PHY 6347 Spring 2018 Homework #10, Due Friday, April 6
|
|
- Philomena McLaughlin
- 6 years ago
- Views:
Transcription
1 PHY 6347 Spring 28 Homework #, Due Friday, April 6. A plane wave ψ = ψ e ik x is incident from z < on an opaque screen that blocks the entire plane z = except for the opening 2 a < x < 2 a, 2 b < y < 2 b. (a) Compute the diffracted field ψ scatt from the scalar Kirchhoff-Neumann relation (.85) in the large-r regime. Asitsays, startingfromtheneumannversionof(.85),or(.83),withintegrandg N ( ψ/ n ) S in place of the Dirichlet ψ S ( G D / n ), the diffracted field is a/2 b/2 ψ diff = dx dy ˆÒ (ie ik x ) e ikr e ikˆr x. a/2 b/2 r The Neumann Green s function onthe boundary is twice e ik x x /4πÜ Ü. The Neumann boundary condition ψ/ n = ˆÒ ψ leads to the angular dependence ˆÞ i = ikcosθ. The area integral factors into x - and y -integrals that can be done separately, a/2 dx e ik(sinθ cosφ sinθcosφ)x = a sin( 2 ka x) a/2 2 ka, x b/2 dy e ik(sinθ sinφ sinθsinφ)y = b sin( 2 kb y) b/2 2 kb, y where x and y are the factors in parentheses in the exponentials, x = sinθcosφ sinθ cosφ, y = sinθsinφ sinθ sinφ. The diffracted wave is then ψ diff = ikabψ cosθ e ikr r sin( 2 ka x) ( 2 ka x) sin( 2 kb y) ( 2 kb y). (b) Compute the angular distribution of power dp/dω. The angular distribution of power is dp dω = r2 ψ 2 = k2 a 2 b 2 ψ 2 cos 2 θ sin( 2 ka x ) 2 sin( 2 kb y ) 2. ( 2 ka x) ( 2 kb y) You could also write these in terms of Cartesian x = a x and y = b y. Write sinx/x = S(x), the so-called sinc function; then, the outgoing power per solid angle is dp dω = k2 a 2 b 2 ψ 2 cos 2 θ S 2 ( 2 ka x)s 2 ( 2 kb y).
2 2 For long wavelength, or small ka, kb, S, For small λ, function S is confined to a small range about the incident direction. Within the first zero of the sin function, 2 ka sinθcosφ sinθ cosφ < π, sinθsinφ sinθ sinφ < λ a, As ka, kb become large S narrows to a δ-function. For what the intensity looks like, see the square diffraction link on the course web page. (c) Write an expression for the total power transmitted through the opening in the forward direction. Compute the transmission coefficient T in the limits ka, kb and ka, kb. Estimate the fraction of the power inside a microwave oven that leaks out through the openings in the screen (a microwave oven operates at f = 245MHz). The incoming power through the opening is P = Ê d 2 aˆþ Ë = ab ψ 2 cosθ ; the transmitted power in the forward direction is the integral over < θ < π 2, π/2 P = dφ and the transmission coefficient T = P/P is T = k2 ab cosθ sinθdθ dp dω ; dωs 2 ( 2 ka x)s 2 ( 2 kb y). The integral cannot be done in general in closed form, but is easy in the two limits. As ka, kb, S, and Ê dω =. and T = k2 abcosθ (ka,kb ). In the Dirichlet version this becomes k 2 ab/4πcosθ, which looks strange, the transmitted power becomes greater for oblique incidence. As ka and kb become large, the functions S 2 ( 2 ka x) and S 2 ( 2 ka y) become narrower and narrower, effectively becoming δ-functions in x and y, or θ θ, φ φ. In the integration over solid angle, it is fairly easy to change variables from θ and φ to x and y, since the Jacobian of the variable change is x θ y θ x φ y φ = sinθ cosθ sinθ cosθ.
