Concept explainers
A horizontal laser beam of wavelength 632.8 nm has a circular cross section 2.00 nun in diameter. A rectangular aperture is to lie placed in the center of the beam so that when the light falls perpendicularly on a wall 4.50 m away, the central maximum fills a rectangle 110 mm wide and 6.00 mm high. The dimensions are measured between the minima bracketing the central maximum. Find the required (a) width and (b) height of the aperture. (c) Is the longer dimension of the central bright patch in the diffraction pattern horizontal or vertical? (d) Is the longer dimension of the aperture horizontal or vertical? (e) Explain the relationship between these two rectangles, using a diagram.
(a)
The width of the aperture.
Answer to Problem 38.4P
The width of the aperture is
Explanation of Solution
Given info: The wavelength of the laser beam is
Write the expression for the destructive interference.
Here,
Write the expression for the distance of the minimum from the central maximum.
Here,
The tangent is approximately equal to the sine if the angle is very small.
Substitute
Write the expression for the width of the central maximum.
Here,
Equate equation (1) and equation (2).
Substitute
Substitute
Conclusion:
Therefore, the width of the aperture is
(b)
The height of the aperture.
Answer to Problem 38.4P
The height of the aperture is
Explanation of Solution
Given info: The wavelength of the laser beam is
Write the expression for the height of the central maximum.
Here,
Substitute
Substitute
Conclusion:
Therefore, the height of the aperture is
(c)
Whether the longer dimension of the central bright patch is horizontal or vertical.
Answer to Problem 38.4P
The longer dimension of the central bright patch is horizontal.
Explanation of Solution
Given info: The wavelength of the laser beam is
From the given information, the width of the rectangle in the central bright patch is
Conclusion:
Therefore, the longer dimension of the central bright patch is horizontal.
(d)
Whether the longer dimension of the aperture is horizontal or vertical.
Answer to Problem 38.4P
The longer dimension of the aperture is vertical.
Explanation of Solution
Given info: The wavelength of the laser beam is
From part (a), the width of the aperture is
Conclusion:
Therefore, the longer dimension of the aperture is vertical
(e)
The relationship between the two rectangles.
Answer to Problem 38.4P
The longer dimension is
Explanation of Solution
Given info: The wavelength of the laser beam is
From part (a), the width of the aperture is
The smaller distance between aperture edges causes a wider diffraction angle.
Write the expression for the ratio of larger dimension to the smaller dimension of the aperture.
Substitute
Thus, the longer dimension is
Conclusion:
Therefore, the longer dimension is
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