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Chapter 25: Problem 32

Ramon has a coherent light source with wavelength \(547 \mathrm{nm} .\) He wishes to send light through a double slit with slit separation of $1.50 \mathrm{mm}\( to a screen \)90.0 \mathrm{cm}$ away. What is the minimum width of the screen if Ramon wants to display five interference maxima?

Short Answer

Expert verified
Answer: The minimum width of the screen needed to display five interference maxima is approximately 32.8 mm.

Step by step solution

01

Determine the formula for the position of bright fringes

We can use the formula for double-slit interference to find the position of bright fringes (interference maxima): $$y = \frac{m \lambda L}{d}$$ where \(y\) is the distance from the central maximum (m=0) to the m-th maximum, \(\lambda\) is the wavelength of light, \(L\) is the distance between the double slits and the screen, \(d\) is the slit separation, and \(m\) is the order number of the maximum (integer value).
02

Calculate the position of the 5th bright fringe

We are asked to find the width of the screen needed to display five interference maxima. The fifth maxima would have an order number \(m = 5\). Plugging in the given values, we get: $$y_5 = \frac{5 (547 \times 10^{-9} \mathrm{m})(0.9 \mathrm{m})}{1.5 \times 10^{-3} \mathrm{m}}$$ Calculating the value, $$y_5 \approx 0.0164 \mathrm{m}$$
03

Calculate the total width of the screen

Since our calculated value, \(0.0164 \mathrm{m}\), represents the distance of the fifth maximum (\(m = 5\)) from the central maximum (\(m = 0\)), the width of the screen would be approximately the sum of the distances of the fifth maximum to the left of the central maximum and the fifth maximum to the right of the central maximum. Therefore, the total width w of the screen would be: $$w = 2 \times y_5$$ Calculating the value, $$w \approx 2 \times 0.0164 \mathrm{m} \approx 0.0328 \mathrm{m}$$ So, the minimum width of the screen needed to display five interference maxima is approximately \(0.0328 \mathrm{m}\) or \(32.8\,\mathrm{mm}\).

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Most popular questions from this chapter

The diffraction pattern from a single slit is viewed on a screen. Using blue light, the width of the central maximum is \(2.0 \mathrm{cm} .\) (a) Would the central maximum be narrower or wider if red light is used instead? (b) If the blue light has wavelength \(0.43 \mu \mathrm{m}\) and the red light has wavelength \(0.70 \mu \mathrm{m},\) what is the width of the central maximum when red light is used?
Two incoherent EM waves of intensities \(9 I_{0}\) and \(16 I_{0}\) travel in the same direction in the same region of space. What is the intensity of EM radiation in this region?
A red line (wavelength \(630 \mathrm{nm}\) ) in the third order overlaps with a blue line in the fourth order for a particular grating. What is the wavelength of the blue line?
A grating spectrometer is used to resolve wavelengths \(660.0 \mathrm{nm}\) and \(661.4 \mathrm{nm}\) in second order. (a) How many slits per centimeter must the grating have to produce both wavelengths in second order? (The answer is either a maximum or a minimum number of slits per centimeter.) (b) The minimum number of slits required to resolve two closely spaced lines is $N=\lambda /(m \Delta \lambda),\( where \)\lambda$ is the average of the two wavelengths, \(\Delta \lambda\) is the difference between the two wavelengths, and \(m\) is the order. What minimum number of slits must this grating have to resolve the lines in second order?
Thin Films At a science museum, Marlow looks down into a display case and sees two pieces of very flat glass lying on top of each other with light and dark regions on the glass. The exhibit states that monochromatic light with a wavelength of \(550 \mathrm{nm}\) is incident on the glass plates and that the plates are sitting in air. The glass has an index of refraction of \(1.51 .\) (a) What is the minimum distance between the two glass plates for one of the dark regions? (b) What is the minimum distance between the two glass plates for one of the light regions? (c) What is the next largest distance between the plates for a dark region? [Hint: Do not worry about the thickness of the glass plates; the thin film is the air between the plates.]
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