• 5 Pages PHYS30642_Exam_2006
    PHYS30642_Exam_2006

    School: East Los Angeles College

    PC3642 ONE HOUR THIRTY MINUTES A list of constants is enclosed. UNIVERSITY Electrodynamics OF MANCHESTER 31st May 2006, 2.00 p.m. - 3.30 p.m. Answer ALL parts of question 1 and TWO other questions Electronic calculators may be used, provided t

  • 8 Pages Harmonically Varying Sources Part 1
    Harmonically Varying Sources Part 1

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 4. Harmonically Varying Sources 4.1 Hertzian Dipole The antenna is of length d and can be thought of as two charges q and q each varying as sint. z P r-r' r y -q +q dz d Thus q ( t ) = q 0 sin ( t ) and I ( t ) = dq = I0 cos (

  • 11 Pages Relativity Part 1
    Relativity Part 1

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 3. Relativistic Electromagnetism 3.1 Lorentz Transformations and Tensor Representation The aim is to demonstrate that the theory of electromagnetism is consistent with the special theory of relativity. Hendrik Lorentz, 1853-1

  • 1 Page Example Sheet 1
    Example Sheet 1

    School: East Los Angeles College

    Examples 1 1. Simplify the following expressions to obtain final results which contain no unnecessary dummy indices ( i.e. repeated indices which have not been assigned a numerical value). a 12 a 2 b 2i a i c ij a j d 3k kj e ij jk f ii 2. Demonstrate the

  • 1 Page Dirac Delta Exercise
    Dirac Delta Exercise

    School: East Los Angeles College

    Exercise on Dirac Delta Function In the following exercises prove various relations relating to Dirac's delta function. The following representation is used (although other representations are of course equally valid): x 1 lim 1 xa a exp-x 2 /a 2 1. d

  • 2 Pages PHYS30642 Examples Sheet 4
    PHYS30642 Examples Sheet 4

    School: East Los Angeles College

    Examples 4 1. Show that the Lorentz transformation matrix - 0 0 0 0 0 0 1 0 0 1 - 0 0 corresponds to a rotation through an angle around the yz plane in the four-dimensional Minkowski space, where tanh . Hint: a general rotation in a two-dimensional Minko

  • 3 Pages Relativity Part 2
    Relativity Part 2

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 3. Relativistic Electromagnetism 3.2 Lorentz 4-Vectors Are there any other genuine 4-vectors other than x = ( ct, x ) ? (Remember that A satisfies: A A = A ' A ' under an arbitrary LT A' = A ). Let us define the incremental:

  • 9 Pages EM Field Equations Part 3
    EM Field Equations Part 3

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 1. Electromagnetic Field Equations 1.3 Electric and Magnetic Multipoles Firstly we will look at the dipole field of a pair of charges and approximate the potential at large distances. We now evaluate the potential at point P. V

  • 3 Pages Example Sheet 2
    Example Sheet 2

    School: East Los Angeles College

    Examples 2 1. Consider an infinitely long positron beam of radius a, with a line charge density e Cm -1 traveling along the z-axis with a velocity v. a. By applying Gauss' law to a surface surrounding the beam, calculate the electric field E both inside a

  • 1 Page PHYS30642 Examples Sheet 3
    PHYS30642 Examples Sheet 3

    School: East Los Angeles College

    Examples 3 1. The retarded scalar potential for a moving charge with velocity v c is q V 1 , 4 0 R1 - . R ret where R is the vector linking the field point to the charge and R R/R. Show that for a charge moving at constant velocity v with position given b

  • 3 Pages Example of Spherical Shell Cos2Theta
    Example Of Spherical Shell Cos2Theta

    School: East Los Angeles College

    Example: Here we solve for the potential for a spherical shell of surface charge density = 0 cos(2) located at r=a and where e=0 for all space. The techniques employed to solve for this charge density are very similar to those used in the Q8 on example sh

  • 4 Pages Tensors Overview
    Tensors Overview

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 Overview of Tensors Contravariant and Covariant Vectors Rotation in 2-D space : x ' = cos x + sin y y ' = - sin x + cos y To facilitate generalization, replace (x, y) with (x1, x2) Prototype contravariant vector: dr = (dx1, dx2

  • 3 Pages Introduction to Greens Functions
    Introduction To Greens Functions

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 Introduction to Green's Functions By way of an introduction, we consider the Green's function for Newton's force equation mx = F The Green's function equation for this is defined by: G=(t-t') The initial conditions are G(t1 , t

