# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

2 Pages

### 329sp10hw7

Course: ECE 329, Spring 2010
School: University of Illinois,...
Rating:

Word Count: 617

#### Document Preview

329 1. ECE Verify that vector identity Homework 7 H Due: March 9, 2010, 5PM EE H= (E H) holds for E = 4e and H = 3e by expanding both sides of the identity. Treat as a real x y constant. You should download the table of vector identities from the ECE 329 web site and examine the list to familiarize yourself with the listed identities they are widely employed in electromagnetics as well as in other...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Illinois >> University of Illinois, Urbana Champaign >> ECE 329

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
329 1. ECE Verify that vector identity Homework 7 H Due: March 9, 2010, 5PM EE H= (E H) holds for E = 4e and H = 3e by expanding both sides of the identity. Treat as a real x y constant. You should download the table of vector identities from the ECE 329 web site and examine the list to familiarize yourself with the listed identities they are widely employed in electromagnetics as well as in other branches of engineering such as uid dynamics. 2. Charge conservation states that the net outward ux of current density from a volume V through its bounding surface S equals the time rate of decrease of net charge contained within the volume: z z J dS = S d dt dV V Note that applying the Divergence theorem yields the dierential form: J= d dt a) Calculate J dS for the surface of a cubic volume V = 8 m centered at the origin if the current density is J = 2xx + 3x y + 4z (y 1) z A/m . b) Is the total charge contained in the cube increasing, decreasing, or neither? c) What are the physical units of the coecients 2, 3, and 4 used in dening J. d) Find the charge density at the origin, (0, 0, 0, t) as a function of time, if (0, 0, 0, 0) = 0. 3. Consider a homogeneous conductor where J = E. a) Use Gauss's law E = and the continuity equation J = to derive the dierential equation: 3 S 2 2 2 2 t o t + =0 for the charge density . b) Find the solution of the dierential equation above for t > 0 if at t = 0 the charge density is (x, y, z, 0) = sin(100x) C/m over all space, where is positive. c) Find the current density J(x, y, z, t) by using the continuity equation, considering the problem's geometry, and assuming that there are no external applied E elds. d) According to solution the found in part (b), how long would it take for to reduce to 0.01 sin(100x) C/m ? Assume that = 5.8 10 S/m. e) Explain why the magnitude of the net charge decreases everywhere, by using the results of part (c) to discuss specically the ow of current at x = 0 and x = . 0 3 o 0 0 3 7 100 1 4. In this problem, we will study the propagation of waves in free space. In a source-free region where J = 0 and = 0, Maxwell's equations reduce to: E=0 B t B=0 E B = 0 0 t E= For E = xE (z, t), where E (z, t) is an arbitrary function of coordinates z and t only, a) Show that Gauss' Law is satised. b) Show that E = y . c) Show that ( E) = x . d) Take the curl of Faraday's law and substitue in Ampere's law to show that: x. ( E) = e) Combine (c) and (d) to derive the wave equation: x x Ex z 2 Ex z 2 2 Ex 0 0 t2 2 Ex 2 Ex 0 0 =0 z 2 t2 5. Let E = xE (z, t) as in problem 4 above, where E (z, t) = E cos(t z + ) and E (amplitude), (radian frequency), (wavenumber), and (phase shift) are positive real scalars. These elds describe waves polarized in the x direction and propagating in the z directions, respectively. a) Show that the specied E satises the wave equation derived above if: x x 0 0 x = 0 0 = c where c 3 10 m/s is the speed of light in free space. b) Substitute the specied E into Faraday's law and solve for B(z, t) assuming that B(0, 0) = 0. What is a possible value for ? Show that your answer can be expressed as B = y (be careful about maintaining the correct order for the upper and lower signs). c) Use B = H to show that H = y in terms of an appropriately dened constant . What is in terms of and ? 8 x Ex c Ex 0 0 0 0 0 0 2
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

