Documents

Phy 331 Advanced elsctrodynamics.pdf

Categories
Published
of 71
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Description
PHY331: Advanced Electrodynamics & Magnetism Part I: Electrodynamics Contents 1 Field theories and vector calculus 3 1.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Second derivatives . . . . .
Transcript
  PHY331: Advanced Electrodynamics & Magnetism Part I: Electrodynamics Contents 1 Field theories and vector calculus 3 1.1 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Second derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Electromagnetic forces, potentials, Maxwell’s equations 13 2.1 Electrostatic forces and potentials . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Magnetostatic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Electrodynamics and Maxwell’s equations . . . . . . . . . . . . . . . . . . . 162.4 Charge conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 The vector potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Electrodynamics with scalar and vector potentials 21 3.1 Scalar and vector potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 A particle in an electromagnetic field . . . . . . . . . . . . . . . . . . . . . . 22 4 Electromagnetic waves and Poynting’s theorem 25 4.1 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2  Complex  fields? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Poynting’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Momentum of the electromagnetic field 316 Multipole expansion and spherical harmonics 34 6.1 Multipole expansion of a charge density . . . . . . . . . . . . . . . . . . . . 346.2 Spherical harmonics and Legendre polynomials . . . . . . . . . . . . . . . 366.3 Multipole expansion of a current density . . . . . . . . . . . . . . . . . . . 37 7 Dipole fields and radiation 41 7.1 Electric dipole radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Magnetic dipole radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.3 Larmor’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  PHY331  PART  I: A DVANCED  E LECTRODYNAMICS  L ECTURE  0 8 Electrodynamics in macroscopic media 47 8.1 Macroscopic Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . 478.2 Polarization and Displacement fields . . . . . . . . . . . . . . . . . . . . . . 478.3 Magnetization and Magnetic induction . . . . . . . . . . . . . . . . . . . . 49 9 Waves in dielectric and conducting media 52 9.1 Waves in dielectric media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529.2 Waves in conducting media . . . . . . . . . . . . . . . . . . . . . . . . . . . 559.3 Waves in plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 10 Relativistic formulation of electrodynamics 59 10.1 Four-vectors and transformations in Minkowski space . . . . . . . . . . . . 5910.2 Covariant Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.3 Invariant quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 A Special coordinates and vector identities 67 A.1 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.2 Integral theorems and vector identities . . . . . . . . . . . . . . . . . . . . . 69A.3 Levi-Civita tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 B Unit systems in electrodynamics 70 Literature The lecture notes are intended to be mostly self-sufficient, but below is a list of recom-mended books for this course:1. R.H. Good,  Classical Electromagnetism , Saunders College Publishing (1999). Mostof the material in this course is covered in this book. It is very accessible andprobably should be your first choice to look something up.2. J.D. Jackson,  Classical Electrodynamics , Wiley (1998). This is the standard work onclassical electrodynamics, and it has everything that is covered in this course. Thelevel is quite high, but it will answer your questions.3. R.P. Feynman,  Lectures on Physics , Addison-Wesley (1964). Probably the best gen-eral books on physics ever. The emphasis is on the physical intuition, and it iswritten in a very accessible narrative.4. J. Schwinger et al.,  Classical Electrodynamics , Westview Press (1998). This is quite ahigh-level textbook, containing many topics. It has detailed mathematical deriva-tions of nearly everything.After each lecture there is a detailed “further reading” section that points you to therelevant chapters in various books.2  PHY331  PART  I: A DVANCED  E LECTRODYNAMICS  L ECTURE  1 1 Field theories and vector calculus 1.1 Field theory Michael Faraday(1791-1867)Electrodynamics is a theory of   fields , and all matter entersthe theory in the form of   densities . All modern physicaltheories are field theories, from general relativity to thequantum fields in the standard model and string theory.Therefore, apart from learning some important topics inelectromagnetism, in this course you will aquire an under-standing of modern field theories without having to dealwith the strangeness of quantum mechanics or the math-ematical difficulty of general relativity. Fields were intro-duced by Michael Faraday, who came up with “lines of force” to describe magnetic phenomena.You will be familliar with particle theories such as clas-sical mechanics, where the fundamental object is char-acterised by a position vector and a momentum vector.Ignoring the possible internal structure of the particles,they have six degrees of freedom (three position and threemomentum components). Fields, on the other hand, arecharacterised by an  infinite  number of degrees of freedom.Let’s look at some examples: A vibrating string:  Every point  x  along the string has a displacement  r , which is adegree of freedom. Since there are an infinite number of points along the string, thedisplacement  r ( x )  is a field. The argument  x  denotes a location on a line, so we call thefield  one-dimensional . Landscape altitude:  With every point on a surface  ( x ,  y ) , we can associate a numberthat denotes the altitude  h . The altitude  h ( x ,  y )  is a two-dimensional field. Since thealtitude is a scalar, we call this a  scalar  field. Temperature in a volume:  At every point  ( x ,  y ,  z )  in the volume we can measure thetemperature  T  , which gives rise to the three-dimensional scalar field  T  ( x ,  y ,  z ) .Mathematically, we denote a field by  F ( r , t ) , where the value of the field at position r  and time  t  is given by the quantity  F . This quantity can be anything: if   F  is a scalar, wespeak of a scalar field, and if   F  is a vector we speak of a vector field. In  quantum  fieldtheory, the mathematical object that makes the quantum field are  operators  acting on avacuum state.In this course we will be mostly dealing with scalar and vector fields, but occasion-ally we will encounter  tensor  fields. A common example of a tensor field that you mayhave encountered is the stress in a material. We will discuss the difference betweentensors and ordinary matrices in a moment.3  PHY331  PART  I: A DVANCED  E LECTRODYNAMICS  L ECTURE  1 1.2 Vector calculus In mechanics, a particle does not randomly jump around in phase space, but followsequations of motion determined by the laws of mechanics and the boundary conditions.These equations of motion typically involve derivatives, namely the velocity and accel-eration of the particle. The derivatives tell you how much a quantity changes. Likewise,fields obey equations of motion, and we need to define the derivatives of fields.First, take the scalar field. The interesting aspect of such a field is how the values of the field change when we move to neighbouring points in space, and in what directionthis change is maximal. For example, in the altitude field (with constant gravity) thischange determines how a ball would roll on the surface, and for the temperature fieldit determines how the heat flows.It is easy to see that both a rolling ball and heat flow have a magnitude and a direc-tion. The measure of change of a scalar must therefore be a  vector . Since the change isdefined at every point  r  (and time  t ), it is a  vector field . Let the scalar field be denoted by  f  ( x ,  y ,  z , t ) . Then the change in the  x  direction (denoted by ˆ i ) is given bylim h → 0  f  ( x  +  h ,  y ,  z , t ) −  f  ( x ,  y ,  z , t ) h ˆ i  =  ∂  f  ( x ,  y ,  z , t ) ∂ x ˆ i . (1.1)  A x ( r  +  l  ˆ i )  A x ( r )  A  y ( r )  A  y  A  z ( r )  A  z ( r  +  l  ˆ k ) lll r Figure 1:  The  divergence  of a vector field  A can be found by considering the total flux of  A throughthefacesofasmallcubeofvolume l 3 at point  r  =  x ˆ i  +  y ˆ j  +  z  ˆ k . Similar expressions hold for the change inthe  y  and  z  direction, and in general the spa-tial change of a scalar field is given by ∂  f  ∂ x ˆ i +  ∂  f  ∂  y ˆ j +  ∂  f  ∂  z ˆ k  = ∇  f   ≡ grad  f   , (1.2)called the  gradient  of   f  . The “nabla” or “del”symbol  ∇  is a differential  operator , and it isalso a vector: ∇ =  ˆ i  ∂∂ x  +  ˆ j  ∂∂  y  +  ˆ k  ∂∂  z  . (1.3)Clearly, this makes a vector field out of ascalar field. Note that we do not includechanges over time in the gradient. We firststudy  static  fields.When we want to describe the behaviourof vector fields, there are two main concepts:the  divergence and the  curl . The divergence isa measure of flow into, or out of, a volumeelement. Consider a volume element  dV   = l 3 at point  r  =  x ˆ i  +  y ˆ j  +  z  ˆ k  and a vector field  A ( x ,  y ,  z ) , as shown in figure 1. Weassume that  l  is very small. The difference of the flux of the field going into the volume4

CPD560AS CPD561ASV

Jul 23, 2017

2006 Syllabus

Jul 23, 2017
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks