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Calculating the highest order possible

Interference patterns do not have an infinite number of lines, since there is a limit to how big m can be. What is the highest-order constructive interference possible with the system described in the preceding example?

Strategy

The equation d sin θ = m λ (for m = 0 , ± 1 , ± 2 , ± 3 ) describes constructive interference from two slits. For fixed values of d and λ , the larger m is, the larger sin θ is. However, the maximum value that sin θ can have is 1, for an angle of 90 ° . (Larger angles imply that light goes backward and does not reach the screen at all.) Let us find what value of m corresponds to this maximum diffraction angle.

Solution

Solving the equation d sin θ = m λ for m gives

m = d sin θ λ .

Taking sin θ = 1 and substituting the values of d and λ from the preceding example gives

m = ( 0.0100 mm ) ( 1 ) 633 nm 15.8 .

Therefore, the largest integer m can be is 15, or m = 15 .

Significance

The number of fringes depends on the wavelength and slit separation. The number of fringes is very large for large slit separations. However, recall (see The Propagation of Light and the introduction for this chapter) that wave interference is only prominent when the wave interacts with objects that are not large compared to the wavelength. Therefore, if the slit separation and the sizes of the slits become much greater than the wavelength, the intensity pattern of light on the screen changes, so there are simply two bright lines cast by the slits, as expected, when light behaves like rays. We also note that the fringes get fainter farther away from the center. Consequently, not all 15 fringes may be observable.

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Check Your Understanding In the system used in the preceding examples, at what angles are the first and the second bright fringes formed?

3.63 ° and 7.27 ° , respectively

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Summary

  • In double-slit diffraction, constructive interference occurs when d sin θ = m λ ( for m = 0 , ± 1 , ± 2 , ± 3 ) , where d is the distance between the slits, θ is the angle relative to the incident direction, and m is the order of the interference.
  • Destructive interference occurs when d sin θ = ( m + 1 2 ) λ for m = 0 , ± 1 , ± 2 , ± 3 ,… .

Conceptual questions

Suppose you use the same double slit to perform Young’s double-slit experiment in air and then repeat the experiment in water. Do the angles to the same parts of the interference pattern get larger or smaller? Does the color of the light change? Explain.

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Why is monochromatic light used in the double slit experiment? What would happen if white light were used?

Monochromatic sources produce fringes at angles according to d sin θ = m λ . With white light, each constituent wavelength will produce fringes at its own set of angles, blending into the fringes of adjacent wavelengths. This results in rainbow patterns.

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Problems

At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?

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Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.

0.997 °

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What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of 30.0 ° ?

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Practice Key Terms 2

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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