Problem 2: (a) Consider the double-slit experiment in which the slit separation, d, is equal to the slit width, a. Explain how this set up is equivalent to a single-slit experiment with slit width 2a. In particular, what happens to the interference peaks of that double-slit experiment (besides the central peak)? What happens to the diffraction peaks of that double-slit experiment (besides the central peak)? Sketch the intensity pattern to elucidate the equivalence. (b) To prove that equivalence even further, show that the intensity function of the double-slit experiment [sin(/2)" (0/2) I(0) = Io cos (8/2) (1) %3D reduces to the correct intensity function of the single-slit experiment with slit width 2a. Recall o = 27 asine and 8 = 27 dsine. (c) Now ignore diffraction. A laser with 600 nanometer wavelength is shines at two slits separated by 2 millimeters. An interference pattern forms on a screen 1.5 meters behind the slits. How many maxima are illuminated on the screen? Assume the screen is very large.
Problem 2: (a) Consider the double-slit experiment in which the slit separation, d, is equal to the slit width, a. Explain how this set up is equivalent to a single-slit experiment with slit width 2a. In particular, what happens to the interference peaks of that double-slit experiment (besides the central peak)? What happens to the diffraction peaks of that double-slit experiment (besides the central peak)? Sketch the intensity pattern to elucidate the equivalence. (b) To prove that equivalence even further, show that the intensity function of the double-slit experiment [sin(/2)" (0/2) I(0) = Io cos (8/2) (1) %3D reduces to the correct intensity function of the single-slit experiment with slit width 2a. Recall o = 27 asine and 8 = 27 dsine. (c) Now ignore diffraction. A laser with 600 nanometer wavelength is shines at two slits separated by 2 millimeters. An interference pattern forms on a screen 1.5 meters behind the slits. How many maxima are illuminated on the screen? Assume the screen is very large.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**Problem 2:**
**(a)** Consider the double-slit experiment in which the slit separation, \(d\), is equal to the slit width, \(a\). Explain how this setup is equivalent to a single-slit experiment with slit width \(2a\). In particular, what happens to the interference peaks of this double-slit experiment (besides the central peak)? What happens to the diffraction peaks of that double-slit experiment (besides the central peak)? Sketch the intensity pattern to elucidate the equivalence.
**(b)** To prove that equivalence even further, show that the intensity function of the double-slit experiment
\[
I(\theta) = I_0 \left[ \frac{\sin(\phi/2)}{(\phi/2)} \right]^2 \cos^2(\delta/2) \tag{1}
\]
reduces to the correct intensity function of the single-slit experiment with slit width \(2a\). Recall \(\phi = 2\pi \frac{a \sin \theta}{\lambda}\) and \(\delta = 2\pi \frac{d \sin \theta}{\lambda}\).
**(c)** Now ignore diffraction. A laser with 600 nanometer wavelength is shined at two slits separated by 2 millimeters. An interference pattern forms on a screen 1.5 meters behind the slits. How many maxima are illuminated on the screen? Assume the screen is very large.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d2fdd51-a813-4b36-89e9-f9581acfc2ee%2F68e557fe-cb0d-4ba2-8753-844fcca30589%2Fd3t2g0i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2:**
**(a)** Consider the double-slit experiment in which the slit separation, \(d\), is equal to the slit width, \(a\). Explain how this setup is equivalent to a single-slit experiment with slit width \(2a\). In particular, what happens to the interference peaks of this double-slit experiment (besides the central peak)? What happens to the diffraction peaks of that double-slit experiment (besides the central peak)? Sketch the intensity pattern to elucidate the equivalence.
**(b)** To prove that equivalence even further, show that the intensity function of the double-slit experiment
\[
I(\theta) = I_0 \left[ \frac{\sin(\phi/2)}{(\phi/2)} \right]^2 \cos^2(\delta/2) \tag{1}
\]
reduces to the correct intensity function of the single-slit experiment with slit width \(2a\). Recall \(\phi = 2\pi \frac{a \sin \theta}{\lambda}\) and \(\delta = 2\pi \frac{d \sin \theta}{\lambda}\).
**(c)** Now ignore diffraction. A laser with 600 nanometer wavelength is shined at two slits separated by 2 millimeters. An interference pattern forms on a screen 1.5 meters behind the slits. How many maxima are illuminated on the screen? Assume the screen is very large.
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