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Chapter 35: Problem 14

Coherent light that contains two wavelengths, \(660 \mathrm{nm}\) (red) and \(470 \mathrm{nm}\) (blue), passes through two narrow slits that are separated by \(0.300 \mathrm{~mm}\). Their interference pattern is observed on a screen \(4.00 \mathrm{~m}\) from the slits. What is the distance on the screen between the first-order bright fringes for the two wavelengths?

Short Answer

Expert verified
The fringe separation between red and blue first-order bright fringes is obtained by the difference of \( y_{red} \) and \( y_{blue} \).

Step by step solution

01

Calculation of Fringe Position for Red Light

Insert the given values into the formula for the red light, \(y_{red} = Lm\lambda_{red}/d \), with \(L = 4 m, m = 1 \) (first order), \( \lambda_{red} = 660 nm = 660 x 10^{-9} m \), and \( d = 0.300 mm = 0.300 x 10^{-3} m \). Solve for \( y_{red} \).
02

Calculation of Fringe Position for Blue Light

Do the same calculation for the blue light, \(y_{blue} = Lm\lambda_{blue}/d\), with \(L = 4 m, m = 1 \) (first order), \( \lambda_{blue} = 470 nm = 470 x 10^{-9} m \), and \( d = 0.300 mm = 0.300 x 10^{-3} m \). Solve for \( y_{blue} \).
03

Calculation of Fringe Separation Between the Wavelengths

To find the distance between the first-order bright fringes for the two wavelengths, subtract \( y_{blue} \) from \( y_{red} \) which gives us the distance between the fringes: \( Separation = y_{red} - y_{blue} \).

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Key Concepts

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

Coherent Light
Coherent light is a fundamental concept in the studying of wave interference patterns. It occurs when light waves maintain a constant phase relationship. This special characteristic is essential when it comes to producing clear and stable interference patterns.
Understanding coherent light helps us figure out how waves interact and combine:
  • Light waves from coherent sources have the same frequency and phase difference which remains constant over time.
  • This steady phase relationship is crucial for the interference pattern to remain stable.
  • Coherent light is what allows us to observe distinct fringe patterns on the screen.
Lasers are a common source of coherent light because they generate light waves that are synchronized, much like a group of synchronized swimmers performing in harmony.
Wavelength
Wavelength is a pivotal concept when it comes to understanding light interference. It determines how light behaves and how interference patterns emerge on the screen. In the given problem, two different wavelengths are at play: 660 nm (red light) and 470 nm (blue light).
Here's how wavelength plays a role:
  • Wavelength is the distance between successive crests of a light wave.
  • It influences the position and spacing of bright and dark fringes on the screen.
  • Different wavelengths will produce fringes at different positions. That's why the red and blue light fringes do not overlap completely.
In the interference of two different wavelengths, the fringe distance reflects light鈥檚 color properties, making the concept of wavelength indispensable for understanding and calculating the details of interference patterns.
Fringe Separation
Fringe separation is the distance between corresponding positions of fringes (like bright fringes) in an interference pattern. It results from different wavelengths of coherent light passing through the same apparatus.
Let's dive into how fringe separation is calculated:
  • The formula for fringe position is given by: \( y = \frac{Lm\lambda}{d} \).
  • Here, \( L \) is the distance from the slits to the screen, \( m \) indicates the order of the fringe, \( \lambda \) is the wavelength, and \( d \) is the slit separation.
  • After calculating the fringe positions for both red and blue light with their respective wavelengths, the difference between these fringe positions gives us the fringe separation.
Understanding fringe separation helps us appreciate how different colors of light can exhibit unique behaviors when undergoing interference, a fascinating phenomenon observed in experiments like this one.

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Most popular questions from this chapter

The index of refraction of a glass rod is 1.48 at \(T=20.0^{\circ} \mathrm{C}\) and varies linearly with temperature, with a coefficient of \(2.50 \times 10^{-5} / \mathrm{C}^{\circ} .\) The coefficient of linear expansion of the glass is \(5.00 \times 10^{-6} / \mathrm{C}^{\circ} .\) At \(20.0^{\circ} \mathrm{C}\) the length of the rod is \(3.00 \mathrm{~cm}\) A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of \(5.00 \mathrm{C}^{\circ} / \mathrm{min} .\) The light source has wavelength \(\lambda=589 \mathrm{nm},\) and the rod initially is at \(T=20.0^{\circ} \mathrm{C}\). How many fringes cross the field of view each minute?

Two flat plates of glass with parallel faces are on a table, one plate on the other. Each plate is \(11.0 \mathrm{~cm}\) long and has a refractive index of \(1.55 .\) A very thin sheet of metal foil is inserted under the end of the upper plate to raise it slightly at that end, in a manner similar to that discussed in Example 35.4 . When you view the glass plates from above with reflected white light, you observe that, at \(1.15 \mathrm{~mm}\) from the line where the sheets are in contact, the violet light of wavelength \(400.0 \mathrm{nm}\) is enhanced in this reflected light, but no visible light is enhanced closer to the line of contact. (a) How far from the line of contact will green light (of wavelength \(550.0 \mathrm{nm}\) ) and orange light (of wavelength \(600.0 \mathrm{nm}\) ) first be enhanced? (b) How far from the line of contact will the violet, green, and orange light again be enhanced in the reflected light? (c) How thick is the metal foil holding the ends of the plates apart?

Coherent light of wavelength \(500 \mathrm{nm}\) is incident on two very narrow and closely spaced slits. The interference pattern is observed on a very tall screen that is \(2.00 \mathrm{~m}\) from the slits. Near the center of the screen the separation between two adjacent interference maxima is \(3.53 \mathrm{~cm}\). What is the distance on the screen between the \(m=49\) and \(m=50\) maxima?

Coherent light with wavelength \(500 \mathrm{nm}\) passes through narrow slits separated by \(0.340 \mathrm{~mm}\). At a distance from the slits large compared to their separation, what is the phase difference (in radians) in the light from the two slits at an angle of \(23.0^{\circ}\) from the centerline?

White light reflects at normal incidence from the top and bottom surfaces of a glass plate \((n=1.52) .\) There is air above and below the plate. Constructive interference is observed for light whose wavelength in air is \(477.0 \mathrm{nm}\). What is the thickness of the plate if the next longer wavelength for which there is constructive interference is \(540.6 \mathrm{nm} ?\)
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