Accelerator Physics. G. A. Krafft, A. Bogacz, and H. Sayed Jefferson Lab Old Dominion University Lecture 9
|
|
- Δάφνη Μέλιοι
- 7 χρόνια πριν
- Προβολές:
Transcript
1 Acceleato Physics G. A. Kafft, A. Bogacz, and H. Sayed Jeffeson Lab Old Dominion Univesity Lectue 9 USPAS Acceleato Physics Jan. 11
2 Synchoton Radiation Acceleated paticles emit electomagnetic adiation. i Emission i fom vey high enegy paticles has unique popeties fo a adiation souce. As such adiation was fist obseved at one of the ealiest electon synchotons, adiation fom high enegy paticles (mainly electons is known geneically as synchoton adiation by the acceleato and HENP communities. The adiation is highly collimated in the beam diection Fom elativity ct ' γ ct γβ z x ' x y' y z' γβ ct + γ z USPAS Acceleato Physics Jan. 11
3 Loentz invaiance of wave phase implies k µ (ω/c,k x,k y,k z is a Loentz 4-vecto ω γω γβkc z k x kx k y ky k γβcω + γk z kx + ky k x + k y k z sin θ sin θ cosθ ω / c ω / c ω / c ω / c γω / c + γβ k γ 1 + β cos θ ω / c z z ( ( USPAS Acceleato Physics Jan. 11
4 θ sinθ sinθ γ β θ ( 1+ β cos Theefoe all adiation with θ' < π /, which is oughly y½ of the emission fo dipole emission fom a tansvese acceleation in the beam fame, is Loentz tansfomed into an angle less than 1/γ. Because of the stong Dopple shift of the photon enegy, highe fo θ, most of the enegy in the photons is within a cone of angula extent 1/γ aound the beam diection. USPAS Acceleato Physics Jan. 11
5 Lamo s Fomula Fo a paticle executing non-elativistic motion, the total powe emitted in electomagnetic adiation is (Lamo 1 q 1 e P ( t a p & 6πε c 6πε m c 3 3 Lienad s elativistic genealization: Note both de and dt ae the fouth component of elativistic 4-vectos when one is dealing with photon emission. Theefoe, thei atio must be an Loentz invaiant. The invaiant that educes to Lamo s fomula in the non-elativistic limit is P µ e du duµ 6πε c dτ dτ USPAS Acceleato Physics Jan. 11
6 ( P t e 6 c γ & πε β β & β 6 Fo acceleation along a line, second tem is zeo and fist tem fo the adiation eaction is small compaed to the acceleation as long as gadient less than 1 14 MV/m. Technically impossible. β Fo tansvese bend acceleation & β c ˆ ρ ec 4 4 ( β γ P t 6πε ρ USPAS Acceleato Physics Jan. 11
7 Factional Enegy Loss δe e Θ β γ 6πε ρ 3 4 Fo one tun with isomagnetic bending fields δ E 4π e 3ρ E beam β γ 3 3 e is the classical electon adius: cm USPAS Acceleato Physics Jan. 11
8 Radiation Powe Distibution Consulting you favoite Classical E&M text (Jackson, Schwinge, Landau and Lifshitz Classical Theoy of Fields dp d 3 e ω γ 8 ω π ε ρ ωc ω / ω c K 5/3 ( xdx USPAS Acceleato Physics Jan. 11
9 Citical Fequency Citical (angula fequency is 3 3 c ω c γ ρ Enegy scaling of citical fequency is undestood fom 1/γ emission cone and fact that 1 β ~ 1/( γ t A t B ρ ρ ρ t t B γ c γ c γ c ρ 3 3 γβ c A B ρ ρ ρ + 3 γβc γc γ c 1/γ USPAS Acceleato Physics Jan. 11
10 Photon Numbe dp 3 e e c c 5/3 ( dω 8π ε ρ 6πε ξ ρ P dω ωγ ξ K x dxdξ γ dn& 1 dp dω hω dω 4 hω dn& hω d ω dω dn& d ω dω hω c n& 5α c 5α e 1 γ δn γ α 3 ρ 3 Θ 4πε hc 137 USPAS Acceleato Physics Jan. 11
