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Unformatted text preview: ECE l34 Final Exam Fall 2009 FINAL EXAM You have 3 HOURS to complete this exam. Two pages of notes are allowed, otherwise there
should be no books or other materials in your purview. You may use a calculator if
necessary. When you are ﬁnished, check that your name appears prominently and leginy on
the front page. The exam is worth 100 points total. IMPORTANT POINTS FOR GRADING: SHOW ALL WORK. ANSWERS GIVEN WITHOUT CLEAR
SUPPORTING EVIDENCE OF INDEPENDENT WORK WILL BE
IGNORED. CIRCLE ANSWERS. THOSE NOT CLEARLY MARKED AS YOUR
FINAL ANSWER WILL BE IGNORED. BE NEAT. ILLEGIBLE OR SLOPPY WORK WILL BE IGNORED. Sometimes I think we’re alone in the universe, and sometimes I
think we’re not. In either case the idea is quite staggering. Arthur C. Clarke Page 1 ofll ECE 134 Final Exam Fall 2009 Problem 1 (10 points) Coaxial cables for highpower transmission typically use
a cross—linked polyethelyne (XLPE) insulator which has a
high breakdown ﬁeld. For this problem we will assume a maximum safe operating ﬁeld of Emax : l OkV/mm. a) Using Gauss' law, ﬁnd an expression for the radial
electric ﬁeld E,,(r) between the conductors (a<r<b)
assuming a surface charge density ,05 on the inner V
conductor: §§<E ‘Ag 3‘ glam/£05911. C“; éfi‘ilgﬁ/e 1 ‘65 {ix/((5 ~—/ b) The ﬁeld is largest near the surface of the inner conductor, i.e. E
fact to eliminate ,05 and express Er(r) in terms of E = E,(a). Use this max max, a , and r only: c) Now use your result of part (b) to relate Emax to the voltage V that is applied between the
inner and outer conductors as shown in the ﬁgure: \I: «a may ~_ Ema lyﬁ/Q 1 d) It can be shown that cable designs satisfying ln(b / a) =1 will support the largest possible
voltage for a given Emax. Combine this constraint and your result from part (c) to ﬁnd the
optimal dimensions of the cable (a and b) that will allow for safe operation at V = l 00kV: gar lmbﬂtyl (/
\f I loO cV
‘3 a: E; “’ lo léVf/nlm "1
1342i c) lotqelEQWQ _/ Page 2 of 11 .63... \ 2 ( “xi
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ECE 134 Final Exam 2' Fall 2009
Problem 2
(10 points) Rubbing a balloon over one's hair transfers a certain amount of charge Q onto the surface.
Assume that the balloon is perfectly spherical with a radius a, centered on the origin, and that
the total charge Q is distributed uniformly over the surface. a) Find the electric field inside and outside the balloon:
Evy (4“ .
G7 gr : 0 am draws; 6M“: (0553 \ . gm, :20. {Mam : .. ( Q .  p A ( C7 .. A J;\ u C) E‘r " ( ' e“ {3 “9 l J.“ij j b) Using your result from (b), ﬁnd the electrostatic energy associated with this charge
distribution. Your answer should be expressed in terms of the total charge Q, NOT the surface charge density: ‘ CQ 2 , l.
. Z r q pﬂwr I by"? CL _
‘\ Uta lélclv = {’13 um) ‘er la A, r @{Z "6?
I $7 Tl’éb ‘1 f x c) The charge on the balloon causes it to expand. Find the outward pressure (pressure 2 force
per unit area) on the surface ofthe balloon under the inﬂuence ofthe electrostatic forces. ,. Q u; a? ( a» W my A __.._. L /_ f A 2
(— r I ’2
k“ 73% C“ f 2 2 c“
Q web 4;. "‘ (32 ﬂ 0K /\ : 011 " ' ° Page3 ofll p (13m aw i ECE 134 _ Final Exam Fall 2009 Problem 3 (10 points) A large parallel plate capacitor is made using a thin Mylar ﬁlm, with a thickness d = l,um
and a dielectric constant of 5,. z3.2. The Mylar ﬁlm is not a perfect insulator; there are some
free charges in the material that “EMA; give rise to a conductivity 0'. The plates are square with a side a} 7 _ at V
length of 10 cm. A measurement J M\ of DC current versus applied '
voltage on the capacitor yields the curve shown below. Below «3;» I
100 Volts the current—versus '
voltage is roughly linear, passing '
through I uA at 100 Volts. But 100V léUV
for applied bias ~160 Volts a large current begins to ﬂow, limited only by the external circuit. ”" LJ U) «Cl; \vn a) Estimate the breakdown ﬁeld strength in the material (s’hoqunits); c) Compute the conductivity of the ﬁlm below breakdown (show units). 100v 3‘ b
3 : l0 .Jl j
'   « er
ﬂy, 7) OT yu’l go {out} Page4 of 11 ECE 134 Final Exam Fall 2009 Problem 4 (10 points) A UCSB student buys some 32—gauge copper wire for a low—power lighting project. The wire
has a diameter d:0.2 mm. The wire is wound on a plastic spool and has an unknown
length. The student ﬁnds information online that suggests a resistivity of 1.76 uQcm (or
1.76 ><10'8 Q  m ) for copper at room temperature. a) She applies a voltage of 100 mV between the ends of the wire, and observes a current of
about 5 mA. What is the approximate length of wire on the spool? 0) 11km” A 3/ A «(4%); e ._ 3: 2 Ti (l0 m
£1 _. ,
<3 T b) The wire is used to connect a 5V wall transformer to a high—intensity DC lamp. The lamp
can be modeled by a resistance of 369. What is the maximum length of wire that can be
used ifthe maximum allowable voltage droop is 10%? {LZL f: ii
2 a Page 5 ofll ECE 134 Final Exam Fall 2009 Problem 5 (10 points) In our derivation of fusing current for bare wires we assumed that thermal radiation is the
dominant cooling mechanism because the melting point of most metals is quite high (ag.
