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Chapter 25: Problem 27

You are given a slide with two slits cut into it and asked how far apart the slits are. You shine white light on the slide and notice the first-order color spectrum that is created on a screen \(3.40 \mathrm{m}\) away. On the screen, the red light with a wavelength of 700 nm is separated from the violet light with a wavelength of 400 nm by 7.00 mm. What is the separation of the two slits?

Short Answer

Expert verified
Based on the given information that the distances between the primary red and violet colors on the screen is 7.00 mm, and the distance from the slide to the screen is 3.40 meters, we calculated the separation between the two slits using the double-slit interference formula. The separation between the two slits is approximately 146 micrometers.

Step by step solution

01

Understand the double-slit interference formula

The double-slit interference formula is given by: \(\sin \theta = \frac{m \lambda}{d}\) where \(\theta\) is the angle between the central maximum and the m-th order maximum on the screen, \(m\) is the order of the interference pattern, \(\lambda\) is the wavelength of light, and \(d\) is the separation between the slits.
02

Calculate the angles for red and violet light

In this problem, we are given the distances between the red and violet light on the screen and the distance from the slide to the screen. We can use the small angle approximation (\(\sin \theta \approx \tan \theta\)) to calculate the angles for red and violet light: \(\tan \theta_{red} = \frac{distance_{red}}{distance_{screen}}\) \(\tan \theta_{violet} = \frac{distance_{violet}}{distance_{screen}}\) \(\tan \theta_{red} - \tan \theta_{violet} = \frac{7.00\,\mathrm{mm}}{3.40\,\mathrm{m}}\)
03

Use the double-slit interference formula to find the separation between slits

Since we are dealing with the first-order maximum (\(m=1\)), we can substitute the wavelengths and the calculated angle difference into the double-slit interference formula: \(\frac{\sin \theta_{red} - \sin \theta_{violet}}{1} = \frac{\lambda_{red} - \lambda_{violet}}{d}\) Using the small angle approximation, we get: \(\frac{\tan \theta_{red} - \tan \theta_{violet}}{1} = \frac{\lambda_{red} - \lambda_{violet}}{d}\) Plugging in the known values, we have: \(\frac{7.00\,\mathrm{mm}}{3.40\,\mathrm{m}} = \frac{700\,\mathrm{nm} - 400\,\mathrm{nm}}{d}\) Now we can solve for the separation between the slits, \(d\).
04

Solve for the slit separation, \(d\)

Rearrange the equation to find \(d\): \(d = \frac{(700\,\mathrm{nm} - 400\,\mathrm{nm})(3.40\,\mathrm{m})}{7.00\,\mathrm{mm}}\) Perform the calculation: \(d = \frac{300\,\mathrm{nm}(3.40\,\mathrm{m})}{7.00\,\mathrm{mm}}\) \(d = 0.000146\,\mathrm{m} = 146\,\mathrm{\mu m}\)
05

Conclusion

The separation between the two slits is approximately \(146\,\mathrm{\mu m}\).

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Most popular questions from this chapter

Light from a red laser passes through a single slit to form a diffraction pattern on a distant screen. If the width of the slit is increased by a factor of two, what happens to the width of the central maximum on the screen?
Two slits separated by \(20.0 \mu \mathrm{m}\) are illuminated by light of wavelength \(0.50 \mu \mathrm{m} .\) If the screen is \(8.0 \mathrm{m}\) from the slits, what is the distance between the \(m=0\) and \(m=1\) bright fringes?
A thin soap film \((n=1.35)\) is suspended in air. The spectrum of light reflected from the film is missing two visible wavelengths of $500.0 \mathrm{nm}\( and \)600.0 \mathrm{nm},$ with no missing wavelengths between the two. (a) What is the thickness of the soap film? (b) Are there any other visible wavelengths missing from the reflected light? If so, what are they? (c) What wavelengths of light are strongest in the transmitted light?
In a double-slit interference experiment, the wavelength is \(475 \mathrm{nm}\), the slit separation is \(0.120 \mathrm{mm},\) and the screen is $36.8 \mathrm{cm}$ away from the slits. What is the linear distance between adjacent maxima on the screen? [Hint: Assume the small-angle approximation is justified and then check the validity of your assumption once you know the value of the separation between adjacent maxima.] (tutorial: double slit 1 ).
A thin film of oil \((n=1.50)\) is spread over a puddle of water \((n=1.33) .\) In a region where the film looks red from directly above $(\lambda=630 \mathrm{nm}),$ what is the minimum possible thickness of the film? (tutorial: thin film).
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