University Physics with Modern Physics Plus Mastering Physics with eText -- Access Card Package (14th Edition)
University Physics with Modern Physics Plus Mastering Physics with eText -- Access Card Package (14th Edition)
14th Edition
ISBN: 9780321982582
Author: Hugh D. Young, Roger A. Freedman
Publisher: PEARSON
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Chapter 36, Problem 36.66CP

Intensity Pattern of N Slits. (a) Consider an arrangement of N slits with a distance d between adjacent slits. The slits emit coherently and in phase at wavelength λ. Show that at a time t, the electric Held at a distant point P is

E P ( t ) = E 0 cos ( k R ω t ) + E 0 cos ( k R ω t + ϕ )              + E 0 cos ( k R ω t + 2 ϕ ) + ...               +E 0 cos ( k R ω t + ( N 1 ) ϕ )

where E0 the amplitude at P of the electric field due to an individual slit, ϕ = (2π/sinθ)/λ, θ is the angle of the rays reaching P (as measured from the perpendicular bisector of the slit arrangement), and R is the distance from P to the most distant slit. In this problem, assume that R is much larger than d. (b) To carry out the sum in part (a), it is convenient to use the complex-number relationship = eiz = cosz + i sin z, where i = 1 . In this expression, cosz is the real port of the complex number eiz, and sinz is its imaginary port. Show that the electric field EP(t) is equal to the real part of the complex quantity

n = 0 N 1 E 0 e i ( k R ω t + n ϕ )

(c) Using the properties of the exponential function that = eAeB = e(A+B) and (eA)nshow = en,A, that the sum in part (b) can be written as

E 0 ( e i N ϕ 1 e i ϕ 1 ) e i ( k R ω t )

= E 0 ( e i N ϕ / 2 e i N ϕ / 2 e i ϕ / 2 e i N ϕ / 2 ) e i [ k R ω t + [ N 1 ] ϕ / 2 ]

Then, using the relationship eiz = cosz + isinz, show that the (real) electric field at point P is

E P ( t ) = [ E 0 sin ( N ϕ / 2 ) sin ( ϕ / 2 ) ] cos [ k R ω t + ( N 1 ) ϕ / 2 ]

The quantity in the first square brackets in this expression is the amplitude of the electric field at P. (d) Use the result for the electric-field amplitude in part (c) to show that the intensity at an angle θ is

I = I 0 [ sin ( N ϕ / 2 ) sin ( ϕ / 2 ) ] 2

where I0 is the maximum intensity for an individual slit, (e) Cheek the result in part (d) for the case N = 2. It will help to recall that sin2A = 2 sin A cosA. Explain why your result differs from Eq. (35.10), the expression for the intensity in two-source interference, by a factor of 4. (Hint: Is I0, defined in the same way in both expressions?)

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Chapter 36 Solutions

University Physics with Modern Physics Plus Mastering Physics with eText -- Access Card Package (14th Edition)

