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East Los Angeles College  PHYS 30642

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

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 rr' r y q +q dz d Thus q ( t ) = q 0 sin ( t ) and I ( t ) = dq = I0 cos (

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, 18531

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

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 expx 2 /a 2 1. d

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 fourdimensional Minkowski space, where tanh . Hint: a general rotation in a twodimensional Minko

Relativity Part 2
School: East Los Angeles College
ELECTRODYNAMICS: PHYS 30642 3. Relativistic Electromagnetism 3.2 Lorentz 4Vectors Are there any other genuine 4vectors other than x = ( ct, x ) ? (Remember that A satisfies: A A = A ' A ' under an arbitrary LT A' = A ). Let us define the incremental:

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

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 zaxis with a velocity v. a. By applying Gauss' law to a surface surrounding the beam, calculate the electric field E both inside a

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

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

Tensors Overview
School: East Los Angeles College
ELECTRODYNAMICS: PHYS 30642 Overview of Tensors Contravariant and Covariant Vectors Rotation in 2D 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

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=(tt') The initial conditions are G(t1 , t

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 nonstationary time dependent potentials and fields are given by solutions of the inhomogeneous wave equation for A and V. . i

PHYS30642_Exam_2007
School: East Los Angeles College

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

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

PHYS 30642 Example Sheet 3 SOLUTIONS
School: East Los Angeles College
ELECTRODYNAMICS: PHYS30642 PHYS 30642 Example Sheet 2 SOLUTIONS 1. Radiation and LienardWiechert 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

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

Proof Of Electric And Magnetic Field Lienard Wiechert
School: East Los Angeles College
ELECTRODYNAMICS: PHYS 30642 Proof of LienardWiechert 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

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 4vectors, but what are they? We again rewrite the fields in terms of the scalar and vector potentia

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 =

Potentials And Time Varying Fields
School: East Los Angeles College
ELECTRODYNAMICS: PHYS 30642 Potentials, TimeVarying 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

Radiation And Lienard Wiechert Potentials Part 2
School: East Los Angeles College
ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials 2.2 LienardWiechert 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

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

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