Conservation Laws. Chapter Conservation of Energy
|
|
- Colleen Singleton
- 5 years ago
- Views:
Transcription
1 20
2 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action of, an reaction on, the sources of the electromagnetic fiels. To be precise, we ask whether there is a correct balance in the exchange of energy, momentum, an angular momentum between the charge particles an the electromagnetic fiels. As we shall see, the Maxwell-Lorentz system as it stans implies the conservation of these mechanical properties, no matter how rapily the charges are moving. 3.1 Conservation of Energy We start with a consieration of the rate at which work is one on the particles, that is, the rate of energy transfer, or the power absorbe by the particles. For one particle, we know that the rate at which work is one on it is F v = ev E + gv B = (r)(j e E + j m B), (3.1) where we have use the Lorentz force law, (2.12), an the expressions for the currents, (1.44) an (2.7), for a point particle. We interpret this equation as meaning, even for general current istributions, that j e E+j m B is the rate of energy transfer from the fiel to the particles, per unit volume. Then through elimination of the currents by use of Maxwell s equations, (2.10), this rate can be rewritten as j e E + j m B = c = t ( B 1 ) c t E E + c ( E 2 + B 2 8π ( E 1 ) c t B B ) ( c E B ). (3.2) The general form of any local conservation law, (1.45) or (1.46), suggests the following interpretations: 21
3 22 CHAPTER 3. CONSERVATION LAWS 1. In the absence of charges (j e = j m = 0), this is the local energy conservation law E 2 + B 2 + t 8π c E B = 0. (3.3) We label the two objects appearing here as energy ensity = U = E2 + B 2, (3.4) 8π energy flux vector = S = c E B. (3.5) [The latter is usually calle the Poynting vector, after John Henry Poynting ( ).] 2. In the presence of charges, the relation (3.2) is t U + S + j e E + j m B = 0, (3.6) which, if we integrate over an arbitrary volume V, boune by a surface S, becomes (r)u + S S + (r)(j e E + j m B) = 0. (3.7) t S V V The three terms here are ientifie, respectively, as the rate of change of the electromagnetic fiel energy within the volume, the rate of flow of electromagnetic energy out of the volume, an the rate of transfer of electromagnetic energy to the charge particles. Thus, (3.6) gives a complete escription of energy conservation. 3.2 Conservation of Momentum Next we consier the force on a particle, (2.12), as the rate of change of momentum, F = e (E + v ) c B + g (B v ) c E ( = (r) ρ e E + 1 c j e B + ρ m B 1 ) c j m E (r)f, (3.8) where f is the force ensity. Removing reference to the (generalize) charge an current ensities by use of Maxwell s equations, (2.10), we rewrite the force ensity f as f = 1 [E( E) + B( B)]
4 3.2. CONSERVATION OF MOMENTUM = t [( 1 c ) ( t E + B B + E 1 )] c t B E E B c + 1 [ E ( E) + E( E) B ( B) + B( B)]. The quaratic structure in E occurring here is (3.9) E ( E) + E( E) = E2 2 + (E )E + E( E) = ( ) 1 E2 2 + EE, (3.10) which introuces yaic notation, incluing the unit yaic 1, with components 1 kl = δ kl = { 1, k = l, 0, k l, (3.11) where δ kl is the Kronecker δ symbol. (See Problem 3.1.) The analogous result hols for B. Accoringly, the force ensity is f = E B t c ( 1 E2 + B 2 ) EE + BB. (3.12) 8π We interpret this equation physically by ientifying an momentum ensity = G = E B c, (3.13) momentum flux (stress tensor) = T = 1 E2 + B 2 8π EE + BB. (3.14) When f = 0, we obtain the local statement of the conservation of momentum of the electromagnetic fiel. A full account of momentum balance is containe in t G + T + f = 0. (3.15) The volume integral of this equation for electromagnetic momentum is interprete analogously to the energy result, (3.7). The components of the stress tensor are given by T kl = δ kl U E ke l + B k B l. (3.16) Notice that the stress tensor is symmetrical, T kl = T lk, which, as we shall see in the next section, is require in orer to obtain a local conservation law for angular momentum. The trace of T, the sum of the iagonal elements T kk, is simply the energy ensity, (3.4), TrT = k T kk = U. (3.17)