University of Illinois, Urbana Champaign - ECE - 329
ECE 329Homework 8Due: March 16, 2010, 5PM1. Free space exists in the upper half space z &gt; 0. An innite dielectric slab with permittivity = 5 0and permeability = 6 0 occupies the lower half space z &lt; 0. There is a surface charge density5S 0 = 4C/m2 a
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Homework 9Due: March 30, 2010, 5PM1. For each of the ve plane waves (in free space) described byza) E = 2 cos(t + y) V/mb) E = 10 cos(t y) 10 sin(t y) V/mxzc) H = cos(t + z + ) + sin(t + z ) A/mxyd) H = 3 cos(t + x + ) + 3cos(t x ) A/m
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Spring 2010Homework 9 - SolutionDue: Mar. 30, 20101.a) For plane waves propagating in free space, E and H are related byE = 0 H ,where is the unit vector in the wave propagation direction. Therefore,H1 =11(y ) E1 =[x 2cos ( t + y )] (A/
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Homework 10Due: April 6, 2010, 5PM1. A monochromatic plane wave described by:E = 3 cos(t z ) + 3 sin(t z )xyVmis propagating in vacuum in the +z direction and is incident on the z = 0 plane which happens to bethe boundary of a perfect die
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Homework 11Due: April 13, 2009, 5PMZo = 75 , length l = 100 m, and propagation velocityf (t) with an internal resistance Rg = Zo is connected toone end of the T.L. (at z = 0) and the other end (z = l ) is terminated by a load resistance RL = 3
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Spring 2010Homework 11 - SolutionDue: Apr. 13, 20101.a) Injection coecient can be calculated according to the denition on Page 4, Lecture 28S Z01Z0==,Rg + Z0Z0 + Z02and the reection coecients are:L =S =RL Z0RL +Z0Rg Z0Rg +Z0=
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Homework 12Due: Tue, Apr 20, 2010, 5 PM1. Consider a lossless TL which is open circuited at the source end and shorted at the load end. Ifm is the length of the line, andv = 2 c = 2 1083l = 20m/s for the line,a) What are all the resonance
University of Illinois, Urbana Champaign - ECE - 329
ECE 329Spring 2010Homework 12 - SolutionDue: Apr. 20, 20101.a) The voltage and current phasors can be assumed asV (z ) = V + ejz + V ejz,V jzV + jzeeI (z ) =Z0whereV+andVZ0correspond to the waves travelling along+ and z directions, res
Western Michigan - FIN - 3800
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - BUS - 4750
Western Michigan - FIN - 3730
\D\OFIN 3730April 10, 2012Kaiser Poll Finds Opinions on Medicare Proposal Are 'Malleable'The Kaiser Family Foundation has recently sent out a poll to 1,519 Americans, askingwhether or not they think that the Medicare system should continue as it is t
Western Michigan - FIN - 3730
,.- .o,0\ \/. &quot;&amp;FIN 3730January 31, 2012New Tensions Over AMR's Pension Plan: Company Says Retiree Benefits to Be CutAMR Corp. has recently filed for Chapter 11 bankruptcy in November of2011.One of the maincriticisms of the corporation is that th
Western Michigan - FIN - 3730
/&gt;:.&gt;FCL3730 Retirement Planning and Employee BenefitsWestern Michigan UniversityProfessor SwisherSpring 2012 Quiz 1Ch 2,3,5,6~2.3.4.\~d\o4True-False (2 points each)1.NamFor someone who is four yea!s ~mretirement income needs:r-AtS&lt;~ret
Western Michigan - FIN - 3730
Problem (9 points. To receive credit, you must show your work.)8.You are assisting your client, John Wiley, with his retirement planning. John would like to have retirement income of75% of his current salary of \$70,000. John is currently 22 years from
Western Michigan - FIN - 3730
/Name~TG,i30FCL 3730 Retirement Planning and Employee BenefitsWestern Michigan UniversityProfessor SwisherSpring 2012 Quiz 3Chapters 16 - 21True-False (2 points each)1. ~2.3.4.Employer contributions to profit sharing plans can be integrated wi
Western Michigan - FIN - 3730
~CL-3730 Retirement Planning and Employee BenefitsWestern Michigan UniversityProfessor SwisherSpring 2012 Quiz 2Chapters 1, 7, 8, 10, 11, 13 - 15/&quot;-~alse.-r(2 points each)T~:30Nam-o-~\~1k1. Young employees and low-paid employees often prefe
Western Michigan - FIN - 3730
I11. Giraffes Unlimited offers its EEs a cash balance pension plan, with an annual pay credit of 9% and an interest creditof 6%. Sara, age 25, just joined the company. Her annual salary is \$50,000. The cost of a retirement annuity is\$112,959 per \$10,00
Western Michigan - FIN - 3730
/FCL 3730 Retirement Planning and EmployeeWestern Michigan UniversityProfessor SwisherSpring 2012 Quiz 4 Ch 28,31 - 38Benefits'(j\.\~O\0-Tru~e(2 points each)1._~_~For many plans, ERISA requires the employer to prepare a Summary Plan Descripti
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3300
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100
Western Michigan - FIN - 3100