  • 6 Pages Radiation and Lienard Wiechert Potentials Part 1
    Radiation And Lienard Wiechert Potentials Part 1

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials 2.1 Introduction to Radiation from Accelerated Charges General non-stationary time dependent potentials and fields are given by solutions of the inhomogeneous wave equation for A and V. . i

  • 6 Pages PHYS30642_Exam_2007
    PHYS30642_Exam_2007

    School: East Los Angeles College

  • 7 Pages Radiation and Lienard Wiechert Potentials Part 3
    Radiation And Lienard Wiechert Potentials Part 3

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials 2.3 Radiation from a charged particle with acceleration parallel to velocity Prior to studying the radiation produced by a moving point charge we present the General theory of radiati

  • 10 Pages Solutions to Example Sheet 2
    Solutions To Example Sheet 2

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS30642 8. a) For a surface charge density = 0 cos, the boundary conditions yield: 1 Vout Vin =- 0 cos r - r 0 r a 1 or : E out,r - E in,r r a = + 0 cos 0 Also, Dout,r - Din,r r a = 0 cos ^ ^ Re call from your notes, D1.n - D 2 .n = s b

  • 7 Pages PHYS 30642 Example Sheet 3 SOLUTIONS
    PHYS 30642 Example Sheet 3 SOLUTIONS

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS30642 PHYS 30642 Example Sheet 2 SOLUTIONS 1. Radiation and Lienard-Wiechert potentials V= 1 q ^ 4 0 R r (1 - .R r ) Re lated coordinate: R r = R p + c(t - t ret ) also, R r = c(t - t ret ) ^ R r = R p + R r or R p = R r (R r - ) The

  • 10 Pages EM Field Equations Part 2
    EM Field Equations Part 2

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 1. Electromagnetic Field Equations 1.2 Laplace and Poisson Equations Recall the Divergence form of Maxwell's equations: .D = and in vacuum D=0 E and .D = 2 V = - / 0 1 3 (r ') d r r -r' 40 V' We already know the solution to thi

  • 3 Pages Proof of Electric and Magnetic Field Lienard Wiechert
    Proof Of Electric And Magnetic Field Lienard Wiechert

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 Proof of Lienard-Wiechert Electric and Magnetic Field Equations for Point charges Starting with: V(r, t) = q 1 40 R - .R ( ) ret A(r, t) = 0 qc []ret 4 R - .R ( ) = ret []ret c V(r, t), where R = r - rr (rr is the retarded pos

  • 7 Pages Relativity Part 3
    Relativity Part 3

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 3. Relativistic Electromagnetism 3.3 Electromagnetic Field Tensor How do E and B fields transform under a LT? They cannot be 4-vectors, but what are they? We again re-write the fields in terms of the scalar and vector potentia

  • 2 Pages Poyntings Theorem
    Poyntings Theorem

    School: East Los Angeles College

    Poynting's Theorem Here we study the energy transported by the e.m. field. The work done dW by the e.m. field on charge dq, contained in volume d3r moving through the field with velocity v when it is displaced through a distance dl: dW = dq E + vxB .d l =

  • 2 Pages Potentials and Time Varying Fields
    Potentials And Time Varying Fields

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 Potentials, Time-Varying Fields and Gauge Invariance We begin by considering Faraday's law and replace the magnetic field by the curl of the vector potential xE = - B = - xA t t Here we have used .B = 0 B=xA. This is in fact no

  • 8 Pages Radiation and Lienard Wiechert Potentials Part 2
    Radiation And Lienard Wiechert Potentials Part 2

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials 2.2 Lienard-Wiechert Potentials and Point Charges Retarded Potentials and the Wave Equation We have arrived at a modified form of the vector and scalar potentials in terms of a charge densit

  • 7 Pages Radiation and Lienard Wiechert Potentials Part 3
    Radiation And Lienard Wiechert Potentials Part 3

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials 2.3 Radiation from a charged particle with acceleration parallel to velocity Prior to studying the radiation produced by a moving point charge we present the General theory of radiation As

  • 10 Pages EM Field Equations Part 1
    EM Field Equations Part 1

    School: East Los Angeles College

    ELECTRODYNAMICS: PHYS 30642 1. Electromagnetic Field Equations 1.1 Maxwell's Equations Analysis in free space (vacuum). Coulomb Born June 14, 1736 Angoulme, France Died August 23, 1806 Paris, France In 1785 Coulomb presented his three reports on Electrici

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