5 24 CHAPTER 3. CONSERVATION LAWS We also note that the Poynting vector, (3.5), is proportional to the momentum ensity, S = c 2 G, (3.18) which has the structure of energy ensity velocity = c 2 (mass ensity velocity). (3.19) This is the first inication of the relativistic connection between energy an mass, E = mc Conservation of Angular Momentum. Virial Theorem Having iscusse momentum, we now turn to angular momentum. We will use tensor notation to write (3.15) in component form, t G k + l T lk + f k = 0, (3.20) where we have also use the summation convention: Whenever an inex is repeate, a sum over all values of that inex is assume, a i b i 3 a i b i = a b. (3.21) i=1 The rate of change of angular momentum is the torque τ, which, for one particle, is τ = r F = (r) r f, (3.22) where the volume-integrate form is no longer restricte to a single particle. The torque ensity, the moment of the force ensity, can be written in component form as (r f) i = ǫ ijk x j f k, (3.23) where we have introuce the totally antisymmetric (Levi-Civita) symbol ǫ ijk, which changes sign uner any interchange of two inices, ǫ ijk = ǫ jik = ǫ kji = ǫ ikj = +ǫ kij = +ǫ jki, (3.24) an is normalize by ǫ 123 = 1. In particular, then, it vanishes if any two inices are equal, ǫ 112 = 0, for example. The torque ensity may be obtaine by first taking the moment of the force ensity equation (3.20), where we have note that t x jg k + l (x j T lk ) T jk + x j f k = 0, (3.25) l x j = δ lj. (3.26)
6 3.4. CONSERVATION LAWS AND THE SPEED OF LIGHT 25 When we now multiply (3.25) with ǫ ijk an sum over repeate inices, we fin that the terms involving spatial erivatives can be written as a ivergence: t (ǫ ijkx j G k ) + l (ǫ ijk x j T lk ) + ǫ ijk x j f k = 0. (3.27) This final step is justifie only because T kl is symmetrical (thus this symmetry is require for the existence of a local conservation law of angular momentum). We therefore ientify the following electromagnetic angular momentum quantities: angular momentum ensity = J = r G (3.28) angular momentum flux tensor = K, K ij = ǫ jkl x k T il. (3.29) The interpretation of (3.27) as a local account of angular momentum conservation for fiels an particles procees as before. (See Problem 3.5.) Another important application of (3.25) results if we set j = k an sum. With the ai of (3.17) this gives t (r G) + (T r) U + r f = 0, (3.30) which we call the electromagnetic virial theorem, in analogy with the mechanical virial theorem of Ruolf Clausius ( ). (See Chapter 8.) 3.4 Conservation Laws an the Spee of Light In this section, we restrict our attention to electromagnetic fiels in omains free of charge particles, specifically, moving, finite regions occupie by electromagnetic fiels, which we will refer to as electromagnetic s. The total electromagnetic energy of such a is constant in time: t E = (r) t U = (r) S = 0, (3.31) inasmuch as the resulting surface integral, conucte over an enclosing surface on which all fiels vanish, equals zero. Similar consierations apply to the total electromagnetic linear an angular momentum, t P = t J = (r) t G = (r) t (r G) = (r) T = 0, (3.32) (r) ( T r) = 0. (3.33) With an eye towar relativity, we consier the space an time moments of (3.6) an (3.15), respectively, combine as a single vector statement: ( ) ( ) 0 = x k t U + S c 2 t t G k + l T lk, (3.34)
