PHZ 3113 Fall 2017 Homework #5, Due Friday, October 13
|
|
- Darleen Williamson
- 5 years ago
- Views:
Transcription
1 PHZ 3113 Fall 2017 Homewok #5, Due Fiday, Octobe Genealize the poduct ule (fg) = f g +f g to wite the divegence Ö (Ù Ú) of the coss poduct of the vecto fields Ù and Ú in tems of the cul of Ù and the cul of Ú. Use this to compute Ö (Öφ Öψ), fo scala fields φ and ψ. This can be done seveal ways. One is to ague as follows: The deivative opeato Ö acts on both Ù and Ú; the poduct ule applies in some fom. The outcome must be a scala, and so cannot contain Ö Ù o Ö Ú, since thee is no way to combine eithe with the emaining vecto to poduce a scala. Thus, the esult must contain Ö Ù and Ö Ú, and to fom a scala must contain Ú (Ö Ù) and Ù (Ö Ú). ompaed to the oiginal, in the fist tem the thee vectos Ö, Ù, and Ú emain in cyclic ode, and so it appeas with a + sign; but in the second tem the cyclic ode is evesed, and so it appeas with a sign: Ö (Ù Ú) = Ú (Ö Ù) Ù (Ö Ú). O, you can wite out all the components (u x,y = u x / y, etc.), in exhaustive detail, Ö (Ù Ú) = u y v z u z v y + u y z v x u x v z + u z x v y u y v x = u y,x v z +u y v z,x u z,x v y u z v y,x +u z,y v x +u z v x,y u x,y v z u x v z,y +u x,z v y +u x v y,z u y,z v x u y v x,z = v x (u z,y u y,z )+v y (u x,z u z,x )+v z (u y,x u x,y ) u x (v z,y v y,z ) u y (v x,z v z,x ) u z (v y,x vx,y) = Ú (Ö Ù) Ù (Ö Ú). O, you can use the ǫ ijk notation: (Ù Ú) i = ǫ ijk u j v k, and Ö (Ù Ú) = Ö i (ǫ ijk u j v k ) = ǫ ijk Ö i (u j v k ) = ǫ ijk [(Ö i u j )v k +u j (Ö i v k )] Fo Ù = Öφ and Ú = Öψ, = v k (ǫ ijk Ö i u j )+u j (ǫ ijk Ö i v k ) = v k (+ǫ kij Ö i u j )+u j ( ǫ jik Ö i v k ) = v k (Ö Ù) k u j (Ö Ú) j = Ú (Ö Ù) Ù (Ö Ú). Ö (Öφ Öψ) = Öψ (Ö Öφ) Öφ (Ö Öψ) = 0, since the cul of a gadient always vanishes fo easons discussed peviously. This also follows fom ǫ ijk Ö k (Ö j φ Ö k ψ).
