Maths
EE311 Summer 2018 Name _____________________________________________ Final Time started _________ Time Finished___________ Please note:
• You can use Calculators all kinds • Please do all of the problems (There are 4 problems) • You have 90 minutes to do these problems • You can use extra paper, and before handing the test in, put it in an
organized form the way you want the grader to see it. • You can use your books and notes and calculators, in most cases you are
better off not giving calculated numbers. ( is a better answer than 1.7….) • Please write clearly and show your work as clearly as you can • If you use other paper, please put the test together in order and identify each
paper that you are adding Problem 1 _________________________________________ ________________ Problem 2 _________________________________________ ________________ Problem 3 _________________________________________ ________________ Problem 4 _________________________________________ ________________ Problem 5 _________________________________________ ________________ Total _______________ Comments: (To be used while grading)
3
EE311 Summer 2018 Name _____________________________________________ Final Problem 1(25 points)
a) (10 points) Find the electric flux density for all points in space for the given charge distribution
b) (5 points) Discuss the boundary conditions of your result for r=a and r=c
Charge distribution=
⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑄 𝐶 𝑟 = 0 )*+ ,
- ./
𝑎 < 𝑟 < 𝑏 0 𝑏 < 𝑟 < 𝑐 𝐾 -
.5 𝑐 < 𝑟 < 𝑑
0 𝑑 < 𝑟
EE311 Summer 2018 Name _____________________________________________ Final Problem 1 continue but this is an independent problem
c) (10 points) A given charge distribution created the following Electric Field intensity for r>b. Can you find the charge distribution that would result in this field? If you cannot please explain why and if you can show the detailed work
𝑀8 ,/ 5𝜖8
�̂� − 𝑆?𝑏@
𝜖8𝑟@ 𝑟 A
𝑉 𝑚
EE311 Summer 2018 Name _____________________________________________ Final Problem 2 (25 points) a) (10 Points) Vector A is the vector that connects P1: (1, 2, 5) to P2: (3,2,5) (the vector starts at P1 and ends on P2.) P1 and P2 are given in Cartesian coordinates. Find Vector A in spherical coordinate, and then evaluated at point (x=0, y=1, z=0)
b) (5 Points) Transform �̂� into Cartesian coordinate on the xy plane, where q=90.
c) (10 Points) ∫ E2�̂� − 3𝜙I + �̂�L 𝑑𝜙 MNO PQO (assume q=90 degrees). Clearly utilize what you know
about these vectors and coordinate systems and evaluate the above integral. Show your work clearly.
EE311 Summer 2018 Name _____________________________________________ Final Problem 3 (25 points) In the following problem you know that the length of the line is 36𝜆/4, the negative traveling part of the voltage at the load is -50 volts. The positive traveling current at the input of the line is 2/3 amperes. The characteristic impedance is 150 ohms. Can you find the following? If you cannot explain why and if you can show your detailed work
a) (5 points) The load impedance b) (5 points) The input impedance of the line c) (5points) Vg d) (5points) The total voltage at the middle of the line
(5points) Average power delivered at the input of the line
Vg
ZL
Rg=75 ohms
Z0
x=xin
EE311 Summer 2018 Name _____________________________________________ Final Problem 4 (25 points)
A coaxial line can be modeled as two thin conducting shells with conducting cylindrical shells at r=c and r=d (c<d). The currents are surface currents. Assuming that the surface current on the inner one at r=c is S A/m in �̂� direction and the one at r=d is X in −�̂� direction. a) (10 points) Find H field for all points in space b) (5 points) Show that the magnetic boundary conditions match at all boundaries
Problem 4 continued, but the next 2 parts are not connected to the first 2 c) (5 point) There is a current sheet of -3𝒚A 𝒎𝑨
𝒎 on the plane -3x+2z=5. Find the field on each side of
the plane indicate H1 on the -3x+2y>5 and H2 on -3+2z<5 d) (5 points) Use the H field in part c and verify if the Magnetic boundary condition for tangential H
field is valid. Show your detailed work
z
c d
EE311 Summer 2018 Name _____________________________________________ Final Problem 5 (25 points) An electromagnetic wave is propagating though a media The electric field is given in time domain evaluated at z= -2 m as
𝐸Z⃗ = 10sin(10a𝑡 − 4)𝑥e .f .
Find a) (5 points) Find beta and the speed of light in the media b) (5 points) Is this a positive or negative traveling wave? Explain your answer clearly. c) (5 point) Find E in phasor form as a function of z d) (5points) Find associated H by using Maxwell’s equations in the time-domain (assume 𝜂) e) (5 points) Find the average Poynting vector for this wave.
EE311 Summer 2018 Name _____________________________________________ Final