7 26 CHAPTER 3. CONSERVATION LAWS outsie the charge an current istributions. Exploiting the connection between S an G [(3.18)], we can rewrite (3.34) as a local conservation law, much as the equality of T jk an T kj lea to the conservation of angular momentum: t (ru c2 tg) + (Sr c 2 tt) = 0. (3.35) When (3.35) is integrate over a volume enclosing the electromagnetic, the surface term oes not contribute, an we fin (r)(ru c 2 tg) = 0. (3.36) t The volume integral of the momentum ensity is the total momentum P, (r)g = P, (3.37) which as note in (3.32) is constant in time. Consequently, we can rewrite (3.36) as (r)ru = c 2 P, (3.38) t where the integral here provies an energy weighting of the position vector, at each instant of time, (r)ru(r, t) = E r E (t), (3.39) where, as in (3.31), the energy E is E = (r) U. (3.40) Thus the motion of this energy-centroi vector is governe by E c 2 t r E(t) = P, (3.41) which is to say that the center of energy, r E (t), moves with constant velocity, t r E(t) = v E, (3.42) the total momentum being that velocity multiplie by a mass, m = E/c 2. (3.43) The application of the virial theorem, (3.30), to an electromagnetic supplies another velocity. We infer that (r)r G = E. (3.44) t
8 3.4. CONSERVATION LAWS AND THE SPEED OF LIGHT 27 By introucing a momentum weighting for the position vector, (r)r G(r, t) = r P (t) P, (3.45) we euce that the center of momentum moves with velocity which is constant in the irection of the momentum, We combine (3.47) with (3.41) to yiel t r P(t) = v P, (3.46) v P P = E. (3.47) v P v E = c 2. (3.48) If the flow of energy an momentum takes place in a single irection, it woul be reasonable to expect that these mechanical properties are being transporte with a common velocity, v E = v P = v, (3.49) which then has a efinite magnitue, v v = c 2, v = c, (3.50) which supplies the physical ientification of c as the spee of light. Of course, this ientification was an input to our inference of Maxwell s equations. We here recover it from a consieration of energy an momentum, thus inicating the consistency of Maxwell s equations. The relation between the momentum an the energy of this electromagnetic is then E = v P, P = E c2v, (3.51) so we learn that E = Pc, v = c P P, (3.52) which results express the mechanical properties of a localize electromagnetic carrying both energy an momentum at the spee of light, in the irection of the momentum. There is another, somewhat more irect, mechanical proof that electromagnetic s propagate at spee c. When no charges or currents are present, the local equation of energy conservation, (3.3), implies [ ] (r 2 c 2 t 2 ) t U + S = 0, (3.53) which can be rewritten, using (3.18), as t [(r2 c 2 t 2 )U] + [(r 2 c 2 t 2 )S] + 2c 2 [tu r G] = 0. (3.54)
9 28 CHAPTER 3. CONSERVATION LAWS Integrating this over all space an using the iea of energy an momentum weighting to efine averages, as before, we obtain [ r 2 E (t) c 2 t 2] E = 2c 2 [ r P (t) P te]. (3.55) t Accoring to (3.47), the combination appearing on the right is a constant of the motion, which we can put equal to zero by ientifying the coorinate origin with r P at t = 0. The time integral of this equation is then which implies, for large times, r 2 E = (ct) 2 + constant, (3.56) ( r 2 E (t) ) 1/2 ct; (3.57) the center of energy of the moves away from the origin at the spee of light. What are the fiels oing to enforce the conitions (3.51) of simple mechanical flow in a single irection? The relation between momentum an energy, (3.52), E = P c, can be expresse in terms of the fiels as (r) E2 + B 2 8π = (r) E B, (3.58) where the volume integrations are extene over the. Now, a sum of vectors of given magnitues is of maximum magnitue when all those vectors are parallel, which is to say here that (r) E2 + B 2 (r) E B, (3.59) 2 where equality hols only when E B everywhere points in the same irection, that of the s total momentum or velocity. On the other han, we note the inequality, (E B) 2 = E 2 B 2 (E B) 2 ( E 2 + B 2 ) 2 [ (E 2 B 2 ) 2 ] ( E = + (E B) B 2 ) 2, (3.60) where the equality hols only if both E B = 0 an E 2 = B 2. So we euce the opposite inequality to (3.59), (r) E B (r) E2 + B 2. (3.61) 2 Comparing (3.59) an (3.61), we see that both equalities must hol, so that E B = 0, E 2 = B 2, (3.62)
10 3.5. PROBLEMS FOR CHAPTER 3 29 E B v Figure 3.1: Electric an magnetic fiels for an electromagnetic propagating with velocity v. an E B is uniirectional, pointing in the irection of propagation. Accoringly, the electric an magnetic fiels in a uniirectional are, everywhere within the, of equal magnitue, mutually perpenicular, an perpenicular to the irection of motion of the. (See Fig. 3.1.) These are the familiar properties of the electromagnetic fiels of a light wave, which are here erive without recourse to explicit solutions to Maxwell s equations. 3.5 Problems for Chapter 3 1. The unit yaic 1 is efine in terms of orthogonal unit vectors i, j, k by 1 = ii + jj + kk. Verify that (A is an arbitrary vector) A 1 = A, 1 A = A, 1 1 = 1. Repeat, using components, i.e., A B = A i B i. Expan the following proucts of vectors with yaics: A (BC), (AB) C, A (BC), (AB) C. 2. Let A(r) an B(r) be vector fiels. Show that (AB) = (A )B + B( A). Let λ(r) further be an arbitrary scalar function. Simplify (λab), (λa B). 3. An infinitesimal rotation is escribe by its effect on an arbitrary vector V by δv = δω V,
11 30 CHAPTER 3. CONSERVATION LAWS where the irection of δω points in the irection of the rotation, an has magnitue equal to the (infinitesimal) amount of the rotation. Check that δ(v 2 ) = 0. The statement that, if B an C are vectors, so is B C, is expresse by δ(b C) = δb C + B δc = δω (B C). Verify irectly the resulting relation among the arbitrary vectors. 4. Verify the following relations for the electromagnetic stress tensor: (a) (b) TrT = T kk = U, TrT 2 = T kl T lk = 3U 2 2(cG) 2 U 2, an (c) ett = U[U 2 (cg) 2 ]. Here the summation convention is employe, an the trace an eterminant refer to T thought of as a 3 3 matrix. 5. Show that the angular momentum conservation law erive in Section 3.3 can be written as t J + K + r f = 0, where the angular momentum ensity is J = r G, an the angular momentum flux tensor is K = T r, the cross prouct referring to the secon vector inex of T. 6. What if r P (0) 0 in (3.55)? Show that the integral of that equation can be interprete in analogy with a group of particles that, at time t = 0, are set off with various positions an velocities, thereafter to move with those constant velocities, r(t) = r(0) + vt. Square an average this position vector, an upon comparison with the solution of (3.55), ientify r(0) v an v.
12 3.5. PROBLEMS FOR CHAPTER As in Problem 2.1, let F = E + ib, F = E ib. Ientify the scalar the vector an the yaic 1 8π F F, 1 8πi F F, 1 8π (FF + F F). What happens to these quantities if F is replace by e iφ F, φ being a constant? 8. Electric charge e is locate at the fixe point 1 2R. Magnetic charge g is statione at the fixe point 1 2R. What is the momentum ensity at the arbitrary point r? Verify that it is ivergenceless by writing it as a curl. Evaluate the electromagnetic angular momentum, the integrate moment of the momentum ensity. Recognize that it is a graient with respect to R. Continue the evaluation to iscover that it epens only on the irection of R, not its magnitue. This is the naive, semiclassical basis for the charge quantization conition of Dirac, eg = n 2 hc.
Chapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More information2.20 Marine Hydrodynamics Lecture 3
2.20 Marine Hyroynamics, Fall 2018 Lecture 3 Copyright c 2018 MIT - Department of Mechanical Engineering, All rights reserve. 1.7 Stress Tensor 2.20 Marine Hyroynamics Lecture 3 1.7.1 Stress Tensor τ ij
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationNoether s theorem applied to classical electrodynamics
Noether s theorem applie to classical electroynamics Thomas B. Mieling Faculty of Physics, University of ienna Boltzmanngasse 5, 090 ienna, Austria (Date: November 8, 207) The consequences of gauge invariance
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationThe continuity equation
Chapter 6 The continuity equation 61 The equation of continuity It is evient that in a certain region of space the matter entering it must be equal to the matter leaving it Let us consier an infinitesimal
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More informationBasic Thermoelasticity
Basic hermoelasticity Biswajit Banerjee November 15, 2006 Contents 1 Governing Equations 1 1.1 Balance Laws.............................................. 2 1.2 he Clausius-Duhem Inequality....................................
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More information1.2 - Stress Tensor Marine Hydrodynamics Lecture 3
13.021 Marine Hyroynamics, Fall 2004 Lecture 3 Copyright c 2004 MIT - Department of Ocean Engineering, All rights reserve. 1.2 - Stress Tensor 13.021 Marine Hyroynamics Lecture 3 Stress Tensor τ ij:. The
More informationFree rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012
Free rotation of a rigi boy 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 1 Introuction In this section, we escribe the motion of a rigi boy that is free to rotate
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationLecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Bair, faculty.uml.eu/cbair University of Massachusetts Lowell 1. Pre-Einstein Relativity - Einstein i not invent the concept of relativity,
More informationPhysics 2212 GJ Quiz #4 Solutions Fall 2015
Physics 2212 GJ Quiz #4 Solutions Fall 215 I. (17 points) The magnetic fiel at point P ue to a current through the wire is 5. µt into the page. The curve portion of the wire is a semicircle of raius 2.
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationAnalytic Scaling Formulas for Crossed Laser Acceleration in Vacuum
October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945
More informationMATHEMATICS BONUS FILES for faculty and students
MATHMATI BONU FIL for faculty an stuents http://www.onu.eu/~mcaragiu1/bonus_files.html RIVD: May 15, 9 PUBLIHD: May 5, 9 toffel 1 Maxwell s quations through the Major Vector Theorems Joshua toffel Department
More informationLecture XVI: Symmetrical spacetimes
Lecture XVI: Symmetrical spacetimes Christopher M. Hirata Caltech M/C 350-17, Pasaena CA 91125, USA (Date: January 4, 2012) I. OVERVIEW Our principal concern this term will be symmetrical solutions of
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationMultivariable Calculus: Chapter 13: Topic Guide and Formulas (pgs ) * line segment notation above a variable indicates vector
Multivariable Calculus: Chapter 13: Topic Guie an Formulas (pgs 800 851) * line segment notation above a variable inicates vector The 3D Coorinate System: Distance Formula: (x 2 x ) 2 1 + ( y ) ) 2 y 2
More informationG4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.
G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationChapter 4. Electrostatics of Macroscopic Media
Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationMomentum and Energy. Chapter Conservation Principles
Chapter 2 Momentum an Energy In this chapter we present some funamental results of continuum mechanics. The formulation is base on the principles of conservation of mass, momentum, angular momentum, an
More information12.5. Differentiation of vectors. Introduction. Prerequisites. Learning Outcomes
Differentiation of vectors 12.5 Introuction The area known as vector calculus is use to moel mathematically a vast range of engineering phenomena incluing electrostatics, electromagnetic fiels, air flow
More informationHow the potentials in different gauges yield the same retarded electric and magnetic fields
How the potentials in ifferent gauges yiel the same retare electric an magnetic fiels José A. Heras a Departamento e Física, E. S. F. M., Instituto Politécnico Nacional, México D. F. México an Department
More informationCenter of Gravity and Center of Mass
Center of Gravity an Center of Mass 1 Introuction. Center of mass an center of gravity closely parallel each other: they both work the same way. Center of mass is the more important, but center of gravity
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationDIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10
DIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10 5. Levi-Civita connection From now on we are intereste in connections on the tangent bunle T X of a Riemanninam manifol (X, g). Out main result will be a construction
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationThe electrodynamics of rotating electrons.
Die Elektroynamik es rotierenen Elektrons, Zeit. f. Phys. 37 (196), 43-6. The electroynamics of rotating electrons. By J. Frenkel 1 ) in Leningra. (Receive on May 196) Translate by D. H. Delphenich The
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More information6 Wave equation in spherical polar coordinates
6 Wave equation in spherical polar coorinates We now look at solving problems involving the Laplacian in spherical polar coorinates. The angular epenence of the solutions will be escribe by spherical harmonics.
More informationLagrangian and Hamiltonian Dynamics
Lagrangian an Hamiltonian Dynamics Volker Perlick (Lancaster University) Lecture 1 The Passage from Newtonian to Lagrangian Dynamics (Cockcroft Institute, 22 February 2010) Subjects covere Lecture 2: Discussion
More informationA Model of Electron-Positron Pair Formation
Volume PROGRESS IN PHYSICS January, 8 A Moel of Electron-Positron Pair Formation Bo Lehnert Alfvén Laboratory, Royal Institute of Technology, S-44 Stockholm, Sween E-mail: Bo.Lehnert@ee.kth.se The elementary
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More information1 M3-4-5A16 Assessed Problems # 1: Do 4 out of 5 problems
D. D. Holm M3-4-5A16 Assesse Problems # 1 Due 1 Nov 2012 1 1 M3-4-5A16 Assesse Problems # 1: Do 4 out of 5 problems Exercise 1.1 (Poisson brackets for the Hopf map) Figure 1: The Hopf map. In coorinates
More informationHomework 7 Due 18 November at 6:00 pm
Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationPhysics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2
Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationTEST 2 (PHY 250) Figure Figure P26.21
TEST 2 (PHY 250) 1. a) Write the efinition (in a full sentence) of electric potential. b) What is a capacitor? c) Relate the electric torque, exerte on a molecule in a uniform electric fiel, with the ipole
More informationPure Further Mathematics 1. Revision Notes
Pure Further Mathematics Revision Notes June 20 2 FP JUNE 20 SDB Further Pure Complex Numbers... 3 Definitions an arithmetical operations... 3 Complex conjugate... 3 Properties... 3 Complex number plane,
More informationMathematical Basics. Chapter Introduction and Definitions
Chapter 2 Mathematical Basics 2.1 Introuction an Definitions Flui mechanics eals with transport processes, especially with the flow- an molecule-epenent momentum transports in fluis. Their thermoynamic
More informationGravitation as the result of the reintegration of migrated electrons and positrons to their atomic nuclei. Osvaldo Domann
Gravitation as the result of the reintegration of migrate electrons an positrons to their atomic nuclei. Osvalo Domann oomann@yahoo.com (This paper is an extract of [6] liste in section Bibliography.)
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationProblem Solving 4 Solutions: Magnetic Force, Torque, and Magnetic Moments
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.0 Spring 004 Problem Solving 4 Solutions: Magnetic Force, Torque, an Magnetic Moments OJECTIVES 1. To start with the magnetic force on a moving
More informationThe Three-dimensional Schödinger Equation
The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation
More informationCalculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS
Calculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS. An isosceles triangle, whose base is the interval from (0, 0) to (c, 0), has its verte on the graph
More informationMath 32A Review Sheet
Review Sheet Tau Beta Pi - Boelter 6266 Contents 1 Parametric Equation 2 1.1 Line.................................................. 2 1.2 Circle................................................. 2 1.3 Ellipse.................................................