2 2. Letthevectofield Ú be Ú = y(y2 +z 2 ) 3 ˆÜ+ x(x2 +z 2 ) 3 ˆÝ xyz 3 ˆÞ,whee = Ô x 2 +y 2 +z 2. (a) ompute Ö Ú. ompute Ö Ú. All calculations can make good use of the elation 1 p = and similaly fo y and z. 1 (x 2 +y 2 +z 2 ) p/2 = p 2 2x (x 2 +y 2 +z 2 ) (p+2)/2 = px p+2, The deivatives that appea in the divegence ae = 3xy(y 2 +z 2 ) 5 y = 3xy(x 2 +z 2 ) 5 z = xy(x 2 +y 2 2z 2 ) and the divegence is Ö Ú = + y + z = 4xy 3. The deivatives that appea in the cul ae y = 3x2 y 2 +x 2 z 2 +y 2 z 2 +z 4 = 3x2 y 2 +x 2 z 2 +y 2 z 2 +z 4 = yz(2x2 y 2 z 2 ) z = yz(2x2 y 2 z 2 ) z = xz(2y2 x 2 z 2 ) y = xz(2y2 x 2 z 2 ) and Ö Ú = ˆÜ vz y v y vx + ˆÝ z z v z vy + ˆÞ v x = 0. y
3 (b) ompute Ö(Ö Ú) and Ö 2 Ú. ompute Ö (Ö Ú) two diffeent ways. Again using ( /)(1/ 3 ) = 3x/ etc., it is staightfowad to obtain Ö(Ö Ú) = 4y(3x2 2 ) 5 ˆÜ+ 4x(3y2 2 ) 5 ˆÝ + 12xyz 5 ˆÞ. The Laplacian of the x-component is 2 = 3y(y2 +z 2 )( 2 5x 2 ) 7, y 2 = 3y(2x4 3x 2 y 2 +x 2 z 2 y 2 z 2 z 4 ) 7, z 2 = y(2x4 +x 2 y 2 y 4 11x 2 z 2 +y 2 z 2 +2z 4 ) 7, Ö 2 v x = 4y(2 3x 2 ) 5. The y-component follows fom exchanging x and y, and This gives the familia-looking Ö 2 v z = 12xyz 5. Ö 2 Ú == 4y(3x2 2 ) 5 ˆÜ+ 4x(3y2 2 ) 5 ˆÝ + 12xyz 5 ˆÞ. The cul of the cul diectly is easy, which is the same as Ö(Ö Ú) Ö 2 Ú. Ö (Ö Ú) = Ö (0) = 0. If you think of it, Ö Ú = 0 means that Ú = Öφ fo some φ; and fom v z it is appaent that this potential function is Then, φ = xy. Ö 2 Ú = Ö 2 (Öφ) = Ö(Ö 2 φ) = Ö 4xy 3.
4 (x y) 3. Let Ú(x,y) be the vecto field Ú = ˆÜ+ (x+y) ˆÝ, whee = Ô x 2 +y 2. Let be the cicumfeence of the cicle of adius = 1 in the x-y plane centeed at the oigin. (a) ompute Ú ˆÒds two diffeent ways. To compute integals, we need to wite eveything in tems of some one thing that we can then integate ove. Let a point on the cicle be specified by pola angle θ, x = cosθ, y = sinθ. Then, position and its deivative ae Ö = xˆü+y ˆÝ = cosθ ˆÜ+sinθ ˆÝ, dö = ( sinθ ˆÜ+cosθ ˆÝ)dθ, ds = dö = dθ; and the unit vecto pependicula to the cicle ( unit nomal ) is ˆÒ = cosθ ˆÜ+sinθˆÝ. The diect integal ove θ then gives Ú ˆÒds = [(cosθ sinθ)ˆü+(cosθ+sinθ)ˆý] (cosθ ˆÜ+sinθˆÝ) dθ = cos 2 θ sinθcosθ+cosθsinθ+sin 2 2π θ dθ = dθ = 2π. 0 The othe way is using the two-dimensional divegence theoem, Ú ˆÒds = (Ö Ú)d 2 a. A The divegence is Ö Ú = (x y) + y (x+y) = 1 x(x y) y(x+y) 3 = 1, and the aea integal is also 1 d 2 a(ö Ú) = ddφ 1 A 0 = 2π
5 (b) ompute Ú dö two diffeent ways. The θ-integal is Ú dö = = [(cosθ sinθ)ˆü+(cosθ+sinθ)ˆý] ( sinθ ˆÜ+cosθˆÝ)dθ [ cosθsinθ+sin 2 θ+cos 2 θ+sinθcosθ]dθ = 2π. By Stoke s theoem, this is also Ú dö = (Ö Ú) ˆÒd 2 a. A The cul is leading to the integal Ö Ú = vy v x ˆÞ = 1 y ˆÞ 1 Ú dö = (Ö Ú) ˆÒd 2 a = A 0 ddφ = 2π. (c) What happens when the denominato in Ú is 2 instead of? The values of the integals aound the cicumfeence of the unit cicle ae the same fo any powe p. Without egulaization, the divegence o cul fo denominato 2 would appea to vanish entiely, but with cutoff ǫ and Ö Ú = 2ǫ 2 ( 2 +ǫ 2 ) 2, Ö Ú = 2ǫ 2 ( 2 +ǫ 2 ) 2 ˆÞ, 1 (Ö Ú)d 2 2ǫ 2 2π2 1 a = A 0 ( 2 +ǫ 2 ) 2 2πd = 2 +ǫ 2 = 2π 0 1+ǫ 2 2π. The only contibution to the aea integal is fom the δ-function hiding in the divegence/cul.
Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk
Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating
More informationAs is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.
Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.
More informationMath 2263 Solutions for Spring 2003 Final Exam
Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationMath Notes on Kepler s first law 1. r(t) kp(t)
Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is
More information2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum
2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aeodynamics I UNIT B: Theoy of Aeodynamics ROAD MAP... B-1: Mathematics fo Aeodynamics B-2: Flow Field Repesentations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis AE301
More informationQualifying Examination Electricity and Magnetism Solutions January 12, 2006
1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and
More informationReview: Electrostatics and Magnetostatics
Review: Electostatics and Magnetostatics In the static egime, electomagnetic quantities do not vay as a function of time. We have two main cases: ELECTROSTATICS The electic chages do not change postion
More informationTHE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2
THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace
More informationEM Boundary Value Problems
EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do
More information1 MathematicalPreliminaries
1 MathematicalPeliminaies The enomous usefulness of mathematics in the natual sciences is something bodeing on the mysteious. Eugene Wigne (1960 1.1 Intoduction This chapte pesents a collection of mathematical
More informationVectors, Vector Calculus, and Coordinate Systems
Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado 80523. Scalas and vectos Any
More informationQuestion 1: The dipole
Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite
More informationVector d is a linear vector function of vector d when the following relationships hold:
Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd
More informationQuestion Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if
Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7
More information(read nabla or del) is defined by, k. (9.7.1*)
9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The
More informationB da = 0. Q E da = ε. E da = E dv
lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the
More information3 VECTOR CALCULUS I. 3.1 Divergence and curl of vector fields. 3.2 Important identities
3 VECTOR CALCULU I 3.1 Divegence and cul of vecto fields Let = ( x, y, z ). eveal popeties of φ fo scala φ wee intoduced in section 1.2. The gadient opeato may also be applied to vecto fields. Let F =
More information(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.
Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed
More information6 Vector Operators. 6.1 The Gradient Operator
6 Vecto Opeatos 6. The Gadient Opeato In the B2 couse ou wee intoduced to the gadient opeato in Catesian coodinates. Fo an diffeentiable scala function f(x,, z), we can define a vecto function though (
More informationMATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates
MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v
More information3D-Central Force Problems II
5.73 ectue # - 1 3D-Cental Foce Poblems II ast time: [x,p] vecto commutation ules: genealize fom 1-D to 3-D conugate position and momentum components in Catesian coodinates Coespondence Pinciple Recipe
More informationB. Spherical Wave Propagation
11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We
More informationHomework # 3 Solution Key
PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in
More informationdq 1 (5) q 1 where the previously mentioned limit has been taken.
