Physics for Scientists and Engineers, Technology Update (No access codes included)
Physics for Scientists and Engineers, Technology Update (No access codes included)
9th Edition
ISBN: 9781305116399
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 38, Problem 38.72AP

(a)

To determine

To show: The point where I=0.5Imax must have ϕ=2sinϕ .

(a)

Expert Solution
Check Mark

Answer to Problem 38.72AP

The point where I=0.5Imax must have ϕ=2sinϕ .

Explanation of Solution

Given info: The equation of the intensity of the light in the diffraction pattern is I=Imaxsin2ϕϕ2 where ϕ=(πasinθ)λ

The formula to calculate the intensity of the light is,

I=Imaxsin2ϕϕ2IImax=sin2ϕϕ2

Here,

Imax is the maximum intensity of the light.

ϕ is the phase constant of the light.

The value of I=0.5Imax .

Substitute 0.5Imax for I in above equation to find the value of ϕ .

0.5ImaxImax=sin2ϕϕ2sin2ϕϕ2=12sinϕ=ϕ2ϕ=2sinϕ

Conclusion

Therefore, the point where I=0.5Imax must have ϕ=2sinϕ .

(b)

To determine

To draw: Plot y1=sinϕ and y2=ϕ2 on the same set of axes over a range from ϕ=1rad to ϕ=π2rad .

(b)

Expert Solution
Check Mark

Answer to Problem 38.72AP

The graph between y1=sinϕ and y2=ϕ2 on the same set of axes over a range from ϕ=1rad to ϕ=π2rad .

Physics for Scientists and Engineers, Technology Update (No access codes included), Chapter 38, Problem 38.72AP , additional homework tip  1

Figure (1)

Explanation of Solution

Given info: The equation of the intensity of the light in the diffraction pattern is I=Imaxsin2ϕϕ2 where ϕ=(πasinθ)λ

The equation of y1 is sinϕ and the equation for y2 is ϕ2 over a range from ϕ=1rad to ϕ=π2rad is shown in the figure below.

Physics for Scientists and Engineers, Technology Update (No access codes included), Chapter 38, Problem 38.72AP , additional homework tip  2

The solution of both the equation to coincide at a point is ,

sinϕ=ϕ2ϕ=1.39rad

So the solution of the transcendental equation is ϕ=1.39rad .

(c)

To determine

To show: The angular full width at half maximum of the central diffraction maximum is θ=0.885λa if the fraction λa is not large.

(c)

Expert Solution
Check Mark

Answer to Problem 38.72AP

The angular full width at half maximum of the central diffraction maximum is θ=0.885λa if the fraction λa is not large.

Explanation of Solution

Given info: The equation of the intensity of the light in the diffraction pattern is I=Imaxsin2ϕϕ2 where ϕ=(πasinθ)λ

The formula to calculate the phase angle is,

ϕ=(πasinθ)λ

Rewrite the above equation for sinθ .

sinθ=(ϕπ)λa

If the value of λa is small then,

θ(ϕπ)λa

The path covered by the light is symmetric so the phase angle is double the initial value.

θ=2(ϕπ)λa

Substitute 1.39rad for ϕ in above equation to find the value of θ .

θ=2(1.39rad3.14rad)λa=0.885λa

Conclusion

Therefore, the angular full width at half maximum of the central diffraction maximum is θ=0.885λa if the fraction λa is not large.

(d)

To determine

The number of steps involved to solve the transcendental equation ϕ=2sinϕ .

(d)

Expert Solution
Check Mark

Answer to Problem 38.72AP

The number of steps involved to solve the transcendental equation ϕ=2sinϕ is around 13 .

Explanation of Solution

Given info: The equation of the intensity of the light in the diffraction pattern is I=Imaxsin2ϕϕ2 where ϕ=(πasinθ)λ

The equation of y1 is 2sinϕ and the equation for y2 is ϕ , the value of ϕ is taken from 1 to 2 to find the solution of the equation and the corresponding values are shown in the table below.

ϕ 2sinϕ
1 1.19
2 1.29
1.5 1.41
1.4 1.394
1.39 1.391
1.392 1.3917
1.3915 1.39154
1.39152 1.39155
1.3916 1.39158
1.39158 1.391563
1.39157 1.391561
1.39156 1.391558
1.3915574 1.3915574

The solution of the transcendental equation ϕ=2sinϕ is 1.3915574 and the solution can be achieved in around 13 steps.