T ~1083°C for copper). However, some metals have relatively low melting points (6g. HIE/I
T z 230°C for tin), and hence convection could be the dominant cooling mechanism. Hie/I a) Ifthe wire has a resistance R at temperatures near the melting point, ﬁnd an expression
for the steady—state fusing current when convection cooling dominates. Assume the wire has
a surface area AS , a convection coefﬁcient h, and is suspended in air at temperature To: b) Now express A: and R in terms of the physical parameters such as the wire diameter d,
length (7, and conductivity 0. If h = 0.01/d [W/m2 °C], show that the fusing current will
depend linearly on the wire diameter such that I‘m =Kd, and ﬁnd the coefﬁcient K in
terms of a, T and To: A5 3 7T l/t “
A l _ A. ‘
{LI 0/ we)? ° D¢Ol z
Gaol 5T1 c) Evaluate K numerically for a tin wire suspended in air at T0 = 25°C . The conductivity
of tin is approximately a z 7.3 X 106 S/m near the melting temperature. Page6 of ll ECE 134 Final Exam Fall 2009 Problem 6 (10 points) A certain coaxial cable is constructed from a pair of hollow coaxial conducting cylinders as
shown below, with radii of a and b for the inner and outer conductors, respectively. In
operation, each conductor carries equal and opposite currents I as shown, forming an inﬁnitely long loop: a) Using Ampere's law, ﬁnd an expression for the magnetic ﬁeld H¢(r) between the
conductors (a < r < b ), assuming L —> oo : . _ $5 f; A E eye/Qua} 'i T C; ‘Z‘Trr/Li; Ayah/1;: \Hy «l; 27:: b) Find the total magnetic ﬂux linked by the currents over a ﬁnite length L (i.e. the ﬂux
through the lined rectangular region in the ﬁgur ). x \L
r: We? w M c) Find an expression for the inductance per unit length for the structure. Page 7 of II ECE 134 Final Exam Fall 2009 Problem 7 (10 points)
Two coils of N. and N2 turns are wound concentrically on a straight cylindrical core of radius
aand permeability y. The windings have length El and £2 respectively. a) Using the long solenoid amtion, find the magnetic ﬂux in the core if a current 1]
ﬂows in the long winding as shown. .  v w i“ N( 'A
C; feted w‘\ W; gala” " {Ell/“L 7“ CAL" ,_..._—— c) Find the mutual inductance between the coils Page 8 ofll ECE 134 Final Exam Fall 2009 Problem 8 (10 points) Large AC currents are often measured in silu using
an inductive pickup as shown (this is sometimes
called a “current—sensing” transformer). A high—
permeability torus of mean radius re and cross—
sectional area A is clamped around the current to
be measured. A secondary winding of N turns is
wrapped on this core to measure the induced emf
(voltage), the strength of which can be related to
the AC current magnitude. a) Assuming the current passes through the center of the torus, find the B—ﬁeld inside the
core at the mean radius r M . Amp—MAJ) ' b) lfthe AC current to be measured is expressed as [(1) : IO cos(a)z), ﬁnd an expression for
the induced secondary voltage V(1.‘). Use the result of (a) and assume that the magnetic field
is approximately constant over the cross section ofthe core, as we often did in the homework. c) How many turns in the secondary winding would be required to insure that the induced
AC voltage amplitude is at least 10 mV for 60 Hz currents of >1 A? Assume core
dimensions of r0 2 2 cm and A : 0.25 cm2, and a relative permeability of yr : 500. (43%} V3 ‘> lOMV “69‘” ijocpl l4 Page9of1] ECE I34 Final Exam Problem 9 i Fall 2009 (10 points) The ﬁgure shows an electromechanical device
that is sometimes just referred to as a
“solenoid”. A winding ofN turns is wrapped
on the central arm of the highpermeability
core. A similarly high—u “plunger” slides freely
in a nonmagnetic sleeve between arms of the
core as shown. The cross section of all
segments of the core and plunger is A. When
the coil is energized with a magnetizing current
I, there is a force on the plunger causing it to
pull inward (to the left in the figure). Highp core Highp
plunger a) Using the magnetic circuit method, find the total ﬂux in the air gap as a function ofx and g (assume that the reluctance of the core and plunger are negligible, i.e. /,1 a co) r—VW
‘7 WM nealuifmla emf s5 {Mm/“2&6 ‘2‘} \ i r._‘ “C; It
PM? hf]ch { . N: oh
6; V L? ’ ‘ x7 b) Find an expression for the magnetic stored energy (hint: ﬁnd the self—inductance first). v“? MHZ/4
Lg— H U I x + 77/2, 7 2?.”
U \ L “z. '\_ ﬁlo/U ff c) Find an expression for the retortthepl‘ungerr 2/4 7 (A [B — \ V/UO Page 10 of 11 ECE 134 Final Exam Fall 2009 Problem 10 (10 points)
A plane wave propagates in a nonmagnetic material with 5r =9 and ,ur =1, and the electric
field vector is given by E: 2395mm 0% —ﬂz) Win a) Find the wave velocity Mfg“; ' Q 5 b) Find the propagation constant ,8 a W A quﬂVJ/s Rm G V/ row/s ’ \. M c) Find the magnitude and direction ofthe magnetic ﬁeld (show units) h f fiﬁdy§ A. 7 'Pagellofll ...
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 Fall '08
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 Alternating Current, Magnetic Field, Electric charge, Final Exam Fall

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