11 Insetion Devices Often peiodic magnetic field magnets ae placed in beam path of high enegy stoage ings. The adiation geneated by electons passing though such insetion devices has unique popeties. Field of the insetion device magnet B x y z B z y B z B z ( ( ˆ ( ( π λ,, cos / ID Vecto potential fo magnet (1 dimensional appoximation B λ,, ˆ ID A x y z A z x A z sin π z/ λ π ( ( ( ( ID USPAS Acceleato Physics Jan. 11
12 Electon Obit Unifomity in x-diection means that canonical momentum in the x-diection is conseved ( ea z K vx ( z csin z/ γm γ ( π λ ID v 1 K λ x z dz z v β γ π x ID ( cos( π / λ Field Stength Paamete z z ID K eb λ ID π mc USPAS Acceleato Physics Jan. 11
13 Aveage Velocity 1 Enegy consevation gives that γ is a constant of the motion ( ( z z x z 1 1 β γ β Aveage longitudinal velocity in the insetion device is 1 1 γ γ β β K z Aveage est fame has 1 γ γ / K β γ USPAS Acceleato Physics Jan. 11
14 Relativistic Kinematics In aveage est fame the insetion device is Loentz contacted, and so its wavelength is λ λ ID / β γ The sinusoidal wiggling motion emits with angula fequency ω πc / λ Loentz tansfomation fomulas fo the wave vecto k k k k x y z γ k k k x y γ k ( 1 β cosθ k sinθ cosϕ k sinθ sinϕ ( cosθ β USPAS Acceleato Physics Jan. 11
15 Insetion Device (FEL Resonance Angle tansfoms as Condition ( cosθ β ( 1 β cosθθ k cos θ z k Wave vecto in lab fame has k γ k 1 β cosθ πβ c 1 β cosθ ( λ ( ID In the fowad diection cos θ 1 λ ID λid ( K λ e 1 / γ γ + USPAS Acceleato Physics Jan. 11
16 Powe Emitted Lab Fame Lamo/Lienad calculation in the lab fame yields P e 4 K π γ β c πε γ λid 1 6 Total enegy adiated afte one passage of the insetion device e δ E π γ β NK 6πε λid USPAS Acceleato Physics Jan. 11
17 Powe Emitted Beam Fame Lamo/Lienad calculation in the beam fame yields P e π 1 ck 6πε λ Total enegy of each photon is ħπc/λ, theefoe the total numbe of photons adiated afte one passage of the insetion device π N π γ αnk 3 USPAS Acceleato Physics Jan. 11
18 Spectal Distibution in Beam Fame Begin with powe distibution in beam fame: dipole adiation patten (single hamonic only when K<<1; eplace γ by γ, β by β dp dω e c k a sin Θ 3πε 4 Numbe distibution in tems of wave numbe Solid angle tansfomation dnγ α k + k dω 4 k d y z NK Ω dω ( γ 1 βcosθ USPAS Acceleato Physics Jan. 11
19 Numbe distibution in beam fame Enegy is simply E ( 4 ( dn γ α sin θsin ϕ+ γ cosθ β NK 4 dω 4 γ 1 βcosθ ( θ πβc h 1 λ ID β θ β θ Eˆ ( θ ( 1 cos ( 1 cos Numbe distibution as a function of nomalized lab-fame enegy dn ˆ γ απ E NK 1 β ˆ 3 + de 4γ β γ USPAS Acceleato Physics Jan. 11
20 Limits of integation Aveage Enegy ˆ 1 ˆ 1 cos θ 1 E cos θ 1 E 1 β 1+ β Aveage enegy is also analytically ll calculable l E dnγ ˆ E de deˆ γ h πβc/ λid dnγ de ˆ de ˆ E max USPAS Acceleato Physics Jan. 11
21 Conventions on Fouie Tansfoms Fo the time dimensions i t f% ω f t e dt ( ω ( 1 iω t f ( t f % ( ω e dω π Results on Diac delta functions % ( iω t ( te dt 1 δ ω δ δ ( 1 iω t π t e d ω USPAS Acceleato Physics Jan. 11
22 Fo the thee spatial dimensions δ f% k f x e d x ( ( ik x 3 ( f x f% k e d k 1 3 ( π ( π ( + ik x 3 x x e d k ik x 3 ( δ ( 3 USPAS Acceleato Physics Jan. 11
23 Geen Function fo Wave Equation Solution to inhomogeneous wave equation x y z c t G( x, t; x, t 4πδ x x δ t t ( ( Will pick out the solution with causal bounday conditions G x, t; x, t t < t (, ;, This choice leads automatically to the so-called Retaded Geen Function USPAS Acceleato Physics Jan. 11
