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East Los Angeles College - PHYS - 10302
Lecture 10: Waves at Boundaries and Standing Waves1. Wave speed Wave motion can be written in various forms sin(kx - t) cos(kx - t) ei(kx-t) sin(k(x - vt) sin(2(x/ - f t).The speed of the wave is always given by the ratio of whatever multiplies t
East Los Angeles College - PHYS - 10302
Lecture 12: Music1. Audible Frequencies Human hearing spans the range from tens of Hz to a few kHz. The lowest audible frequency is around 16 Hz, and the piano goes down to 27 Hz. The human voice, for an exceptional bass singer, can get to 44 Hz, an
East Los Angeles College - PHYS - 10302
Lecture 20: Interference and Diffraction1. Double (Young's) slits.d d SinConsider a plane wave falling on two slits. The circular wave patterns from the two slits interfere. The key to understanding is to consider the signal that goes in a parti
Cal Poly - CIS - 122
Naval Research LaboratoryWashington, DC 20375-5320UNIX Tools Course NotesInstructor: Michael G. Vonk Center for Computational Science (202)767-3884 michael.vonk@nrl.navy.milUNIX Tools1. 2. Introduction ..1 Bourne Shell Scripts.2 2.1. Shell Sc
East Los Angeles College - LCC - 12671
PROGRAMME DETAIL SPECIFICATION Programme Summary1 Awarding institution 2 Teaching institution university 3a Programme accredited by: 3b Description of accreditation 4 Final award 5 Programme title 6 UCAS code 7 Subject benchmark statement 8 Educati
East Los Angeles College - LSS - 11398
PROGRAMME DETAIL SPECIFICATION Programme Summary1 Awarding institution 2 Teaching institution university 3a Programme accredited by: 3b Description of accreditation 4 Final award 5 Programme title 6 UCAS code 7 Subject benchmark statement 8 Educati
Colorado - HIST - 1010
Lecture 8: The Roman Empire and Its Frontiers Recap: Augustus and the Making of Empire 1. Pax Romana Peace and its Effects A. Stability in an Imperial State B. The Five Good Emperor C. The Spread of Roman Culture 2. Being Roman: A Mediterranean Iden
Cal Poly - AGB - 452
Project Requirements AGB 452 Dr. Sean Hurley Due: 5/28/09 Project Requirements You will be required to submit at the end of week nine (5/28/09) a group paper, which will represent a culmination of the lessons learned from the course. This will be an
Colorado - CVEN - 3313
CVEN 3313 Theoretical Fluid MechanicsProblem Set #3Assigned: 2/2/06 Due: 2/9/06 Show all of your work, and please turn in a neat problem set. Some of the problems here will require that you have done this week's reading. Note: Answers to even-numbe
East Los Angeles College - PHYS - 30642
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 1. Electromagnetic Field Equations1.1 Maxwell's Equations Analysis in free space (vacuum). Coulomb Born June 14, 1736 Angoulme, France Died August 23, 1806 Paris, FranceIn 1785 Coulomb presented his three reports on Ele
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 Note on the Relation Between Vector Potential and Biot-SavartFrom the postulate that the divergence of the magnetic field is zero, or .B = 0, then B can be expressed as the curl of another vector, say A:B = xAThe vecto
East Los Angeles College - PHYS - 30642
Poynting's TheoremHere 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 .
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 Potentials, Time-Varying Fields and Gauge InvarianceWe begin by considering Faraday's law and replace the magnetic field by the curl of the vector potentialxE = - B = - xA t tHere we have used .B = 0 B=xA. This is in
East Los Angeles College - PHYS - 30642
Exercise on Dirac Delta FunctionIn 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 aexp-x 2 /a 2
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 Introduction to Green's FunctionsBy way of an introduction, we consider the Green's function for Newton's force equationmx = F The Green's function equation for this is defined by: G=(t-t') The initial conditions are G(t
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 1. Electromagnetic Field Equations1.2 Laplace and Poisson EquationsRecall 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 solut
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 1. Electromagnetic Field Equations1.3 Electric and Magnetic MultipolesFirstly 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
East Los Angeles College - PHYS - 30642
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 exam
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials2.1 Introduction to Radiation from Accelerated ChargesGeneral non-stationary time dependent potentials and fields are given by solutions of the inhomogeneous wave equation for A and V.
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials2.2 Lienard-Wiechert Potentials and Point Charges Retarded Potentials and the Wave EquationWe have arrived at a modified form of the vector and scalar potentials in terms of a charge d
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 Proof of Lienard-Wiechert Electric and Magnetic Field Equations for Point chargesStarting with:V(r, t) = q 1 40 R - .R()retA(r, t) = 0 qc []ret 4 R - .R()=ret[]retcV(r, t),where R = r - rr (rr is
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 2. Radiation and Retarded Potentials2.3 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
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 3. Relativistic Electromagnetism3.1 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, 1
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS 30642 3. Relativistic Electromagnetism3.2 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 th
East Los Angeles College - PHYS - 30642
Examples 11. 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. De
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS306428. 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 n
East Los Angeles College - PHYS - 30642
Examples 31. The retarded scalar potential for a moving charge with velocity v c is q V 1 , 4 0 R1 - . Rretwhere 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 posi
East Los Angeles College - PHYS - 30642
ELECTRODYNAMICS: PHYS30642PHYS 30642 Example Sheet 2 SOLUTIONS1. Radiation and Lienard-Wiechert potentialsV= 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
East Los Angeles College - PHYS - 30642
Examples 41. Show that the Lorentz transformation matrix - 0 0 0 0 0 0 1 0 0 1- 0 0corresponds 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-di
Colorado - ECEN - 5458
HOMEWORK ASSIGNMENT #1ECEN 5458 Digital Control Prof D. G. Meyer September 11, 2008Due: See website 1. The Fibonacci sequence is f (k) = 1, 1, 2, 3, 5, 8, 13, 21, . . . where each value is the sum of the proceeding two (after the initial samples o