More informationTorque OBJECTIVE INTRODUCTION APPARATUS THEORY
Torque OBJECTIVE To verify the rotational an translational conitions for equilibrium. To etermine the center of ravity of a rii boy (meter stick). To apply the torque concept to the etermination of an
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationECE 6310 Spring 2012 Exam 1 Solutions. Balanis The electric fields are given by. E r = ˆxe jβ 0 z
ECE 6310 Spring 2012 Exam 1 Solutions Balanis 1.30 The electric fiels are given by E i ˆxe jβ 0 z E r ˆxe jβ 0 z The curl of the electric fiels are the usual cross prouct E i jβ 0 ẑ ˆxe jβ 0 z jβ 0 ŷe
More informationFundamental Laws of Motion for Particles, Material Volumes, and Control Volumes
Funamental Laws of Motion for Particles, Material Volumes, an Control Volumes Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambrige, MA 02139, USA August 2001
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationEuler Equations: derivation, basic invariants and formulae
Euler Equations: erivation, basic invariants an formulae Mat 529, Lesson 1. 1 Derivation The incompressible Euler equations are couple with t u + u u + p = 0, (1) u = 0. (2) The unknown variable is the
More informationLeast-Squares Regression on Sparse Spaces
Least-Squares Regression on Sparse Spaces Yuri Grinberg, Mahi Milani Far, Joelle Pineau School of Computer Science McGill University Montreal, Canaa {ygrinb,mmilan1,jpineau}@cs.mcgill.ca 1 Introuction
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationTOWARDS THERMOELASTICITY OF FRACTAL MEDIA
ownloae By: [University of Illinois] At: 21:04 17 August 2007 Journal of Thermal Stresses, 30: 889 896, 2007 Copyright Taylor & Francis Group, LLC ISSN: 0149-5739 print/1521-074x online OI: 10.1080/01495730701495618
More information8.022 (E&M) Lecture 19
8. (E&M) Lecture 19 Topics: The missing term in Maxwell s equation Displacement current: what it is, why it s useful The complete Maxwell s equations An their solution in vacuum: EM waves Maxwell s equations
More informationWitten s Proof of Morse Inequalities
Witten s Proof of Morse Inequalities by Igor Prokhorenkov Let M be a smooth, compact, oriente manifol with imension n. A Morse function is a smooth function f : M R such that all of its critical points
More informationIntroduction to. Crystallography
M. MORALES Introuction to Crystallography magali.morales@ensicaen.fr Classification of the matter in 3 states : Crystallise soli liqui or amorphous gaz soli Crystallise soli : unique arrangement of atoms
More informationConvective heat transfer
CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case,
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this
More informationPhysics 2212 K Quiz #2 Solutions Summer 2016
Physics 1 K Quiz # Solutions Summer 016 I. (18 points) A positron has the same mass as an electron, but has opposite charge. Consier a positron an an electron at rest, separate by a istance = 1.0 nm. What
More informationA look at Einstein s clocks synchronization
A look at Einstein s clocks synchronization ilton Penha Departamento e Física, Universiae Feeral e Minas Gerais, Brasil. nilton.penha@gmail.com Bernhar Rothenstein Politehnica University of Timisoara,
More informationObjective: To introduce the equations of motion and describe the forces that act upon the Atmosphere
Objective: To introuce the equations of motion an escribe the forces that act upon the Atmosphere Reaing: Rea pp 18 6 in Chapter 1 of Houghton & Hakim Problems: Work 1.1, 1.8, an 1.9 on pp. 6 & 7 at the
More informationTEMPORAL AND TIME-FREQUENCY CORRELATION-BASED BLIND SOURCE SEPARATION METHODS. Yannick DEVILLE
TEMPORAL AND TIME-FREQUENCY CORRELATION-BASED BLIND SOURCE SEPARATION METHODS Yannick DEVILLE Université Paul Sabatier Laboratoire Acoustique, Métrologie, Instrumentation Bât. 3RB2, 8 Route e Narbonne,
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationCalculus in the AP Physics C Course The Derivative
Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.