1 Vecto Calculus And Continuum Consevation Equations In Cuvilinea Othogonal Coodinates Robet Maska: Novembe 25, 2008 In ode to ewite the consevation equations(continuit, momentum, eneg) to some cuvilinea
More informationKEPLER S LAWS AND PLANETARY ORBITS
KEPE S AWS AND PANETAY OBITS 1. Selected popeties of pola coodinates and ellipses Pola coodinates: I take a some what extended view of pola coodinates in that I allow fo a z diection (cylindical coodinates
More informationworking pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50
woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,
More informationPHYS 110B - HW #7 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #7 Sping 2004, Solutions by David Pace Any efeenced euations ae fom Giffiths Poblem statements ae paaphased. Poblem 0.3 fom Giffiths A point chage,, moves in a loop of adius a. At time t 0
More informationKEPLER S LAWS OF PLANETARY MOTION
EPER S AWS OF PANETARY MOTION 1. Intoduction We ae now in a position to apply what we have leaned about the coss poduct and vecto valued functions to deive eple s aws of planetay motion. These laws wee
More information( ) Make-up Tests. From Last Time. Electric Field Flux. o The Electric Field Flux through a bit of area is
Mon., 3/23 Wed., 3/25 Thus., 3/26 Fi., 3/27 Mon., 3/30 Tues., 3/31 21.4-6 Using Gauss s & nto to Ampee s 21.7-9 Maxwell s, Gauss s, and Ampee s Quiz Ch 21, Lab 9 Ampee s Law (wite up) 22.1-2,10 nto to
More informationSo, if we are finding the amount of work done over a non-conservative vector field F r, we do that long ur r b ur =
3.4 Geen s Theoem Geoge Geen: self-taught English scientist, 793-84 So, if we ae finding the amount of wok done ove a non-consevative vecto field F, we do that long u b u 3. method Wok = F d F( () t )
More informationVectors, Vector Calculus, and Coordinate Systems
! Revised Apil 11, 2017 1:48 PM! 1 Vectos, Vecto Calculus, and Coodinate Systems David Randall Physical laws and coodinate systems Fo the pesent discussion, we define a coodinate system as a tool fo descibing
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationLecture 8 - Gauss s Law
Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.
More informationTUTORIAL 9. Static magnetic field
TUTOIAL 9 Static magnetic field Vecto magnetic potential Null Identity % & %$ A # Fist postulation # " B such that: Vecto magnetic potential Vecto Poisson s equation The solution is: " Substitute it into
More informationEFFECTS OF FRINGING FIELDS ON SINGLE PARTICLE DYNAMICS. M. Bassetti and C. Biscari INFN-LNF, CP 13, Frascati (RM), Italy
Fascati Physics Seies Vol. X (998), pp. 47-54 4 th Advanced ICFA Beam Dynamics Wokshop, Fascati, Oct. -5, 997 EFFECTS OF FRININ FIELDS ON SINLE PARTICLE DYNAMICS M. Bassetti and C. Biscai INFN-LNF, CP
More informationMath 124B February 02, 2012
Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial
More informationAST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1
Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be
More informationLecture 1a: Satellite Orbits
Lectue 1a: Satellite Obits Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion) Scala & Vectos Scala: a physical quantity
More informationANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. D = εe. For a linear, homogeneous, isotropic medium µ and ε are contant.