24 In geneal G ( x, t ; x, t t < t G x, t; x, t ( ( i( k x t ( i( k x t 3 A k e ω ω B k e + + d k t > t because thee ae two possible signs of the fequency fo each value of the wave vecto. To solve the homogeneous wave equation it is necessay that ω k k c ( i.e., thee is no dispesion in fee space. USPAS Acceleato Physics Jan. 11
25 Continuity of G implies ω A k e B k e ( ( i t iω t t t + ε Integate the inhomogeneous equation between and t t ε 1 G( x, t; x, t 4πδ ( x x c t t + ε ( i( k x ωt ( i( k x+ ωt 3 iω A k e + iωb k e d k 4π c δ x x c A( k e e ω π iω ( ik x i t ( USPAS Acceleato Physics Jan. 11
26 G ( x, t; x, t c ( π i ik x x c e iω ( t t c e dk π x x π δ ( x x / c t + t + x x 1 i( k ( x x ω( t t i( k ( x x + ω( t t 3 e e d k ω t > t e x ik x x x ( > + iω t t e dk t t Called etaded because the influence at time t is due to the souce evaluated at the etaded time t t x x / c USPAS Acceleato Physics Jan. 11
27 Retaded Solutions fo Fields 1 ρ + + φ z x y c t ε A µ J x y z c t 1 3 ρ ( x, t φ ( x, t d x dt δ x x / c t + t 4πε x x µ 3 J ( x, t A ( x, t d x dt δ ( x x / c t + t 4π x x ( Tip: Leave the delta function in it s integal fom to do deivations. Don t have to emembe complicated delta-function ules USPAS Acceleato Physics Jan. 11
28 φ (, Retaded Solutions fo Fields 1 3 x t d x dt dω e 8π ε x x (, 3 A x t d x dt d e ρ µ J ω 8 π x x ( x, t iω x x / c ( t t ( x, t iω x x / c ( t t Evaluation can be expedited by noting and using the symmety of the Geen function and using elations such as x f ( t t f ( t t t t f x x f x x x ( ( USPAS Acceleato Physics Jan. 11
29 φ (, Retaded Solutions fo Fields 1 3 x t d x dt dω e 8π ε x x (, 3 A x t d x dt dω e ρ µ J 8 π x x ( x, t iω x x / c ( t t ( x, t iω x x / c ( t t Evaluation can be expedited by noting and using the symmety of the Geen function and using elations such as x f ( t t f ( t t t t f x x f x x x ( ( USPAS Acceleato Physics Jan. 11
30 Radiation Fom Relativistic Electons Fom discussion ealy in the couse, in the Loenz gauge the equation fo the potentials is 1 ρ + + φ x y z c t ε A µ J x y z c t The solution, using the etaded Geen Function is 1 3 ρ ( x, t φ ( x, t d x dt δ ( x x / c t + t 4πε x x µ 3 J ( x, t A ( x, t d x dt δ ( x x / c t + t 4π x x USPAS Acceleato Physics Jan. 11
31 φ (, Delta Function Repesentation 1 ( x, t i x x / c ( t t 3 x t d x dt d e ω ω 8π ε x x µ 3 J ( x, t i x x / c ( t t A x t d x dt dω e ω (, 8π x x 3 3 ρ x, t qδ x t J x, t qv t δ x t ( ( ρ ( ( ( ( ( q 1 φ( x, t dt dω e 8π ε x qµ v A( x, t dt dω e 8π x ( t ( t ( t iω x t c t t ( / ( iω x t c t t ( / ( USPAS Acceleato Physics Jan. 11
32 Lienad-Weichet Potentials φ ( x, t A x t φ (, ( x, t A x t (, q 4πε qµ 4π dt δ ( / ( x ( t ( x t c t t v / dt ( t δ x ( t c ( t t x ( t ( q 1 4πε ( ( ( x t 1 nˆ β t qµ v ( t 4π x ( t ( 1 nˆ β ( t et et USPAS Acceleato Physics Jan. 11
33 φ (, π ε EM Field Radiated q 1 x t dt dω e 8 x ( t v ( t x ( t iω x t c t t ( / ( qµ iω x ( t / c ( t t A( x, t dt dω e 8 π x t A E φ B A t ˆ nˆ & q n β q E + 4πε ( 3 γ 1 nˆ β R 4πε c ( 1 nˆ 3 β R et B nˆ E / c {( nˆ β β} et USPAS Acceleato Physics Jan. 11