More informationThe Principle of Least Action and Designing Fiber Optics
University of Southampton Department of Physics & Astronomy Year 2 Theory Labs The Principle of Least Action an Designing Fiber Optics 1 Purpose of this Moule We will be intereste in esigning fiber optic
More informationSOLUTIONS for Homework #3
SOLUTIONS for Hoework #3 1. In the potential of given for there is no unboun states. Boun states have positive energies E n labele by an integer n. For each energy level E, two syetrically locate classical
More informationAll s Well That Ends Well: Supplementary Proofs
All s Well That Ens Well: Guarantee Resolution of Simultaneous Rigi Boy Impact 1:1 All s Well That Ens Well: Supplementary Proofs This ocument complements the paper All s Well That Ens Well: Guarantee
More informationThe Press-Schechter mass function
The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationCharacterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
Characterizing Real-Value Multivariate Complex Polynomials an Their Symmetric Tensor Representations Bo JIANG Zhening LI Shuzhong ZHANG December 31, 2014 Abstract In this paper we stuy multivariate polynomial
More informationFundamental Laws of Motion for Particles, Material Volumes, and Control Volumes
1 Funamental Laws of Motion for Particles, Material Volumes, an Control Volumes Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambrige, MA 02139, USA March 2003
More informationALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS
ALGEBRAIC AND ANALYTIC PROPERTIES OF ARITHMETIC FUNCTIONS MARK SCHACHNER Abstract. When consiere as an algebraic space, the set of arithmetic functions equippe with the operations of pointwise aition an
More informationLecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.
b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference
More informationThe Kepler Problem. 1 Features of the Ellipse: Geometry and Analysis
The Kepler Problem For the Newtonian 1/r force law, a miracle occurs all of the solutions are perioic instea of just quasiperioic. To put it another way, the two-imensional tori are further ecompose into
More informationTime-of-Arrival Estimation in Non-Line-Of-Sight Environments
2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor
More informationLower Bounds for the Smoothed Number of Pareto optimal Solutions
Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.uni-bonn.e, heiko@roeglin.org Abstract.
More informationPH 132 Exam 1 Spring Student Name. Student Number. Lab/Recitation Section Number (11,,36)
PH 13 Exam 1 Spring 010 Stuent Name Stuent Number ab/ecitation Section Number (11,,36) Instructions: 1. Fill out all of the information requeste above. Write your name on each page.. Clearly inicate your
More informationPhysics 115C Homework 4
Physics 115C Homework 4 Problem 1 a In the Heisenberg picture, the ynamical equation is the Heisenberg equation of motion: for any operator Q H, we have Q H = 1 t i [Q H,H]+ Q H t where the partial erivative
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More information3 The variational formulation of elliptic PDEs
Chapter 3 The variational formulation of elliptic PDEs We now begin the theoretical stuy of elliptic partial ifferential equations an bounary value problems. We will focus on one approach, which is calle
More informationMany problems in physics, engineering, and chemistry fall in a general class of equations of the form. d dx. d dx
Math 53 Notes on turm-liouville equations Many problems in physics, engineering, an chemistry fall in a general class of equations of the form w(x)p(x) u ] + (q(x) λ) u = w(x) on an interval a, b], plus
More informationChapter 1. Maxwell s Equations
Chapter 1 Maxwell s Equations 11 Review of familiar basics Throughout this course on electrodynamics, we shall use cgs units, in which the electric field E and the magnetic field B, the two aspects of
More informationPhysics 110. Electricity and Magnetism. Professor Dine. Spring, Handout: Vectors and Tensors: Everything You Need to Know
Physics 110. Electricity and Magnetism. Professor Dine Spring, 2008. Handout: Vectors and Tensors: Everything You Need to Know What makes E&M hard, more than anything else, is the problem that the electric
More informationEnergy Splitting Theorems for Materials with Memory
J Elast 2010 101: 59 67 DOI 10.1007/s10659-010-9244-y Energy Splitting Theorems for Materials with Memory Antonino Favata Paolo Poio-Guiugli Giuseppe Tomassetti Receive: 29 July 2009 / Publishe online:
More information