ANTNNAS Vecto and Scala Potentials Maxwell's quations jωb J + jωd D ρ B (M) (M) (M3) (M4) D ε B Fo a linea, homogeneous, isotopic medium and ε ae contant. Since B, thee exists a vecto A such that B A and
More informationCOORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit
More informationMath 259 Winter Handout 6: In-class Review for the Cumulative Final Exam
Math 259 Winte 2009 Handout 6: In-class Review fo the Cumulative Final Exam The topics coveed by the cumulative final exam include the following: Paametic cuves. Finding fomulas fo paametic cuves. Dawing
More informationNow we just need to shuffle indices around a bit. The second term is already of the form
Depatment of Physics, UCSD Physics 5B, Geneal Relativity Winte 05 Homewok, solutions. (a) Fom the Killing equation, ρ K σ ` σ K ρ 0 taking one deivative, µ ρ K σ ` µ σ K ρ 0 σ µ K ρ σ ρ K µ 0 ρ µ K σ `
More information16.1 Permanent magnets
Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and
More informationSUPPLEMENTARY MATERIAL CHAPTER 7 A (2 ) B. a x + bx + c dx
SUPPLEMENTARY MATERIAL 613 7.6.3 CHAPTER 7 ( px + q) a x + bx + c dx. We choose constants A and B such that d px + q A ( ax + bx + c) + B dx A(ax + b) + B Compaing the coefficients of x and the constant
More informationPhysics 506 Winter 2006 Homework Assignment #9 Solutions
Physics 506 Winte 2006 Homewok Assignment #9 Solutions Textbook poblems: Ch. 12: 12.2, 12.9, 12.13, 12.14 12.2 a) Show fom Hamilton s pinciple that Lagangians that diffe only by a total time deivative
More informationECE 3318 Applied Electricity and Magnetism. Spring Prof. David R. Jackson ECE Dept. Notes 13
ECE 338 Applied Electicity and Magnetism ping 07 Pof. David R. Jackson ECE Dept. Notes 3 Divegence The Physical Concept Find the flux going outwad though a sphee of adius. x ρ v0 z a y ψ = D nˆ d = D ˆ
More informationLecture 1a: Satellite Orbits
Lectue 1a: Satellite Obits Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Univesal Gavita3on 3. Calcula3ng satellite obital paametes (assuming cicula mo3on) Scala & Vectos Scala: a physical quan3ty
More informationClassical Mechanics Homework set 7, due Nov 8th: Solutions
Classical Mechanics Homewok set 7, due Nov 8th: Solutions 1. Do deivation 8.. It has been asked what effect does a total deivative as a function of q i, t have on the Hamiltonian. Thus, lets us begin with
More informationPendulum in Orbit. Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ (December 1, 2017)
1 Poblem Pendulum in Obit Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 08544 (Decembe 1, 2017) Discuss the fequency of small oscillations of a simple pendulum in obit, say,
More information3. Electromagnetic Waves II
Lectue 3 - Electomagnetic Waves II 9 3. Electomagnetic Waves II Last time, we discussed the following. 1. The popagation of an EM wave though a macoscopic media: We discussed how the wave inteacts with
More informationPhysics 2A Chapter 10 - Moment of Inertia Fall 2018
Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.
More informationF Q E v B MAGNETOSTATICS. Creation of magnetic field B. Effect of B on a moving charge. On moving charges only. Stationary and moving charges
MAGNETOSTATICS Ceation of magnetic field. Effect of on a moving chage. Take the second case: F Q v mag On moving chages only F QE v Stationay and moving chages dw F dl Analysis on F mag : mag mag Qv. vdt
More informationJ. N. R E DDY ENERGY PRINCIPLES AND VARIATIONAL METHODS APPLIED MECHANICS
J. N. E DDY ENEGY PINCIPLES AND VAIATIONAL METHODS IN APPLIED MECHANICS T H I D E DI T IO N JN eddy - 1 MEEN 618: ENEGY AND VAIATIONAL METHODS A EVIEW OF VECTOS AND TENSOS ead: Chapte 2 CONTENTS Physical
More informationSources of Magnetic Fields (chap 28)
Souces of Magnetic Fields (chap 8) In chapte 7, we consideed the magnetic field effects on a moving chage, a line cuent and a cuent loop. Now in Chap 8, we conside the magnetic fields that ae ceated by
More information3D-Central Force Problems I
5.73 Lectue #1 1-1 Roadmap 1. define adial momentum 3D-Cental Foce Poblems I Read: C-TDL, pages 643-660 fo next lectue. All -Body, 3-D poblems can be educed to * a -D angula pat that is exactly and univesally
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More informationOLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.
OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto
More informationReview. Electrostatic. Dr. Ray Kwok SJSU
Review Electostatic D. Ray Kwok SJSU Paty Balloons Coulomb s Law F e q q k 1 Coulomb foce o electical foce. (vecto) Be caeful on detemining the sign & diection. k 9 10 9 (N m / C ) k 1 4πε o k is the Coulomb
More informationMATH Homework #1 Solution - by Ben Ong
MATH 46 - Homewok #1 Solution - by Ben Ong Deivation of the Eule Equations We pesent a fist pinciples deivation of the Eule Equations fo two-dimensional fluid flow in thee-dimensional cylndical cooodinates
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationLecture 23. Representation of the Dirac delta function in other coordinate systems
Lectue 23 Repesentation of the Diac delta function in othe coodinate systems In a geneal sense, one can wite, ( ) = (x x ) (y y ) (z z ) = (u u ) (v v ) (w w ) J Whee J epesents the Jacobian of the tansfomation.
More information-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r.
The Laplace opeato in pola coodinates We now conside the Laplace opeato with Diichlet bounday conditions on a cicula egion Ω {(x,y) x + y A }. Ou goal is to compute eigenvalues and eigenfunctions of the
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More informationGreen s Identities and Green s Functions
LECTURE 7 Geen s Identities and Geen s Functions Let us ecall The ivegence Theoem in n-dimensions Theoem 7 Let F : R n R n be a vecto field ove R n that is of class C on some closed, connected, simply
More informationsinγ(h y > ) exp(iωt iqx)dωdq
Lectue 9/28/5 Can we ecove a ay pictue fom the above G fo a membane stip? Such a pictue would be complementay to the above expansion in a seies of integals along the many banches of the dispesion elation.
More informationVector Calculus Identities
The Coope Union Depatment of Electical Engineeing ECE135 Engineeing Electomagnetics Vecto Analysis Mach 8, 2012 Vecto Calculus Identities A B B A A B C B C A C A B A B C B A C C A B fg f g + g f fg f G
More informationENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi
ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationMultipole Radiation. February 29, The electromagnetic field of an isolated, oscillating source
Multipole Radiation Febuay 29, 26 The electomagnetic field of an isolated, oscillating souce Conside a localized, oscillating souce, located in othewise empty space. We know that the solution fo the vecto
More information15 Solving the Laplace equation by Fourier method
5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the
More informationc 2011 Faith A. Morrison, all rights reserved. 1 August 30, 2011
c 20 Faith A. Moison, all ights eseved. August 30, 20 2 c 20 Faith A. Moison, all ights eseved. An Intoduction to Fluid Mechanics: Supplemental Web Appendices Faith A. Moison Associate Pofesso of Chemical
More informationA proof of the binomial theorem
A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative
More informationChapter 22: Electric Fields. 22-1: What is physics? General physics II (22102) Dr. Iyad SAADEDDIN. 22-2: The Electric Field (E)
Geneal physics II (10) D. Iyad D. Iyad Chapte : lectic Fields In this chapte we will cove The lectic Field lectic Field Lines -: The lectic Field () lectic field exists in a egion of space suounding a
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,
More informationMath 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3
Math : alculus I Math/Sci majos MWF am / pm, ampion Witten homewok Review: p 94, p 977,8,9,6, 6: p 46, 6: p 4964b,c,69, 6: p 47,6 p 94, Evaluate the following it by identifying the integal that it epesents:
More informationLab #9: The Kinematics & Dynamics of. Circular Motion & Rotational Motion
Reading Assignment: Lab #9: The Kinematics & Dynamics of Cicula Motion & Rotational Motion Chapte 6 Section 4 Chapte 11 Section 1 though Section 5 Intoduction: When discussing motion, it is impotant to
More informationFaraday s Law (continued)
Faaday s Law (continued) What causes cuent to flow in wie? Answe: an field in the wie. A changing magnetic flux not only causes an MF aound a loop but an induced electic field. Can wite Faaday s Law: ε
More informationLecture 7: Angular Momentum, Hydrogen Atom
Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z
More information221B Lecture Notes Scattering Theory I
Why Scatteing? B Lectue Notes Scatteing Theoy I Scatteing of paticles off taget has been one of the most impotant applications of quantum mechanics. It is pobably the most effective way to study the stuctue
More informationFinal Review of AerE 243 Class
Final Review of AeE 4 Class Content of Aeodynamics I I Chapte : Review of Multivaiable Calculus Chapte : Review of Vectos Chapte : Review of Fluid Mechanics Chapte 4: Consevation Equations Chapte 5: Simplifications
More informationGeometry of the homogeneous and isotropic spaces
Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant
More informationDynamic Visualization of Complex Integrals with Cabri II Plus
Dynamic Visualiation of omplex Integals with abi II Plus Sae MIKI Kawai-juu, IES Japan Email: sand_pictue@hotmailcom Abstact: Dynamic visualiation helps us undestand the concepts of mathematics This pape
More informationS7: Classical mechanics problem set 2
J. Magoian MT 9, boowing fom J. J. Binney s 6 couse S7: Classical mechanics poblem set. Show that if the Hamiltonian is indepdent of a genealized co-odinate q, then the conjugate momentum p is a constant
More informationClass #16 Monday, March 20, 2017
D. Pogo Class #16 Monday, Mach 0, 017 D Non-Catesian Coodinate Systems A point in space can be specified by thee numbes:, y, and z. O, it can be specified by 3 diffeent numbes:,, and z, whee = cos, y =
More informationRight-handed screw dislocation in an isotropic solid
Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We
More informationGauss Law. Physics 231 Lecture 2-1
Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationarxiv: v1 [physics.pop-ph] 3 Jun 2013
A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,
More informationPHYS 705: Classical Mechanics. Small Oscillations
PHYS 705: Classical Mechanics Small Oscillations Fomulation of the Poblem Assumptions: V q - A consevative system with depending on position only - The tansfomation equation defining does not dep on time
More informationREVIEW Polar Coordinates and Equations
REVIEW 9.1-9.4 Pola Coodinates and Equations You ae familia with plotting with a ectangula coodinate system. We ae going to look at a new coodinate system called the pola coodinate system. The cente of
More informationFI 2201 Electromagnetism
FI Electomagnetim Aleande A. Ikanda, Ph.D. Phyic of Magnetim and Photonic Reeach Goup ecto Analyi CURILINEAR COORDINAES, DIRAC DELA FUNCION AND HEORY OF ECOR FIELDS Cuvilinea Coodinate Sytem Cateian coodinate:
More informationPhysics 122, Fall October 2012
hsics 1, Fall 1 3 Octobe 1 Toda in hsics 1: finding Foce between paallel cuents Eample calculations of fom the iot- Savat field law Ampèe s Law Eample calculations of fom Ampèe s law Unifom cuents in conductos?
More informationINTRODUCTION. 2. Vectors in Physics 1
INTRODUCTION Vectos ae used in physics to extend the study of motion fom one dimension to two dimensions Vectos ae indispensable when a physical quantity has a diection associated with it As an example,
More informationUniform Circular Motion
Unifom Cicula Motion Intoduction Ealie we defined acceleation as being the change in velocity with time: a = v t Until now we have only talked about changes in the magnitude of the acceleation: the speeding
More informationPhysics 862: Atoms, Nuclei, and Elementary Particles
Physics 86: Atoms, Nuclei, and Elementay Paticles Bian Bockelman Septembe 11, 008 Contents 1 Cental Field Poblems 1.1 Classical Teatment......................... 1. Quantum Teatment.........................
More informationThe geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
More informationElectromagnetism Physics 15b
lectomagnetism Physics 15b Lectue #20 Dielectics lectic Dipoles Pucell 10.1 10.6 What We Did Last Time Plane wave solutions of Maxwell s equations = 0 sin(k ωt) B = B 0 sin(k ωt) ω = kc, 0 = B, 0 ˆk =
More informationNewton s Laws, Kepler s Laws, and Planetary Orbits
Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion
More information