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Unformatted text preview: easiestesÂ» DO NOT BEGlN THIS EXAM UNTIL TOLD TO START Name: ___________Soiution _____ _________________
Student Number: ___ ____â€ť_______________________
ECE 2317
Applied Electricity and Magnetism
Exam 2
November 8, 2003 . This exam is closed book and closed notes. A calculator and two formula sheets (two 8.5â€ť X 11â€ť pieces of paper) are allowed. . Show all of your work. No credit will be given if the work required to obtain the solutions is not shown. Perform all your work on the paper provided. . Write neatly. Your work must be legible to receive credit. Leave all answers in terms of the parameters given in the problem.
Show units in ail of your ď¬nal answers. Put answers in the blanks provided. . If you have any questions, ask the instructors. You will not be given credit for work based on incorrect assumptions. . You have a total of 90 minutes to work the entire exam. _____/25 Prob. 1
____/25 Prob. 2
_____ l25 Prob. 3 ___I25 Prob. 4 Data â€”â€”â€”â€”â€”â€”â€”â€”dx: mxlxz +a :ď¬‚â€”aâ€”2~ln(x+xlx2+a2) uď¬dxzagw
(x +a 2m): 3522+â€ś X22411 corrections and additions added in red: VG): Eggp+iÂ§Â˘+Â§z 6p [78Â˘ (3â€2
am] 162(1) 62(1) V2(D:â€”â€” m +w +
[,0 6p p2 6452 822 Problem 1 (25 pts) a) Find the potential d3 along the z~axis ď¬eld
due to a constant line charge density pro on an equilateral triangle (all sides of
length L) lying in the 2:0 plane and
centered about the zâ€”axis. The side of the
triangle parallel to the  yaxis passes
through the x~axis at x=a. Since potential needed is along zâ€”axis, by symmetry we can treat one leg, of triangle and multiply the result by three. RZJHâ€E + :12 + y'2
I 2 1,". 3 . :>d)=3 mpâ€ś) _ = ď¬‚'â€ť ln(y'+â€/al+zz+yâ€™2)
I"â€™ 2 2 I" â€ tum â€ť47:50 a +2 +y' 4m?â€ś vâ€ť 132 b) From the potential found in part a), ď¬nd the electric field along the 2..
axis. l,
3 +2 32 l , . L 3 1 I 2 Ill
(IL â€™0'â€ś in 2"â€ť 1n" al+zl+ â€” +â€”â€ ~ln 212+23+ 4 â€”i';
411.4?n L 4H8â€ť L 2 2 2 2 l
_. _ J:
2 . â€”
E :gtlgwm __1_____WJW
: (3)2 â€ . '2 ' 2 2
4313â€,â€™az+22+[LJ al+22+[ÂŁ] +L az+zz+ L] eelâ€”1W
2 2 2 2 2 l
1.
31.1.0211 Problem 2 (25 pts) A cylindrical capacitor of length L with inner conductor of radius a and
Outer conductor or radius b is half ď¬tted with a dielectric material of
relative permittivity a, as shown. The other region is airď¬lled. The potential difference between the inner and outer conductors is V0.
Neglect fringing at the ends of the capacitor. a) Starting from Laplace's or Poissonâ€™s equation,
determine the potential between the
conductors that satisfies the boundary
conditions shown. (I) : (])(;))0nly._ so , 1 1f . ' i. â€
V'41=~;f if}: :0 ::> pď¬‚z/l :3 (I):Alnp+B
p dpl rip dp @pah, Â«13:02Alnb+B :sazâ€”Ainra :> (Ilzxď¬‚nf: b
I)
, V1113 Vain"
_ a. it _ 0 b p
(iii/72:2, rt)=li,=Alnâ€” :2 ,4:â€”â€”â€™â€”:> ti): =eeer
h a a b
ln Inâ€” low
I) I) a
Vulnď¬ Vuhiji
(D: , {IKEâ€”â€ś551 [V ], :23pr
lnâ€” lnâ€”
!) a b) From the potential in part 3), determine the electric ď¬eld between
the conductors in terms of the voltage V0 ' n V A
Esz(I}:â€”(Â§ÂŁpm â€śb p [V/m]
"0 pinâ€”
a
_ V0 A
E. h p [Wm] ,aÂŁp<b
phrâ€” (I c) Determine (in terms of V0) the surface charge densities on the inner conductor in both the dielectric and the air (free Space).
regions.