34 1 nˆ x t nˆ ( x ( t x ( t 1 ˆ β ˆ d / dt + nˆ( nˆ d / dt x ( t x ( t dt x ( t d n c dn dt q nˆ 1 i x t / c x t dt dω e ω (, φ 8π ε ω ( x ( t ( i x ( t / c ( t t qµ v i x ( t / c ( t t A x t dt d e ω (, ω t 8π x ( t iω ( t d ω ( ( e i ω 1 β t n ˆ t e dt i x t / c t t i ω x ( t / c ( t t ( ( ( L USPAS Acceleato Physics Jan. 11
35 q i x ( t / c ( t t nˆ i n ω ω β E( x, t dt dω e + 8π ε integate by pats to get final esult iω x t c t t ( / ( q e E( x, t dt dω vel 8π ε ˆ ( 1 β n x ( t ( ˆ x ( t cx ( t 1 nˆ ( nˆ β + β + β nˆ β nˆ + β nˆ β nˆ nˆ β + β n ˆ ( β 1 β nˆ + β nˆ ( β nˆ β + ( β nˆ ( ( ( USPAS Acceleato Physics Jan. 11
36 i x ( t / c ( t t ω q e (, ω acc ( 8π ε c 1 β nˆ x ( t E x t dt d ( & nˆ β nˆ & ˆ ˆ ( ( & β 1 β n β β n ( / ( iω x t c t t q e ˆ & 8 c dt dω n n β β π ε 1 β n ˆ x t ( ( { ( ˆ } USPAS Acceleato Physics Jan. 11
37 Lamo s Fomula Fo small velocities can neglect etadation q E( x, t ˆ { n nˆ & β } / R acc 4πε c dp dω P q nˆ 16π ε µ c 3 q 16π ε c q 6πε c 3 3 v & v & sin & { } nˆ β θ USPAS Acceleato Physics Jan. 11
38 Relativistic Peaking In fa field afte shot acceleation nˆ nˆ & β ( ( dp t q dω 16π ε c 1 nˆ β dp t θ ( dω max 1 γ { β } ( 5 q & β sin θ 16π ε c 1 β cosθ Fo cicula motions ( ( & sin cos 1 Ω 16π ε c ( 1 β cosθ γ ( 1 β cosθ dp t q β θ ϕ d 5 3 USPAS Acceleato Physics Jan. 11
39 Spectum Radiated by Motion de dp 1 dt E H nr ( ˆ dt E E R dt dω dω cµ { & ( } {( &} ˆ ˆ ˆ ˆ 1 q n n β β n n β β ( t ( t ( ( cµ 8π ε c 1 nˆ β 1 nˆ β e iω R 1 n ˆ ( t / R+ ( n ˆ ( t / R / c t+ t iω 1 n ˆ ( t ( ( / R+ nˆ t / R / c t+ t cleaing the unpimed time integal and omega pime e dt dωdt dω dt integal with delta epesntation {( & } ( ˆ ˆ ˆ ˆ & π q n n β β n n β β ( t c ( µ 8π ε c 1 n ˆ β 1 n ( ˆ β iω nˆ ( t / c t+ t iω nˆ ( t / c t+ t e e dt dt dω { } t ( USPAS Acceleato Physics Jan. 11
40 ˆ d E q n 3 d ω dω 3 π ε c 1 nˆ β {( nˆ β β} ( 1 & e iω nˆ ( t / c t+ t dt d E q ω iω t nˆ ( t / c ˆ ( ˆ n n β e dt 3 dωdω 3π ε c Facto of two diffeence fom Jackson because he combines positive fequency and negative fequency contibutions in one positive fequency integal. I don't like because Paseval's fomula, etc. don't wok! I've witten papes about pefoming this calculation in new egimes of high intensity pulsed lases. USPAS Acceleato Physics Jan. 11
Oscillating dipole system Suppose we have two small spheres separated by a distance s. The charge on one sphere changes with time and is described by
5 Radiation (Chapte 11) 5.1 Electic dipole adiation Oscillating dipole system Suppose we have two small sphees sepaated by a distance s. The chage on one sphee changes with time and is descibed by q(t)
Space Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines
Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the
Analytical Expression for Hessian
Analytical Expession fo Hessian We deive the expession of Hessian fo a binay potential the coesponding expessions wee deived in [] fo a multibody potential. In what follows, we use the convention that
Accelerator Physics Synchrotron Radiation. A. Bogacz, G. A. Krafft, and T. Zolkin Jefferson Lab Colorado State University Lecture 8
Acclato Physics Synchoton Radiation A. Bogacz, G. A. Kafft, and T. Zolkin Jffson Lab Coloado Stat Univsity Lctu 8 USPAS Acclato Physics Jun 13 Synchoton Radiation Acclatd paticls mit lctomagntic adiation.
