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Unformatted text preview: ECE 311 — Spring Quarter 2010
Final Exam June 9th, 2010 Write your name below and sign the honor pledge “No aid given, received, or observed” if it
applies. There are 3 problems on this exam. Exam is open book and notes. Please box or underline your
ﬁnal answers, and remember to include units. Be sure to show all work clearly if you wish to
obtain any partial credit. Try to keep your work within the provided space; use the back of a previous sheet if necessary.
A partial list of useful formulas is included below for your reference. Name: ' Egg “The Pledge”: Gauss” Law (integral form) 4[551'75' = [Hp VdV differential form: V  D = pV E ﬁeld path independence 41:? (TL = 0 differential form: V x E = 0
Potential difference: V(P2)— V(P1) = —f2E (71 80 = 8.854 x 10—12 F/m
P1 Capacitance Deﬁnition C = % Energy inacapacitor éCV2 Dielectric media D = 8E
Electrostatic boundary conditions E,1 = Et2 Dnl —Dn2 = pS
Ohm’s Law .7 = of? Resistance  Deﬁnition R = ; Uniform conductors: RC = 8/0' . . — _ 6 . . _ — _ 6
Charge conservat10n(1ntegral) d — ——a—IIHdeV dIfferentlal form. V J — —a—t(pV) Power dissipation in a volume P =  .7dV = oHﬂElde = gruff/lde Deﬁnitions of curl and divergence in cylindrical and spherical coordinates: see PS 7 or book
fix = —1 (—1 —I—1 ) forn>l = me)
n n _ 1 n — 1 — l a x rdesinefndrb = 4n rdefndtb = 21:2 0 0 0 0 Problem 1 (3 parts, 35 pointsl A capacitor is created from two concentric (and hollow) perfectly conducting spheres. The
inner sphere has radius 5 cm, while the outer has radius 15 cm. The space between the spheres is __ W30 ﬁlled with a dielectric material having a nonconstant permittivity of s —a 2
g R
spherical coordinate in meters. Assume total charges of +Q and —Q are‘distributed on the
surfaces of the inner and outer spheres, respectively. , where R is the Inner radius
5 cm Outer radius
15 cm (a) What components of the electric ﬁeld should exist in this problem, and what coordinates do
those components depend on? Be sure to specify the coordinate system you are using. (3’ points) (b) Find the electric ﬁeld vector in the region between the spheres in terms of Q  be sure to
specify both magnitude and direction. M points) ML (c) Find the capacitance of this device. Give a numerical answer. ($6 points)
C= &
Sb \x ‘
\/ “M 1* Rb v. “ e:— o an» (amp,
V=g%ﬁ Rea (Axum Ell/tpé‘dg h L“
UN: (wankM gawk wké‘
:D.\< .
\I —  “I " A
a Q Q .QML
K=M< "
:2 %E_(b,\)= Eg
an 3%
C: g e, 2 SC. “
ﬂit» b L3 W
5 f \x ‘ m
{5 5V 3‘, mbéctﬁnﬂ‘D‘ MY] wk ‘
W‘kﬁﬁ)“ “kw % \D h) *U‘Q‘L
w» mmxouuﬂ (8 9M3)
e— ' L, _
m w vim $9 5‘, w \‘D‘év/L
C: CQ°KK 3° VN—zcgﬂ 7— ‘0 Go 5 Problem 2 (4 parts, 30 points: The xy plane is an interface between free space (2 > 0) and a PEC material (2 < 0 ). An electric potential V = yz + z2 (Volts) exists in the free space region. (a) Find the electric ﬁeld associated with this potential in the free space region (8 points). TE; 5N ‘' ‘3?  gkal— 1%) v[m_ (b) Find the surface charge density on the PEC surface. (8 points) E‘Wm“ S’s/es on: 1:0 M£=Dl E:* a SQ §S=t€°a> (vw‘aﬁlASqu> N K J
W\V\O©W3v ‘QKXX 90V}; \owwxo ((Qr 37k) (c) Find the volumetric charge density in the free space region. (8 points) Q‘fD" iv $1 Eu: = %(‘3 7c 41, (wag) Q ‘5‘ 979131 3v 9v: ‘kb > (d) A point charge of 1 nC is moved from the point (x=0,y=0, z=1) to a point that is 10 m away.
Specify (x,y,z) coordinates for this point so that no work is required in performing this motion (6
points, note there are many correct answers.) Problem 3 7 arts 5 oints each : Circle the answer which is alwa s true.
(1* ch of the following is NOT true about the method of images? t can be used to ﬁeld the ﬁelds produced by charges in the presence of any PEC material
 when applicable, it shows that the ﬁeld outside the PEC material arises from the original charges and a set of “image” charges
(c) when considering a point charge above an inﬁnite PEC plane, it shows that ﬁeld above the plane is that of 3113 electric dipole
(d) when applicable, it ensures that there are no tangential electric ﬁelds on the PEC surface (e) none of the above (2) Given 2 = J‘cy, V x Z is ﬁx)
(a) 0 (b) 1 (c) 5c + j) I\‘(d) —§, (e) none of the above (3) The xy plane is a boundary between two dielectric materials: region 1 (z > 0 , ﬁeld E] ) that has a = 280 and region 2 (z < 0 , ﬁeld E2 ) that has a = 680. Which set of ﬁelds below satisﬁes
the appropriate boundary conditions?
a E1 = 2, E2 = 32
(b) ‘1 = 32,1?2 = 2
1 = 5:, E2 = 35c
(d)1721= 32,3 = 2 (e) none of the above (4) Which of the following statements is NOT true about electrostatic ﬁelds? (a) when computing a line integral to evaluate the potential difference between two points in free
space, any path connecting the two [email protected] can be used (b) the curl of the electric ﬁeld is zero (c) the divergence of the electric ﬁeld can be nonzero d the electric ﬁeld inside a perfectly conducting material must be zero
@me of the above
(5) An airﬁlled coaxial capacitor has length 10 cm, inner radius 2 mm and outer radius 1 cm. U r:  the nofringing approximation, the capacitance of this geometry is:
w: .46 pF (b) 10.7 pF (0) 824.3 pF (d) 0.35 nF (e) none of the above (6) A medium has conductivity 0' = 0.25 S/m. If an electric ﬁeld E = 45c + 22 V/m exists in this
medium, the resulting current density (in Amps/m2) is (a)5c+22 (b) 165c+82 (d) 42+ (7) The volume V bounded by surface S contains an amount of charge, Q
time. Which of the following statements must be true? (e) none of the above NIN) we , that is constant in . 6 current density on surface S must be zero e net current ﬂowing out of surface S must be zero
0) the current density must be only in the normal direction on surface S (d) the current density must be only in tangential directions on surface S
(e) none of the above Have a great Summer break! Good luck in ECE 312. ...
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 Fall '08
 Johnson,J
 Electromagnet

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