Conclusion

Therefore, the number of steps involved to solve the transcendental equation ϕ=2sinϕ is around 13 .

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
How much diffraction spreading does a light beam undergo? One quantitative answer is the full width at half maximum of the central maximum of the single-slit Fraunhofer diffraction pattern. You can evaluate this angle of spreading in this problem. (a) as shown, define φ = πa sin φ/λ and show that at the point where I = 0.5Imax we must have φ = √2 sin φ. (b) Let y1 = sin φ and y2 = φ = /√2. Plot y1 and y2 on the same set of axes over a range from φ = 1 rad to φ = π/2 rad. Determine φ from the point of intersection of the two curves. (c) Then show that if the fraction λ/a is notlarge, the angular full width at half maximum of the central diffraction maximum is θ = 0.885λ/a. (d) What If? Another method to solve the transcendental equation φ = √2 sin φ in part (a) is to guess a first value of φ, use a computer or calculator to see how nearly it fits, and continue to update your estimate until the equation balances. How many steps(iterations) does this process take?
Q1/ A/ Consider a resonator consisting of two concave spherical mirrors both with radius of curvature 4 m and separated by a distance of 1 m. Calculate the minimum beam diameter of the TEMy mode at the resonator center and on the mirrors when the laser oscillation is Art laser wavelength 2=514.5 nm. Then, Find if this cavity is stable or not. B/ One of the mirrors in A is replaced by a concave mirror of 1.5m radius of curvature, calculate the position of minimum beam radius. Then calculate the beam waist and radius of curvature at 150 cm from M1.
Problem 5: Consider light that has its third minimum at an angle of 24.4° when it falls on a single slit of width 3.55 µm . Randomized Variables e = 24.4 ° w = 3.55 µm Find the wavelength of the light in nanometers. 2 = 789 E AAL 4 |5 | 6 1| 2 sin() cos() tan() HOME cotan() asin() acos() atan() acotan() sinh() 3 cosh() tanh() cotanh() END O Degrees O Radians vol BACKSPACE DEL CLEAR Submit I give up! Hint Feedback

Chapter 38 Solutions

Physics for Scientists and Engineers, Technology Update (No access codes included)