Physics 401 Final Exam Cheat Sheet, 17 April t = 0 = 1 c 2 ε 0. = 4π 10 7 c = SI (mks) units. = SI (mks) units H + M
Maxwell' s Equations in vauum E ρ ε Physis 4 Final Exam Cheat Sheet, 7 Apil E B t B Loent Foe Law: F q E + v B B µ J + µ ε E t Consevation of hage: J + ρ t µ ε ε 8.85 µ 4π 7 3. 8 SI ms) units q eleton.6
Matrix Hartree-Fock Equations for a Closed Shell System
atix Hatee-Fock Equations fo a Closed Shell System A single deteminant wavefunction fo a system containing an even numbe of electon N) consists of N/ spatial obitals, each occupied with an α & β spin has
ANTENNAS and WAVE PROPAGATION. Solution Manual
ANTENNAS and WAVE PROPAGATION Solution Manual A.R. Haish and M. Sachidananda Depatment of Electical Engineeing Indian Institute of Technolog Kanpu Kanpu - 208 06, India OXFORD UNIVERSITY PRESS 2 Contents
Example 1: THE ELECTRIC DIPOLE
Example 1: THE ELECTRIC DIPOLE 1 The Electic Dipole: z + P + θ d _ Φ = Q 4πε + Q = Q 4πε 4πε 1 + 1 2 The Electic Dipole: d + _ z + Law of Cosines: θ A B α C A 2 = B 2 + C 2 2ABcosα P ± = 2 ( + d ) 2 2
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
e t e r Cylindrical and Spherical Coordinate Representation of grad, div, curl and 2
Cylindical and Spheical Coodinate Repesentation of gad, div, cul and 2 Thus fa, we have descibed an abitay vecto in F as a linea combination of i, j and k, which ae unit vectos in the diection of inceasin,
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Tutorial Note - Week 09 - Solution
Tutoial Note - Week 9 - Solution ouble Integals in Pola Coodinates. a Since + and + 5 ae cicles centeed at oigin with adius and 5, then {,θ 5, θ π } Figue. f, f cos θ, sin θ cos θ sin θ sin θ da 5 69 5
dx x ψ, we should find a similar expression for rθφ L ψ. From L = R P and our knowledge of momentum operators, it follows that + e y z d
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Coveed: Obital angula momentum, cente-of-mass coodinates Some Key Concepts: angula degees of feedom, spheical hamonics 1. [20 pts] In
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Laplace s Equation in Spherical Polar Coördinates
Laplace s Equation in Spheical Pola Coödinates C. W. David Dated: Januay 3, 001 We stat with the pimitive definitions I. x = sin θ cos φ y = sin θ sin φ z = cos θ thei inveses = x y z θ = cos 1 z = z cos1
ECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
4.2 Differential Equations in Polar Coordinates
Section 4. 4. Diffeential qations in Pola Coodinates Hee the two-dimensional Catesian elations of Chapte ae e-cast in pola coodinates. 4.. qilibim eqations in Pola Coodinates One wa of epesg the eqations
Curvilinear Systems of Coordinates
A Cuvilinea Systems of Coodinates A.1 Geneal Fomulas Given a nonlinea tansfomation between Catesian coodinates x i, i 1,..., 3 and geneal cuvilinea coodinates u j, j 1,..., 3, x i x i (u j ), we intoduce
1 3D Helmholtz Equation
Deivation of the Geen s Funtions fo the Helmholtz and Wave Equations Alexande Miles Witten: Deembe 19th, 211 Last Edited: Deembe 19, 211 1 3D Helmholtz Equation A Geen s Funtion fo the 3D Helmholtz equation
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0
TRIGONOMETRIC IDENTITIES (a,b) Let s eview the geneal definitions of tig functions fist. (See back cove of you book) θ b/ θ a/ tan θ b/a, a 0 θ csc θ /b, b 0 sec θ /a, a 0 cot θ a/b, b 0 By doing some
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
VEKTORANALYS. CURVILINEAR COORDINATES (kroklinjiga koordinatsytem) Kursvecka 4. Kapitel 10 Sidor
VEKTORANALYS Kusvecka 4 CURVILINEAR COORDINATES (koklinjiga koodinatstem) Kapitel 10 Sido 99-11 TARGET PROBLEM An athlete is otating a hamme Calculate the foce on the ams. F ams F F ma dv a v dt d v dt
Derivation of Optical-Bloch Equations
Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be
Second Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
21. Stresses Around a Hole (I) 21. Stresses Around a Hole (I) I Main Topics
I Main Topics A Intoducon to stess fields and stess concentaons B An axisymmetic poblem B Stesses in a pola (cylindical) efeence fame C quaons of equilibium D Soluon of bounday value poblem fo a pessuized
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
[1] P Q. Fig. 3.1
1 (a) Define resistance....... [1] (b) The smallest conductor within a computer processing chip can be represented as a rectangular block that is one atom high, four atoms wide and twenty atoms long. One