Ch. 36 - Light of wavelength and frequency f passes...Ch. 36 - In a diffraction experiment with waves of...Ch. 36 - An interference pattern is produced by four...Ch. 36 - Phasor Diagram for Eight Slits. An interference...Ch. 36 - A rainbow ordinarily shows a range of colors (see...Ch. 36 - Some loudspeaker horns for outdoor concerts (at...Ch. 36 - Figure 31.12 (Section 31.2) shows a loudspeaker...Ch. 36 - Prob. 36.11DQCh. 36 - With which color of light can the Hubble Space...Ch. 36 - At the end of Section 36.4, the following...Ch. 36 - Prob. 36.14DQCh. 36 - Why is a diffraction grating better than a...Ch. 36 - One sometimes sees rows of evenly spaced radio...Ch. 36 - Prob. 36.17DQCh. 36 - Prob. 36.18DQCh. 36 - Ordinary photographic film reverses black and...Ch. 36 - Monochromatic light from a distant source is...Ch. 36 - Parallel rays of green mercury light with a...Ch. 36 - Light of wavelength 585 nm falls on a slit 0.0666...Ch. 36 - Light of wavelength 633 nm from a distant source...Ch. 36 - Diffraction occurs for all types of waves,...Ch. 36 - CP Tsunami! On December 26, 2004, a violent...Ch. 36 - Prob. 36.7ECh. 36 - Monochromatic electromagnetic radiation with...Ch. 36 - Doorway Diffraction. Sound of frequency 1250 Hz...Ch. 36 - Figure 31.12 (Section 31.2) shows a loudspeaker...Ch. 36 - Red light of wavelength 633 nm from a helium neon...Ch. 36 - Public Radio station KXPR-FM in Sacramento...Ch. 36 - Monochromatic light of wavelength 580 nm passes...Ch. 36 - Monochromatic light of wavelength = 620 nm from a...Ch. 36 - A slit 0.240 mm wide is illuminated by parallel...Ch. 36 - Monochromatic light of wavelength 592 nm from a...Ch. 36 - A single-slit diffraction pattern is formed by...Ch. 36 - Parallel rays of monochromatic light with...Ch. 36 - Number of Fringes in a Diffraction Maximum. In...Ch. 36 - Diffraction and Interference Combined. Consider...Ch. 36 - An interference pattern is produced by light of...Ch. 36 - Laser light of wavelength 500.0 nm illuminates two...Ch. 36 - When laser light of wavelength 632.8 nm passes...Ch. 36 - Monochromatic light is at normal incidence on a...Ch. 36 - If a diffraction grating produces its third-order...Ch. 36 - If a diffraction grating produces a third-order...Ch. 36 - Visible light passes through a diffraction grating...Ch. 36 - The wavelength range of the visible spectrum is...Ch. 36 - (a) What is the wavelength of light that is...Ch. 36 - CDs and DVDs as Diffraction Gratings. A laser beam...Ch. 36 - A typical laboratory diffraction grating has 5.00 ...Ch. 36 - Identifying Isotopes by Spectra. Different...Ch. 36 - The light from an iron arc includes many different...Ch. 36 - If the planes of a crystal are 3.50 (1 = 1010 m...Ch. 36 - Prob. 36.35ECh. 36 - Monochromatic x rays are incident on a crystal for...Ch. 36 - Monochromatic light with wavelength 620 nm passes...Ch. 36 - Monochromatic light with wavelength 490 nm passes...Ch. 36 - Two satellites at an altitude of 1200 km are...Ch. 36 - BIO If you can read the bottom row of your doctors...Ch. 36 - The VLBA (Very Long Baseline Array) uses a number...Ch. 36 - Searching for Planets Around Other Stars. If an...Ch. 36 - Hubble Versus Arecibo. The Hubble Space Telescope...Ch. 36 - Photography. A wildlife photographer uses a...Ch. 36 - Observing Jupiter. You are asked to design a space...Ch. 36 - Coherent monochromatic light of wavelength passes...Ch. 36 - BIO Thickness of Human Hair. Although we have...Ch. 36 - CP A loudspeaker with a diaphragm that vibrates at...Ch. 36 - Laser light of wavelength 632.8 nm falls normally...Ch. 36 - Grating Design. Your boss asks you to design a...Ch. 36 - Measuring Refractive Index. A thin slit...Ch. 36 - Underwater Photography. An underwater camera has a...Ch. 36 - CALC The intensity of light in the Fraunhofer...Ch. 36 - A slit 0.360 mm wide is illuminated by parallel...Ch. 36 - CP CALC In a large vacuum chamber, monochromatic...Ch. 36 - CP In a laboratory, light from a particular...Ch. 36 - What is the longest wavelength that can be...Ch. 36 - It has been proposed to use an array of infrared...Ch. 36 - A diffraction grating has 650 slits/mm. What is...Ch. 36 - Quasars, an abbreviation for quasi-stellar radio...Ch. 36 - A glass sheet is covered by a very thin opaque...Ch. 36 - BIO Resolution of the Eye. The maximum resolution...Ch. 36 - DATA While researching the use of laser pointers,...Ch. 36 - DATA Your physics study partner tells you that the...Ch. 36 - DATA At the metal fabrication company where you...Ch. 36 - Intensity Pattern of N Slits. (a) Consider an...Ch. 36 - CALC Intensity Pattern of N Silts, Continued. Part...Ch. 36 - CALC It is possible to calculate the intensity in...Ch. 36 - Prob. 36.69PPCh. 36 - BRAGG REFLECTION ON A DIFFERENT SCALE. A colloid...Ch. 36 - Prob. 36.71PP
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