Ch. 38 - A Fraunhofer diffraction pattern is produced on a...Ch. 38 - Consider a wave passing through a single slit....Ch. 38 - Assume Figure 38.1 was photographed with red light...Ch. 38 - If plane polarized light is sent through two...Ch. 38 - Why is it advantageous to use a large-diameter...Ch. 38 - What combination of optical phenomena causes the...Ch. 38 - Prob. 38.10OQCh. 38 - When unpolarized light passes through a...Ch. 38 - Off in the distance, you see the headlights of a...Ch. 38 - Prob. 38.1CQCh. 38 - Holding your hand at arms length, you can readily...Ch. 38 - Prob. 38.3CQCh. 38 - (a) Is light from the sky polarized? (b) Why is it...Ch. 38 - Prob. 38.5CQCh. 38 - If a coin is glued to a glass sheet and this...Ch. 38 - Fingerprints left on a piece of glass such as a...Ch. 38 - A laser produces a beam a few millimeters wide,...Ch. 38 - Prob. 38.9CQCh. 38 - John William Strutt, Lord Rayleigh (1842-1919),...Ch. 38 - Prob. 38.11CQCh. 38 - Prob. 38.12CQCh. 38 - Light of wavelength 587.5 nm illuminates a slit of...Ch. 38 - Heliumneon laser light ( = 632.8 nm) is sent...Ch. 38 - Sound with a frequency 650 Hz from a distant...Ch. 38 - A horizontal laser beam of wavelength 632.8 nm has...Ch. 38 - Coherent microwaves of wavelength 5.00 cm enter a...Ch. 38 - Light of wavelength 540 nm passes through a slit...Ch. 38 - A screen is placed 50.0 cm from a single slit,...Ch. 38 - A screen is placed a distance L from a single slit...Ch. 38 - Assume light of wavelength 650 nm passes through...Ch. 38 - What If? Suppose light strikes a single slit of...Ch. 38 - A diffraction pattern is formed on a screen 120 cm...Ch. 38 - Coherent light of wavelength 501.5 nm is sent...Ch. 38 - Prob. 38.13PCh. 38 - The pupil of a cats eye narrows to a vertical slit...Ch. 38 - The angular resolution of a radio telescope is to...Ch. 38 - A pinhole camera has a small circular aperture of...Ch. 38 - The objective lens of a certain refracting...Ch. 38 - Yellow light of wavelength 589 nm is used to view...Ch. 38 - What is the approximate size of the smallest...Ch. 38 - A heliumneon laser emits light that has a...Ch. 38 - To increase the resolving power of a microscope,...Ch. 38 - Narrow, parallel, glowing gas-filled tubes in a...Ch. 38 - Impressionist painter Georges Seurat created...Ch. 38 - A circular radar antenna on a Coast Guard ship has...Ch. 38 - Prob. 38.25PCh. 38 - Prob. 38.26PCh. 38 - Consider an array of parallel wires with uniform...Ch. 38 - Three discrete spectral lines occur at angles of...Ch. 38 - The laser in a compact disc player must precisely...Ch. 38 - A grating with 250 grooves/mm is used with an...Ch. 38 - A diffraction grating has 4 200 rulings/cm. On a...Ch. 38 - The hydrogen spectrum includes a red line at 656...Ch. 38 - Light from an argon laser strikes a diffraction...Ch. 38 - Show that whenever white light is passed through a...Ch. 38 - Light of wavelength 500 nm is incident normally on...Ch. 38 - A wide beam of laser light with a wavelength of...Ch. 38 - Prob. 38.37PCh. 38 - Prob. 38.38PCh. 38 - Potassium iodide (Kl) has the same crystalline...Ch. 38 - Prob. 38.40PCh. 38 - Prob. 38.41PCh. 38 - Why is the following situation impossible? A...Ch. 38 - Prob. 38.43PCh. 38 - The angle of incidence of a light beam onto a...Ch. 38 - Unpolarized light passes through two ideal...Ch. 38 - Prob. 38.46PCh. 38 - You use a sequence of ideal polarizing niters,...Ch. 38 - An unpolarized beam of light is incident on a...Ch. 38 - The critical angle for total internal reflection...Ch. 38 - For a particular transparent medium surrounded by...Ch. 38 - Three polarizing plates whose planes are parallel...Ch. 38 - Two polarizing sheets are placed together with...Ch. 38 - In a single-slit diffraction pattern, assuming...Ch. 38 - Laser light with a wavelength of 632.8 nm is...Ch. 38 - Prob. 38.55APCh. 38 - Prob. 38.56APCh. 38 - Prob. 38.57APCh. 38 - Two motorcycles separated laterally by 2.30 m are...Ch. 38 - The Very Large Array (VLA) is a set of 27 radio...Ch. 38 - Two wavelengths and + (with ) are incident on...Ch. 38 - Review. A beam of 541-nm light is incident on a...Ch. 38 - Prob. 38.62APCh. 38 - Prob. 38.63APCh. 38 - Prob. 38.64APCh. 38 - Prob. 38.65APCh. 38 - Prob. 38.66APCh. 38 - Prob. 38.67APCh. 38 - A pinhole camera has a small circular aperture of...Ch. 38 - Prob. 38.69APCh. 38 - (a) Light traveling in a medium of index of...Ch. 38 - The intensity of light in a diffraction pattern of...Ch. 38 - Prob. 38.72APCh. 38 - Two closely spaced wavelengths of light are...Ch. 38 - Light of wavelength 632.8 nm illuminates a single...Ch. 38 - Prob. 38.75CPCh. 38 - A spy satellite can consist of a large-diameter...Ch. 38 - Suppose the single slit in Figure 38.4 is 6.00 cm...Ch. 38 - In Figure P37.52, suppose the transmission axes of...Ch. 38 - Consider a light wave passing through a slit and...
Knowledge Booster
Background pattern image
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
University Physics Volume 3
Physics
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:OpenStax
Diffraction of light animation best to understand class 12 physics; Author: PTAS: Physics Tomorrow Ambition School;https://www.youtube.com/watch?v=aYkd_xSvaxE;License: Standard YouTube License, CC-BY