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Theoretical Competition: 12 July 2011 Question 1 Page 1 of 2
Theoetical Competition: July Question Page of. Ένα πρόβλημα τριών σωμάτων και το LISA μ M O m EIKONA Ομοεπίπεδες τροχιές των τριών σωμάτων. Δύο μάζες Μ και m κινούνται σε κυκλικές τροχιές με ακτίνες και,
Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Srednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Jackson 2.25 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 2.25 Hoework Proble Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: Two conducting planes at zero potential eet along the z axis, aking an angle β between the, as
Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Written Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Synchrotron Radiation. G. Wang
Synhoton Radiation G. Wang What is synhoton adiation Stati field fo a hage at est When a patile moves with a onstant veloity, field moves with patile. When a patile gets aeleated, some pat of the field
Chapter 7a. Elements of Elasticity, Thermal Stresses
Chapte 7a lements of lasticit, Themal Stesses Mechanics of mateials method: 1. Defomation; guesswok, intuition, smmet, pio knowledge, epeiment, etc.. Stain; eact o appoimate solution fom defomation. Stess;
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Problems in curvilinear coordinates
Poblems in cuvilinea coodinates Lectue Notes by D K M Udayanandan Cylindical coodinates. Show that ˆ φ ˆφ, ˆφ φ ˆ and that all othe fist deivatives of the cicula cylindical unit vectos with espect to the
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Lifting Entry 2. Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYLAND U N I V E R S I T Y O F
ifting Entry Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion MARYAN 1 010 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu ifting Atmospheric
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Lifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Forced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
The Laplacian in Spherical Polar Coordinates
Univesity of Connecticut DigitalCommons@UConn Chemisty Education Mateials Depatment of Chemisty -6-007 The Laplacian in Spheical Pola Coodinates Cal W. David Univesity of Connecticut, Cal.David@uconn.edu
CHAPTER (2) Electric Charges, Electric Charge Densities and Electric Field Intensity
CHAPTE () Electric Chrges, Electric Chrge Densities nd Electric Field Intensity Chrge Configurtion ) Point Chrge: The concept of the point chrge is used when the dimensions of n electric chrge distriution
r = x 2 + y 2 and h = z y = r sin sin ϕ
Homewok 4. Solutions Calculate the Chistoffel symbols of the canonical flat connection in E 3 in a cylindical coodinates x cos ϕ, y sin ϕ, z h, b spheical coodinates. Fo the case of sphee ty to make calculations
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
wave energy Superposition of linear plane progressive waves Marine Hydrodynamics Lecture Oblique Plane Waves:
3.0 Marine Hydrodynamics, Fall 004 Lecture 0 Copyriht c 004 MIT - Department of Ocean Enineerin, All rihts reserved. 3.0 - Marine Hydrodynamics Lecture 0 Free-surface waves: wave enery linear superposition,
Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Graded Refractive-Index
Graded Refractive-Index Common Devices Methodologies for Graded Refractive Index Methodologies: Ray Optics WKB Multilayer Modelling Solution requires: some knowledge of index profile n 2 x Ray Optics for
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)
Q1. (a) A fluorescent tube is filled with mercury vapour at low pressure. In order to emit electromagnetic radiation the mercury atoms must first be excited. (i) What is meant by an excited atom? (1) (ii)
PARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
ST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Solutions - Chapter 4
Solutions - Chapter Kevin S. Huang Problem.1 Unitary: Ût = 1 ī hĥt Û tût = 1 Neglect t term: 1 + hĥ ī t 1 īhĥt = 1 + hĥ ī t ī hĥt = 1 Ĥ = Ĥ Problem. Ût = lim 1 ī ] n hĥ1t 1 ī ] hĥt... 1 ī ] hĥnt 1 ī ]
Problem 7.19 Ignoring reflection at the air soil boundary, if the amplitude of a 3-GHz incident wave is 10 V/m at the surface of a wet soil medium, at what depth will it be down to 1 mv/m? Wet soil is
Physics 505 Fall 2005 Practice Midterm Solutions. The midterm will be a 120 minute open book, open notes exam. Do all three problems.