I 30V?) p5, in ailregion : 3013!); = f [C/mz]
â€ rm! )
â€śInâ€”â€”
a
' (â€śIn .' . .1 ' ._ E m 5130K: 2.
pi m (it. emu. legion m area f, m? , h [C/m ]
alnâ€”
a I  z a". 1 lâ€ť â€ 1
i in dielectric "câ€ś â€ś C/nr
p; h a In â€”
a d) Determine the total capacitance of the capacitor. Q: [52an in airrcgion +pJTď¬'L in dielectric region
f.)(s,+l)%ll C] Ink
:1
C â€ś9 , mâ€ś (I 1â€”1111 [F]
Vâ€ś in h
a
mâ€ś gr+l L
C â€śâ€0.â€ś n( M7)___)_m_ [F l Problem 3 (25 pts) Parts a b and c of this roblem are
fndegendent! a) The D ď¬eld at point a above the conductor
surface shown is given by D = 2i+2JÂ§y.
Find the surface charge density ps and the
surface tilt angle a at a. px = Dâ€ť = #22 +22 3 m 4 {Cf/n12} (D is in the normal direction!) a : Ian" Ax]?â€” = 600
2â€™
p5 ;____w____m_ 4 [ C/m2 ]
a ;_________%___ 60"â€™[ deg ] b) The electric field at point a just above the dielectric interface shown is El=3i+4y. grlzs . a iy X Find E2 at b, just below the interface. a}: =4 Â° â€5 xcomponcntis tangent tointcrlâ€ace, hcncc Eh : Eâ€ť = 3.
Vincomponentisnorrnal tointcrtâ€acc, hence ad 0 E2] = aâ€ť of)?â€ś
a 3 ::> Eâ€ś. minEâ€ť =mâ€”4:3V:â€>E? =3yt+3Â§n
â€ 8'1 .. 4 A E2: 3fi+3Â§r [Vim] c) An electric ď¬eld is given by E = 3y2ď¬+ 6xy5r+ď¬z. Give a short proof that
the ď¬eld is conservative, i.e. (ind}? = 0 for any closed path C . _ 2w:w*_ÂŁÂŁÂŁ_~_ A: .. ..
VxEâ€”Vx(3y Moxyy+z)â€”lm W 6714ny 6y): 0 <:> (LE (Ifâ€”0
319 6X37 1â€; [Id = 1â€"n Probiem 4 (25 pts) a) A grounded conducting slab of thickness t is positioned between two
conductors with surface charges pâ€ś and pď¬‚ as shown. What are the surface charge values p;, and piz on opposite sides of the grounded conductor? (Ignore fringing.)
The two charged conductors will attract enough charge from. the ground plane to cancel all their flux lines. Hence. by inspection, p5. = ~17â€ś; [2:2 2 â€”pâ‚¬2 [CT/r112]
pg. : â€” 17â€™s: : â€”.90 [ C/m2 ] .4: w
p;. :__â€”p.. :jncmr 1 13) Find the electric field in all regions. Use Gaussâ€™s law where
appropriate. E, x < 412: (l ' [Wm ] E,  h 2< x < 0 : 3};ng m 2%)} (by GRUSS'S Law)_[ V/m ]
in
E,0<x<t: 0 [Wm]
E, t< x < r + h l: â€” â‚¬ng 2 â€”Ilft (by Goose's Law) _[V/m ]
*0
E,x>r+h!: 0 [Wm ] 6) Plot the potential in all regions as a function of x on the graph below.
Label the vertical axis scale as appropriate. Since geometry is planar, assume potential is oncÂ»dimcnsional and grounded conductor is at zero potential. Then the upper
conductor is at a potential 1[V/m] x 1 ho]: +1[V] higher than the
grounded conductor and the voltage varies linearly from zero if we
assume the grounded conductor is at zero potential. The lower
conductor is at a potential l/â€2[V/m]>< l /2[m] : +l/4[V] higher than the grounded conductor and also varies linearly. The potential must be continuous everywhere and is constant in regions where the ď¬eld is zero. 1 [W 1/4 [V] 10 ...
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