3 3 When ka becomes large, any finite integration range in θ, φ is effectively infinite in x, y, and the projected transmission is T = k2 abcosθ = k2 ab cosθ dθdφ sinθ S 2 ( 2 ka x)s 2 ( 2 ka y) d x d y sinθ cosθ sinθ S 2 ( 2 ka x)s 2 ( 2 ka y) Believe it or not. = k2 ab sin 2 ( d 2 ka x) x ( 2 ka x) 2 =. kb = k2 ab ka d y sin 2 ( 2 kb y) ( 2 kb y) 2 This calculation tells you how it is that you can look through the holes in the screen of your microwave oven to see what s happening inside: visible light with λ = 5nm passes easily through holes of size mm (ka 2,, T ); while the microwave cooking power (f = 2.45GHz, λ = 2.2cm) mostly remains confined in the interior, with only a fraction k 2 a 2 /.4 (or k 2 a 2 /8 =.3, using the area of a circle of diameter a) leaking out. A microwave oven might operate at a power of kw. Spread over the area 25cm 2 of the door, the leakage per area is of order The published safety limit is 5mWcm 2. (kw).4 25cm 2 =.6mWcm 2.
4 4 2. This is another classic problem: An opaque screen occupies the x < half of the x-y plane. A plane wave ψ = Ô I e i(kz ωt) is incident on the screen from z <. (a) Show that in the usual scalar Kirchhoff approximation, for kz and for z x, y the diffracted field in the forward direction is ψ(x,y,z) = 2I π +i e i(kz ωt) e iξ 2 dξ, 2i ξ ξ = kx Ô 2kz. In the limit kr, and with cosθ = cosθ, the scalar Kirchhoff approximation (.79) and its Dirichlet and Neumann variations all give ψ diff = dx dy eikr 4π x > R 2ikψ, where ψ S = ψ = I /2. In Cartesian coordinates, the distance from source to observer is R = Ü Ü = (x x ) 2 +(y y ) 2 +z 2 /2 z + 2z [(x x ) 2 +(y y ) 2 ], where the last holds for x x, y y z. Thus, integrated over the illuminated portion of the x-y plane, ψ = ki/2 iz eikz dx e ik(x x) 2 /2z dy e ik(y y) 2 /2z. Change integration variables to ξ 2 = k(x x) 2 /2z and η = k(y y) 2 /2z; then ψ = I/2 iπ eikz dξ e iξ 2 dη e iη 2. ξ The answer is independent of y. Use Ê e iη 2 dη = Ô iπ = (+i) Ô π/2 [analytic continuation of the Gaussian integral; or see the two lines above (.32)]. The choice of sign of the square root follows because the real and imaginary parts of the integral Ê dη(cosη 2 +isinη 2 ) are both positive, and ψ = I /2 ikz +i e i Ô dξ e iξ 2. ξ You are of course worried about expanding for small x x when x is integrated to infinity, but the integral falls off exponentially when the exponent becomes larger than, and where the exponent is, k( x) 2 /z is of order, so x is of order Ô z/k, and x/z is of order / Ô kz.
5 (b) Show that the diffracted intensity I = ψ 2 can be written I = I(ξ) = 2 I C(ξ) S(ξ)+ 2 2, 5 where C(ξ) and S(ξ) are one representation of the Fresnel cosine and sine integrals in the form 2 ξ C(ξ) = dξ cos(ξ 2 2 ξ ), S(ξ) = dξ sin(ξ 2 ). π π What is I()? What are the behaviors of I(ξ) for large positive and negative values of ξ? Plot I(ξ). On consulting various sources, including Abramowitz and Stegun, Handbook of Mathematical Functions, and MathWorld, as well as Jackson, one finds that there are at least three forms of the Fresnel integrals. The perhaps standard form is defined as ξ C (ξ) = cos( π ξ 2 t2 )dt, S (ξ) = sin( π 2 t2 )dt, with two alternative versions, and C (ξ) = C 2 (ξ) = Ô ξ 2 ξ cost 2 dt, S π (ξ) = 2 ξ sint 2 dt π cost Ô t dt, S 2 (ξ) = Ô ξ sint Ô t dt, related by the obvious changes of variable C (ξ) = C ( Ô π 2 ξ) = C 2 ( π 2 ξ2 ) and S (ξ) = S ( Ô π 2 ξ) = S 2 ( π 2 ξ2 ). In his Problem. Jackson uses C, S ; so for the remainder of this problem, take C = C (ξ) and S = S (ξ). The integrands are even, so both the functions are odd, C( ξ) = C(ξ) and S( ξ) = S(ξ). For large argument both approach C(½) = S(½) = 2. Expressed using the Fresnel integrals, the phase integral is 2 e iξ 2 dξ 2 = cosξ 2 dξ 2 +i sinξ 2 dξ = π ξ π ξ π 2 +C(ξ) +i 2 +S(ξ), ξ and the diffracted amplitude and intensity are ψ = I /2 ikz +i e 2i As ξ, ξ ½, and ξ ½: [ 2 +C(ξ)]+i 2 +S(ξ), I = ψ 2 = 2 I 2 +C(ξ) S(ξ) 2. I [] I 4 + Ô ξ + ξ2 ; I π [ ] I + sinξ2 cosξ 2 Ô ; I [ ] I ξ ξ 2.
6 6 Fresnel Integrals: Diffracted Intensity:
7 7 (c) At the end of the eclipse of 2 August 27, an observer noticed that bands of light and shadow appeared to sweep over him with the return of the sun. What (in meters) is the scale of the width of these bands? The first few intensity maxima at ξ =.525, 2.938, and 3.862, with minima at ξ =.346, The separations are all different, with smaller contrast and closer spacing at larger ξ, but are all of order ξ =. Visible light has wavelengths between 4nm and 7nm, and the distance to the moon is 38km. This gives x = 2z λz k ξ = π = (55nm)(38km) π = 8m m as the scale of the modulation. A factor of 2 either way is also OK. This is consistent with the impressions of the observer.
Fourier Approach to Wave Propagation
Phys 531 Lecture 15 13 October 005 Fourier Approach to Wave Propagation Last time, reviewed Fourier transform Write any function of space/time = sum of harmonic functions e i(k r ωt) Actual waves: harmonic
More informationLecture notes 5: Diffraction
Lecture notes 5: Diffraction Let us now consider how light reacts to being confined to a given aperture. The resolution of an aperture is restricted due to the wave nature of light: as light passes through
More informationContents. Diffraction by 1-D Obstacles. Narrow Slit. Wide Slit. N Slits. 5 Infinite Number of Slits
Diffraction Contents 1 2 Narrow Slit 3 Wide Slit 4 N Slits 5 Infinite Number of Slits - geometric arrangement diffraction pattern amplitude Fk ( ) ik r F( k)= f( r) dr all r f( r) : amplitude function
More informationSolutions: Homework 7
Solutions: Homework 7 Ex. 7.1: Frustrated Total Internal Reflection a) Consider light propagating from a prism, with refraction index n, into air, with refraction index 1. We fix the angle of incidence
More informationPhysics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010
Physics 216 Problem Set 4 Spring 2010 DUE: MAY 25, 2010 1. (a) Consider the Born approximation as the first term of the Born series. Show that: (i) the Born approximation for the forward scattering amplitude
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 32
ECE 6345 Spring 215 Prof. David R. Jackson ECE Dept. Notes 32 1 Overview In this set of notes we extend the spectral-domain method to analyze infinite periodic structures. Two typical examples of infinite
More informationPhysics 214 Midterm Exam Solutions Winter 2017
Physics 14 Midterm Exam Solutions Winter 017 1. A linearly polarized electromagnetic wave, polarized in the ˆx direction, is traveling in the ẑ-direction in a dielectric medium of refractive index n 1.
More informationInterference, Diffraction and Fourier Theory. ATI 2014 Lecture 02! Keller and Kenworthy
Interference, Diffraction and Fourier Theory ATI 2014 Lecture 02! Keller and Kenworthy The three major branches of optics Geometrical Optics Light travels as straight rays Physical Optics Light can be
More informationcauchy s integral theorem: examples
Physics 4 Spring 17 cauchy s integral theorem: examples lecture notes, spring semester 17 http://www.phys.uconn.edu/ rozman/courses/p4_17s/ Last modified: April 6, 17 Cauchy s theorem states that if f
More informationPhysics 505 Homework No. 12 Solutions S12-1
Physics 55 Homework No. 1 s S1-1 1. 1D ionization. This problem is from the January, 7, prelims. Consider a nonrelativistic mass m particle with coordinate x in one dimension that is subject to an attractive
More informationLECTURE 18: Horn Antennas (Rectangular horn antennas. Circular apertures.)
LCTUR 18: Horn Antennas (Rectangular horn antennas. Circular apertures.) 1 Rectangular Horn Antennas Horn antennas are popular in the microwave bands (above 1 GHz). Horns provide high gain, low VSWR (with
More informationGoal: The theory behind the electromagnetic radiation in remote sensing. 2.1 Maxwell Equations and Electromagnetic Waves
Chapter 2 Electromagnetic Radiation Goal: The theory behind the electromagnetic radiation in remote sensing. 2.1 Maxwell Equations and Electromagnetic Waves Electromagnetic waves do not need a medium to
More informationIf the wavelength is larger than the aperture, the wave will spread out at a large angle. [Picture P445] . Distance l S
Chapter 10 Diffraction 10.1 Preliminary Considerations Diffraction is a deviation of light from rectilinear propagation. t occurs whenever a portion of a wavefront is obstructed. Hecht; 11/8/010; 10-1
More information221B Lecture Notes Scattering Theory II
22B Lecture Notes Scattering Theory II Born Approximation Lippmann Schwinger equation ψ = φ + V ψ, () E H 0 + iɛ is an exact equation for the scattering problem, but it still is an equation to be solved
More informationr p = r o r cos( φ ) cos( α )
Section 4. : Sound Radiation Pattern from the Mouth of a Horn In the previous section, the acoustic impedance at the mouth of a horn was calculated. Distributed simple sources were used to model the mouth
More informationInterference by Wavefront Division
nterference by Wavefront Division One of the seminal experiments in physics was conducted in 1801 by Thomas Young, an English physicist who cut a small hole in an opaque screen, set a second screen in
More informationLet b be the distance of closest approach between the trajectory of the center of the moving ball and the center of the stationary one.
Scattering Classical model As a model for the classical approach to collision, consider the case of a billiard ball colliding with a stationary one. The scattering direction quite clearly depends rather
More informationWaves, the Wave Equation, and Phase Velocity. We ll start with optics. The one-dimensional wave equation. What is a wave? Optional optics texts: f(x)
We ll start with optics Optional optics texts: Waves, the Wave Equation, and Phase Velocity What is a wave? f(x) f(x-) f(x-) f(x-3) Eugene Hecht, Optics, 4th ed. J.F. James, A Student's Guide to Fourier
More informationVI. Local Properties of Radiation
VI. Local Properties of Radiation Kirchhoff-Huygens Approximation Error for Shadowing by a Circular Window Relation to Fresnel Zone September 3 3 by H.L. Bertoni 1 Kirchhoff-Huygen Approximation y α dz
More informationkg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.
II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that
More informationSolutions to Laplace s Equation in Cylindrical Coordinates and Numerical solutions. ρ + (1/ρ) 2 V
Solutions to Laplace s Equation in Cylindrical Coordinates and Numerical solutions Lecture 8 1 Introduction Solutions to Laplace s equation can be obtained using separation of variables in Cartesian and
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 10
ECE 6345 Spring 215 Prof. David R. Jackson ECE Dept. Notes 1 1 Overview In this set of notes we derive the far-field pattern of a circular patch operating in the dominant TM 11 mode. We use the magnetic
More informationToday in Physics 218: Fresnel s equations
Today in Physics 8: Fresnel s equations Transmission and reflection with E parallel to the incidence plane The Fresnel equations Total internal reflection Polarization on reflection nterference R 08 06
More informationA family of closed form expressions for the scalar field of strongly focused
Scalar field of non-paraxial Gaussian beams Z. Ulanowski and I. K. Ludlow Department of Physical Sciences University of Hertfordshire Hatfield Herts AL1 9AB UK. A family of closed form expressions for
More informationDavid J. Starling Penn State Hazleton PHYS 214
All the fifty years of conscious brooding have brought me no closer to answer the question, What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself. -Albert
More informationPHY 5246: Theoretical Dynamics, Fall Assignment # 7, Solutions. Θ = π 2ψ, (1)
PHY 546: Theoretical Dynamics, Fall 05 Assignment # 7, Solutions Graded Problems Problem ψ ψ ψ Θ b (.a) The scattering angle satisfies the relation Θ π ψ, () where ψ is the angle between the direction
More informationIntroduction to Condensed Matter Physics
Introduction to Condensed Matter Physics Diffraction I Basic Physics M.P. Vaughan Diffraction Electromagnetic waves Geometric wavefront The Principle of Linear Superposition Diffraction regimes Single
More informationNondiffracting Waves in 2D and 3D
Nondiffracting Waves in 2D and 3D A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics from the College of William and Mary by Matthew Stephen
More informationPhys102 Lecture Diffraction of Light
Phys102 Lecture 31-33 Diffraction of Light Key Points Diffraction by a Single Slit Diffraction in the Double-Slit Experiment Limits of Resolution Diffraction Grating and Spectroscopy Polarization References
More informationSolution Set of Homework # 2. Friday, September 09, 2017
Temple University Department of Physics Quantum Mechanics II Physics 57 Fall Semester 17 Z. Meziani Quantum Mechanics Textboo Volume II Solution Set of Homewor # Friday, September 9, 17 Problem # 1 In
More information3 December Lesson 5.5
Preparation Assignments for Homework #8 Due at the start of class. Reading Assignments Please see the handouts for each lesson for the reading assignments. 3 December Lesson 5.5 A uniform plane wave is
More informationMAT 211 Final Exam. Spring Jennings. Show your work!
MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),
More information31. Diffraction: a few important illustrations
31. Diffraction: a few important illustrations Babinet s Principle Diffraction gratings X-ray diffraction: Bragg scattering and crystal structures A lens transforms a Fresnel diffraction problem into a
More informationWaves in Linear Optical Media
1/53 Waves in Linear Optical Media Sergey A. Ponomarenko Dalhousie University c 2009 S. A. Ponomarenko Outline Plane waves in free space. Polarization. Plane waves in linear lossy media. Dispersion relations
More information3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2
AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using
More informationComplex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,
Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )
More informationTHE WAVE EQUATION (5.1)
THE WAVE EQUATION 5.1. Solution to the wave equation in Cartesian coordinates Recall the Helmholtz equation for a scalar field U in rectangular coordinates U U r, ( r, ) r, 0, (5.1) Where is the wavenumber,
More informationChapter 10: QUANTUM SCATTERING
Chapter : QUANTUM SCATTERING Scattering is an extremely important tool to investigate particle structures and the interaction between the target particle and the scattering particle. For example, Rutherford
More informationProblem Set 5 Math 213, Fall 2016
Problem Set 5 Math 213, Fall 216 Directions: Name: Show all your work. You are welcome and encouraged to use Mathematica, or similar software, to check your answers and aid in your understanding of the
More information2.3 Oscillation. The harmonic oscillator equation is the differential equation. d 2 y dt 2 r y (r > 0). Its solutions have the form
2. Oscillation So far, we have used differential equations to describe functions that grow or decay over time. The next most common behavior for a function is to oscillate, meaning that it increases and
More informationPhysics I : Oscillations and Waves Prof. S. Bharadwaj Department of Physics and Meteorology Indian Institute of Technology, Kharagpur
Physics I : Oscillations and Waves Prof. S. Bharadwaj Department of Physics and Meteorology Indian Institute of Technology, Kharagpur Lecture - 21 Diffraction-II Good morning. In the last class, we had
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More information2.710 Optics Spring 09 Solutions to Problem Set #6 Due Wednesday, Apr. 15, 2009
MASSACHUSETTS INSTITUTE OF TECHNOLOGY.710 Optics Spring 09 Solutions to Problem Set #6 Due Wednesday, Apr. 15, 009 Problem 1: Grating with tilted plane wave illumination 1. a) In this problem, one dimensional
More informationComparative study of scattering by hard core and absorptive potential
6 Comparative study of scattering by hard core and absorptive potential Quantum scattering in three dimension by a hard sphere and complex potential are important in collision theory to study the nuclear
More informationMassachusetts Institute of Technology Physics 8.03 Practice Final Exam 3
Massachusetts Institute of Technology Physics 8.03 Practice Final Exam 3 Instructions Please write your solutions in the white booklets. We will not grade anything written on the exam copy. This exam is
More informationThe laser oscillator. Atoms and light. Fabry-Perot interferometer. Quiz
toms and light Introduction toms Semi-classical physics: Bohr atom Quantum-mechanics: H-atom Many-body physics: BEC, atom laser Light Optics: rays Electro-magnetic fields: Maxwell eq. s Quantized fields:
More informationThe laser oscillator. Atoms and light. Fabry-Perot interferometer. Quiz
toms and light Introduction toms Semi-classical physics: Bohr atom Quantum-mechanics: H-atom Many-body physics: BEC, atom laser Light Optics: rays Electro-magnetic fields: Maxwell eq. s Quantized fields:
More information1 Polarization of Light
J. Rothberg April, 014 Outline: Introduction to Quantum Mechanics 1 Polarization of Light 1.1 Classical Description Light polarized in the x direction has an electric field vector E = E 0 ˆx cos(kz ωt)
More informationChapter 2 Basic Optics
Chapter Basic Optics.1 Introduction In this chapter we will discuss the basic concepts associated with polarization, diffraction, and interference of a light wave. The concepts developed in this chapter
More informationAndy Buffler Department of Physics University of Cape Town
University of Cape Town Department of Physics PHY014F Vibrations and Waves Part 3 Travelling waves Boundary conditions Sound Interference and diffraction covering (more or less) French Chapters 7 & 8 Andy
More informationWAVES CP4 REVISION LECTURE ON. The wave equation. Traveling waves. Standing waves. Dispersion. Phase and group velocities.
CP4 REVISION LECTURE ON WAVES The wave equation. Traveling waves. Standing waves. Dispersion. Phase and group velocities. Boundary effects. Reflection and transmission of waves. !"#$%&''(%)*%+,-.%/%+,01%
More informationScattering. 1 Classical scattering of a charged particle (Rutherford Scattering)
Scattering 1 Classical scattering of a charged particle (Rutherford Scattering) Begin by considering radiation when charged particles collide. The classical scattering equation for this process is called
More informationPhysics 505 Homework No. 4 Solutions S4-1
Physics 505 Homework No 4 s S4- From Prelims, January 2, 2007 Electron with effective mass An electron is moving in one dimension in a potential V (x) = 0 for x > 0 and V (x) = V 0 > 0 for x < 0 The region
More informationWave Phenomena Physics 15c. Lecture 15 Reflection and Refraction
Wave Phenomena Physics 15c Lecture 15 Reflection and Refraction What We (OK, Brian) Did Last Time Discussed EM waves in vacuum and in matter Maxwell s equations Wave equation Plane waves E t = c E B t
More informationTHE PHYSICS OF WAVES CHAPTER 1. Problem 1.1 Show that Ψ(x, t) = (x vt) 2. is a traveling wave.
CHAPTER 1 THE PHYSICS OF WAVES Problem 1.1 Show that Ψ(x, t) = (x vt) is a traveling wave. Show thatψ(x, t) is a wave by substitutioninto Equation 1.1. Proceed as in Example 1.1. On line version uses Ψ(x,
More informationJackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 7.6 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: A plane wave of frequency ω is incident normally from vacuum on a semi-infinite slab of material
More informationKirchhoff, Fresnel, Fraunhofer, Born approximation and more
Kirchhoff, Fresnel, Fraunhofer, Born approximation and more Oberseminar, May 2008 Maxwell equations Or: X-ray wave fields X-rays are electromagnetic waves with wave length from 10 nm to 1 pm, i.e., 10
More informationA Statistical Kirchhoff Model for EM Scattering from Gaussian Rough Surface
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 187 A Statistical Kirchhoff Model for EM Scattering from Gaussian Rough Surface Yang Du 1, Tao Xu 1, Yingliang Luo 1,
More informationCMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation
CMPT 889: Lecture 2 Sinusoids, Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University September 26, 2005 1 Sinusoids Sinusoids
More informationIntroduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu.50 Introduction to Seismology Spring 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. .50 Introduction to Seismology
More informationIntroduction to Seismology
1.510 Introduction to Seismology Lecture 5 Feb., 005 1 Introduction At previous lectures, we derived the equation of motion (λ + µ) ( u(x, t)) µ ( u(x, t)) = ρ u(x, t) (1) t This equation of motion can
More informationin Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD
2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website: Light
More informationQuantum Physics 130A. April 1, 2006
Quantum Physics 130A April 1, 2006 2 1 HOMEWORK 1: Due Friday, Apr. 14 1. A polished silver plate is hit by beams of photons of known energy. It is measured that the maximum electron energy is 3.1 ± 0.11
More informationMechanics Physics 151
Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential
More informationPHYS 110B - HW #5 Fall 2005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #5 Fall 005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased [.] Imagine a prism made of lucite (n.5) whose cross-section is a
More informationIntroduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 12.510 Introduction
More informationMultipole Expansion for Radiation;Vector Spherical Harmonics
Multipole Expansion for Radiation;Vector Spherical Harmonics Michael Dine Department of Physics University of California, Santa Cruz February 2013 We seek a more systematic treatment of the multipole expansion
More informationOPSE FINAL EXAM Fall 2016 YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT.
CLOSED BOOK. Equation Sheet is provided. YOU MUST SHOW YOUR WORK. ANSWERS THAT ARE NOT JUSTIFIED WILL BE GIVEN ZERO CREDIT. ALL NUMERICAL ANSERS MUST HAVE UNITS INDICATED. (Except dimensionless units like
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More informationElectromagnetic Waves. Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition)
PH 222-3A Spring 2007 Electromagnetic Waves Lecture 22 Chapter 33 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1 Chapter 33 Electromagnetic Waves Today s information age is based almost
More informationThere is light at the end of the tunnel. -- proverb. The light at the end of the tunnel is just the light of an oncoming train. --R.
A vast time bubble has been projected into the future to the precise moment of the end of the universe. This is, of course, impossible. --D. Adams, The Hitchhiker s Guide to the Galaxy There is light at
More informationSpring /2/ pts 1 point per minute
Physics 519 MIDTERM Name: Spring 014 6//14 80 pts 1 point per minute Exam procedures. Please write your name above. Please sit away from other students. If you have a question about the exam, please ask.
More informationA Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets
A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets George A. Hagedorn Happy 60 th birthday, Mr. Fritz! Abstract. Although real, normalized Gaussian wave packets minimize the product
More informationSUPPLEMENTARY INFORMATION
doi: 1.138/nature5677 An experimental test of non-local realism Simon Gröblacher, 1, Tomasz Paterek, 3, 4 Rainer Kaltenbaek, 1 Časlav Brukner, 1, Marek Żukowski,3, 1 Markus Aspelmeyer, 1, and Anton Zeilinger
More informationFinal Exam - PHYS 611 Electromagnetic Theory. Mendes, Spring 2013, April
NAME: Final Exam - PHYS 611 Electromagnetic Theory Mendes, Spring 2013, April 24 2013 During the exam you can consult your textbooks (Melia, Jackson, Panofsky/Phillips, Griffiths), the print-outs of classnotes,
More informationChapter 5. Diffraction Part 2
EE 430.43.00 06. nd Semester Chapter 5. Diffraction Part 06. 0. 0. Changhee Lee School of Electrical and Computer Engineering Seoul National niv. chlee7@snu.ac.kr /7 Changhee Lee, SN, Korea 5.5 Fresnel
More informationCHEM-UA 127: Advanced General Chemistry I
1 CHEM-UA 127: Advanced General Chemistry I Notes for Lecture 11 Nowthatwehaveintroducedthebasicconceptsofquantummechanics, wecanstarttoapplythese conceptsto build up matter, starting from its most elementary
More informationHomework 3. 1 Coherent Control [22 pts.] 1.1 State vector vs Bloch vector [8 pts.]
Homework 3 Contact: jangi@ethz.ch Due date: December 5, 2014 Nano Optics, Fall Semester 2014 Photonics Laboratory, ETH Zürich www.photonics.ethz.ch 1 Coherent Control [22 pts.] In the first part of this
More informationDiffraction. S.M.Lea. Fall 1998
Diffraction.M.Lea Fall 1998 Diffraction occurs wen EM waves approac an aperture (or an obstacle) wit dimension d > λ. We sall refer to te region containing te source of te waves as region I and te region
More informationFourier Optics - Exam #1 Review
Fourier Optics - Exam #1 Review Ch. 2 2-D Linear Systems A. Fourier Transforms, theorems. - handout --> your note sheet B. Linear Systems C. Applications of above - sampled data and the DFT (supplement
More information- 1 - θ 1. n 1. θ 2. mirror. object. image
TEST 5 (PHY 50) 1. a) How will the ray indicated in the figure on the following page be reflected by the mirror? (Be accurate!) b) Explain the symbols in the thin lens equation. c) Recall the laws governing
More informationHigh-Resolution. Transmission. Electron Microscopy
Part 4 High-Resolution Transmission Electron Microscopy 186 Significance high-resolution transmission electron microscopy (HRTEM): resolve object details smaller than 1nm (10 9 m) image the interior of
More informationChapter 33. Electromagnetic Waves
Chapter 33 Electromagnetic Waves Today s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is own of
More informationPhysics 142 Wave Optics 1 Page 1. Wave Optics 1. For every complex problem there is one solution that is simple, neat, and wrong. H.L.
Physics 142 Wave Optics 1 Page 1 Wave Optics 1 For every complex problem there is one solution that is simple, neat, and wrong. H.L. Mencken Interference and diffraction of waves The essential characteristic
More informationFoundations of Scalar Diffraction Theory(advanced stuff for fun)
Foundations of Scalar Diffraction Theory(advanced stuff for fun The phenomenon known as diffraction plays a role of the utmost importance in the branches of physics and engineering that deal with wave
More informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in
More informationElectromagnetic Waves
Physics 8 Electromagnetic Waves Overview. The most remarkable conclusion of Maxwell s work on electromagnetism in the 860 s was that waves could exist in the fields themselves, traveling with the speed
More informationLecture 9: Introduction to Diffraction of Light
Lecture 9: Introduction to Diffraction of Light Lecture aims to explain: 1. Diffraction of waves in everyday life and applications 2. Interference of two one dimensional electromagnetic waves 3. Typical
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not
More informationWeek 7: Interference
Week 7: Interference Superposition: Till now we have mostly discusssed single waves. While discussing group velocity we did talk briefly about superposing more than one wave. We will now focus on superposition
More informationTransmission across potential wells and barriers
3 Transmission across potential wells and barriers The physics of transmission and tunneling of waves and particles across different media has wide applications. In geometrical optics, certain phenomenon
More informationLecture 16 February 25, 2016
MTH 262/CME 372: pplied Fourier nalysis and Winter 2016 Elements of Modern Signal Processing Lecture 16 February 25, 2016 Prof. Emmanuel Candes Scribe: Carlos. Sing-Long, Edited by E. Bates 1 Outline genda:
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationMassachusetts Institute of Technology Physics 8.03 Fall 2004 Final Exam Thursday, December 16, 2004
You have 3 hours Do all eight problems You may use calculators Massachusetts Institute of Technology Physics 8.03 Fall 004 Final Exam Thursday, December 16, 004 This is a closed-book exam; no notes are
More information27 Fraunhofer Diffraction
27 Fraunhofer Diffraction Contents 27. Fraunhofer approximation 27.2 Rectangular aperture Keywords: Fraunhofer diffraction, Obliquity factor. Ref: M. Born and E. Wolf: Principles of Optics; R.S. Longhurst:
More informationPHYS 408, Optics. Problem Set 1 - Spring Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015.
PHYS 408, Optics Problem Set 1 - Spring 2016 Posted: Fri, January 8, 2015 Due: Thu, January 21, 2015. 1. An electric field in vacuum has the wave equation, Let us consider the solution, 2 E 1 c 2 2 E =
More information6. LIGHT SCATTERING 6.1 The first Born approximation
6. LIGHT SCATTERING 6.1 The first Born approximation In many situations, light interacts with inhomogeneous systems, in which case the generic light-matter interaction process is referred to as scattering
More informationWave Phenomena Physics 15c. Lecture 11 Dispersion
Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed
More information