Physics 55 Fll 25 Pctice Midtem Solutions The midtem will e 2 minute open ook, open notes exm. Do ll thee polems.. A two-dimensionl polem is defined y semi-cicul wedge with φ nd ρ. Fo the Diichlet polem,
Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
the total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Strain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Statistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Parametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΦΥΣΙΚΗΣ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΧΑΟΤΙΚΕΣ ΚΙΝΗΣΕΙΣ ΓΥΡΩ ΑΠΟ ΜΑΥΡΕΣ ΤΡΥΠΕΣ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΦΥΣΙΚΗΣ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΧΑΟΤΙΚΕΣ ΚΙΝΗΣΕΙΣ ΓΥΡΩ ΑΠΟ ΜΑΥΡΕΣ ΤΡΥΠΕΣ Γιουνανλής Παναγιώτης Επιβλέπων: Γ.Βουγιατζής Επίκουρος Καθηγητής
Answer sheet: Third Midterm for Math 2339
Answer sheet: Third Midterm for Math 339 November 3, Problem. Calculate the iterated integrals (Simplify as much as possible) (a) e sin(x) dydx y e sin(x) dydx y sin(x) ln y ( cos(x)) ye y dx sin(x)(lne
6.4 Superposition of Linear Plane Progressive Waves
.0 - Marine Hydrodynamics, Spring 005 Lecture.0 - Marine Hydrodynamics Lecture 6.4 Superposition of Linear Plane Progressive Waves. Oblique Plane Waves z v k k k z v k = ( k, k z ) θ (Looking up the y-ais
Integrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Finite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
3.7 Governing Equations and Boundary Conditions for P-Flow
.0 - Maine Hydodynaics, Sping 005 Lectue 10.0 - Maine Hydodynaics Lectue 10 3.7 Govening Equations and Bounday Conditions fo P-Flow 3.7.1 Govening Equations fo P-Flow (a Continuity φ = 0 ( 1 (b Benoulli
Assalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
Inflation and Reheating in Spontaneously Generated Gravity
Univesità di Bologna Inflation and Reheating in Spontaneously Geneated Gavity (A. Ceioni, F. Finelli, A. Tonconi, G. Ventui) Phys.Rev.D81:123505,2010 Motivations Inflation (FTV Phys.Lett.B681:383-386,2009)
ECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
Durbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님
상대론적고에너지중이온충돌에서 제트입자와관련된제동복사 박가영 인하대학교 윤진희교수님, 권민정교수님 Motivation Bremsstrahlung is a major rocess losing energies while jet articles get through the medium. BUT it should be quite different from low energy
STEADY, INVISCID ( potential flow, irrotational) INCOMPRESSIBLE + V Φ + i x. Ψ y = Φ. and. Ψ x
STEADY, INVISCID ( potential flow, iotational) INCOMPRESSIBLE constant Benolli's eqation along a steamline, EQATION MOMENTM constant is a steamline the Steam Fnction is sbsititing into the continit eqation,
CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Solutions Ph 236a Week 2
Solutions Ph 236a Week 2 Page 1 of 13 Solutions Ph 236a Week 2 Kevin Bakett, Jonas Lippune, and Mak Scheel Octobe 6, 2015 Contents Poblem 1................................... 2 Pat (a...................................
If we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο