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Chapter 34: Problem 38

An instructor uses light of wavelength \(633 \mathrm{nm}\) to create a diffraction pattern with a slit of width \(0.135 \mathrm{~mm} .\) How far away from the slit must the instructor place the screen in order for the full width of the central maximum to be \(5.00 \mathrm{~cm} ?\)

Short Answer

Expert verified
Answer: The distance between the slit and the screen should be approximately 5319.15 meters.

Step by step solution

01

1. Understand the problem and given values

We are given the wavelength of light (λ), the width of the slit (a), and the width of the central maximum (W). We need to find the distance (D) between the slit and the screen. Given values: λ = 633 nm a = 0.135 mm W = 5.00 cm First, let's convert these values into consistent units (meters). λ = 633 * 10^{-9} m a = 0.135 * 10^{-3} m W = 5.00 * 10^{-2} m
02

2. Derive the formula for the width of the central maximum

We can use the formula for single-slit diffraction patterns, which states that the angular width of the central maximum (θ) is given by: θ = 2 * arctan(λ / (2 * a))
03

3. Determine the relationship between θ, W, and D using the small-angle approximation

We can use the small-angle approximation, sin(θ) ≈ tan(θ) ≈ θ, to relate the angular width of the central maximum (θ), the width of the central maximum (W), and the distance between the slit and the screen (D): θ ≈ (W / 2) / D
04

4. Express the formula for the distance between the slit and the screen

We can re-write the small-angle approximation and diffraction pattern formulas to solve for the distance between the slit and the screen (D): D ≈ (W / 2) / θ = (W / 2) / [2 * arctan(λ / (2 * a))]
05

5. Calculate the distance D using the given values

Now we can plug the values for λ, a, and W into our formula and calculate the distance (D): D ≈ [(5.00 * 10^{-2} m) / 2] / [2 * arctan((633 * 10^{-9} m) / (2 * 0.135 * 10^{-3} m))] D ≈ (2.50 * 10^{-2} m) / [2 * arctan(2.34 * 10^{-6} m)] D ≈ (2.50 * 10^{-2} m) / 4.70 * 10^{-6} m D ≈ 5319.15 m
06

6. Conclusion

The instructor must place the screen approximately 5319.15 meters away from the slit in order for the full width of the central maximum to be 5.00 cm.

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Most popular questions from this chapter

A Young's interference experiment is performed with monochromatic green light \((\lambda=540 \mathrm{nm}) .\) The separation between the slits is \(0.100 \mathrm{~mm},\) and the interference pattern on a screen shows the first side maximum \(5.40 \mathrm{~mm}\) from the center of the pattern. How far away from the slits is the screen?

In Young's double-slit experiment, both slits were illuminated by a laser beam and the interference pattern was observed on a screen. If the viewing screen is moved farther from the slit, what happens to the interference pattern? a) The pattern gets brighter. b) The pattern gets brighter and closer together. c) The pattern gets less bright and farther apart. d) There is no change in the pattern. e) The pattern becomes unfocused. f) The pattern disappears.

A diffraction grating has \(4.00 \cdot 10^{3}\) lines \(/ \mathrm{cm}\) and has white light \((400 .-700 . \mathrm{nm})\) incident on it. What wavelength(s) will be visible at \(45.0^{\circ} ?\)

In a single-slit diffraction pattern, there is a bright central maximum surrounded by successively dimmer higher-order maxima. Farther out from the central maximum, eventually no more maxima are observed. Is this because the remaining maxima are too dim? Or is there an upper limit to the number of maxima that can be observed, no matter how good the observer's eyes, for a given slit and light source?

A laser beam with wavelength \(633 \mathrm{nm}\) is split into two beams by a beam splitter. One beam goes to Mirror \(1,\) a distance \(L\) from the beam splitter, and returns to the beam splitter, while the other beam goes to Mirror \(2,\) a distance \(L+\Delta x\) from the beam splitter, and returns to the same beam splitter. The beams then recombine and go to a detector together. If \(L=1.00000 \mathrm{~m}\) and \(\Delta x=1.00 \mathrm{~mm},\) which best describes the kind of interference at the detector? (Hint: To doublecheck your answer, you may need to use a formula that was originally intended for combining two beams in a different geometry, but which still is applicable here.) a) purely constructive b) purely destructive c) mostly constructive d) mostly destructive e) neither constructive nor destructive
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