angle of \(\theta_{\mathrm{B}}=56^{\circ} .\) Find the ratio \(W_{\mathrm{A}} / W_{\mathrm{B}}\) of the widths of the two slits.

Point A is the midpoint of one of the sides of a square. On the side opposite this spot, two in-phase loudspeakers are located at adjacent corners, as shown in the figure. Standing at point A you hear a loud sound because of constructive interference between the identical sound waves coming from the speakers. As you walk along the side of the square toward either empty corner, the loudness diminishes gradually to nothing and then increases again until you hear a maximally loud sound at the corner. If the length of each side of the square is \(4.6 \mathrm{m},\) find the wavelength of the sound waves.

In a setup like that in Figure \(27.7,\) a wavelength of \(625 \mathrm{nm}\) is used in a Young's double-slit experiment. The separation between the slits is \(d=\) \(1.4 \times 10^{-5} \mathrm{m} .\) The total width of the screen is \(0.20 \mathrm{m} .\) In one version of the setup, the separation between the double slit and the screen is \(L_{\mathrm{A}}=0.35 \mathrm{m}\) whereas in another version it is \(L_{\mathrm{B}}=0.50 \mathrm{m} .\) On one side of the central bright fringe, how many bright fringes lie on the screen in the two versions of the setup? Do not include the central bright fringe in your counting.

Light that has a wavelength of 668 nm passes through a slit \(6.73 \times 10^{-6} \mathrm{m}\) wide and falls on a screen that is \(1.85 \mathrm{m}\) away. What is the distance on the screen from the center of the central bright fringe to the third dark fringe on either side?
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