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Unformatted text preview: B .V/ft/lw i, cadmium sure FOLYTECHMC'UNNERSETL FOMONA I L. Ferguson . ESE , 3&2 . 
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G Let a point charge Q1 = 25nC be located at P1(4, —2, 7) and a charge / Q2 = 60 nC be at P2(—3, 4, —2). (a) If e = 60, ﬁnd E at P3(1, 2, 3). (b) At
WhatpointontheyaxisisEx=O? .u ._.€
'35:: A 2 ,LLC point charge is located at AG, 3,5) in free space. Find E p, E45, E; at P(8, 12,2). i344? » L '7 H j e ‘7 r I“. ~ ‘ QQ 11C p0int charge is located at A(— I, 1, 3) in free space. (a) Find the
lens of all points P(x, y, z) at which E, = 500 Wm. (19) Find y1 if
“(l—2‘, y}, 3) lies on that locus. .. g, 4 , ~ ; \ Hun charge Q0 located at the origin in free space produces a ﬁeld for which 4 LE, = 1 kV/m at point P(—2, 1, 1). (0) Find Q0. FindE at M(1, 6, 5) in — 'v rectangular coordinates/5 (c) cylindn‘calcoordinates; (d) spherical
coordinates. “ 'k' I F ' V" —/23/;/"  / uniform volume charge density of 0.2 /.LC/m3 is present throughout the Elsemeal a (Dementia Engineering Department A l \) D a f/ "K "  ~ I “ fSpherical shell extending from r = 3 cm to r = 5 cm. H p, = 0 elsewhere, , , f'fﬁnd: (a) the total charge present throughout the shell, and (b) r1 if half the
total charge is located in the region 3 cm < r < r1. 7 I (6 > A uniform line charge of 16 nC/m is located along the line deﬁned by y =
—2, z = 5. If s = so: (a) ﬁnd E at P(1, 2, 3). (b) ﬁnd E at that point in the z = 0 plane where thedirection of E is given by (1/3021y — (2 /3)az. An inﬁnite uniform line charge pL = 2 nC/m lies along the x axis in free
space, while point charges of 8 nC each are located at (0,0, 1) and (O, 0, —1). (a) Find E at (2, 3, —4). (b) To What value should pL be changed to cause E
to be zero at (0, 0, 3)? i @ Spherical surfaces at r = 2, 4, and 6_ m carry uniform surface charge
densities of 20 nC/mz, —4 nC/mz, and pm, respectively. (a) Find D at r = 1,
3, and 5 In. (b) Determine p30 such that D = O at r = 7 n1. Volume charge density is located as follows: ,0U = 0 for p < 1 mm and for
p’ > 2 mm, p, = 4p tie/m3 for 1 < ,0 < 2 mm. ((1) Calculate the total charge
intheregionO < p < p1,0 < z < L,wherel< p1<2rnm.(b)Use
Gauss’s law to determine DP at p = p1. (c) Evaluate D, at ,0 = 0.8 mm, 1.6 mm, and 2.4 mm. ‘ in = 5.00r2a, mC/mz for r s 0.08 m and D «$0.205 a,/r2 uC/mz for >_" 0.08 m. ((1) Find p, for): F 0.06m. (b) Find p, for r = 0.1 m (c) What
ace charge density coiﬂd be located at r = 0.08 m to cause D = O for a 0.03 m? in the region of free space that includes the volume, 2 < x, y, z < 3, D =
{$012 a: + xz a,  ny az) C/mz. (a) Evaluate the volume integral side of
tithe divergence theorem for the volume deﬁned here. (b) Evaluate the surface
{integral side for the corresponding closed surface. ' Avert/«22C.
b) ya: —3.;?5’ mc/M
a) E: 2;.1+,357,Lt’; v/M a = I
(\ Z t) A uniform surface charge density of_20 nC/m2 is present on the spherical \ 5m ’ C
surface r = 0.6 cm in free space. (a) Find the absolute potential at P(r = 1 62
u » 6111.9 = 25°, ¢ = 50°). (b) Find VAR, given points A(r = 2 cm, 9 = 30°, ‘3.) 3° “1 \t/
» ¢=60°)andB(r=3cm,6}=45°,¢v= 90°). V V ‘_ ) 12:) \/ Let a uniform surface charge density of 5 nC/m2 be present at the z = O . f _ a
plane, a uniform line charge density of 8 nC/m be located at x = 0, z = 4, QS W'U" '
and a point charge of 2 “C be present at P(2, O, 0). If V = 0 at M(0, 0,5), ﬁndVatN(1,2,3). V '2 iﬂXK V In spherical coordinates, E = 2r / (r2 + a2)zar V/IIL Find the potential at any
point, using the reference (a)V = O at inﬁnity; gb) V = O.at r = 0;
(C)V = 100 V atr = a. 1 i x = —l, y = 2 in free space. Ifthe potential at the origin is 100 V, ﬁnd V at
P(4, 1,3). 7 Uniform surface charge densities of 6 and 2 nC/m2 are present at p = 2 and 6 cm, respectively, in free space. Assume V = O at ,0 = 4 cm, and calculate V at:(a)p=50m;(b)p=7cm. .) 5/ g .. ‘, / A certain potential ﬁeld is given in spherical coordinates by V = I Q n a Wt: _ _ 7' ~
& Vo(r/a) sine. Find the total charge contained within the region r < a. Q “ “TV “ eovo C
. Within the cylinder p = _2, 0 < z < 1, the potential is given by V = 100 + 50p + 150p sin ¢V. (a) Find V, E, D, and p” at P(1, 60“, 0.5) in free space.
(b) How much charge lies within the cylinder? ( l 3" 3' . J! $5 @ TWO uniform line charges, 8 nC/m each, are located at x = 1, z = 2, and at
I ~ a: .. , I ‘ _ u _ _ . Awire thatlias a uniform linear charge density is
O A 59mm“ “hat?” “mum ‘5 51m _ . ‘ bent into the shape shown in Figure l . Fina $e
' ., . ' V . a M) F electrical potential at point 0. ’ I a:
O,’ ' r > a
(a) FindE and Vfor r 2 a;
(b) Find E andeorr's a.
(c) Show that the maximum value of E is. at r = 0.745a. , (d) FmdhhereViemaximumandcaleulatethatmaxirnumvalue. " ' 'V _‘ Wk“ " 54 l Pv i ® A ring of radius R carries a uniformly distributed ..
positivecharge, as in 2. . The linear
\‘ ' L charge density of the ring is , and an electron is _
x ' located a distance at above the plme of the ring on
" thecentral perpendicular axis If this electron is re.
leased'ﬂom rest, what is its speed; when it reaches the
‘Icenter 9? £11.? __ ,._ ‘ @ Two identical raindrops. each carrying surplus elec«
trons on its surface to give a net charge — q on eacR.
collide and form a single drop of larger size. Before
the collision, the characteristics of each droo are as
follows"; (a).surface charge density; , (‘9) Leleczric
ﬁeld E0 at Lhe sulface, (c) electric pageantial V0 at the 
surface (where f = 0 atr= 00). For the combined
drop, ﬁnd these three quantities in terms of their
original values. . . _ ' :On‘planet Tehar, the gravitational ﬁeld strength is
the same as that on Earth but on Tehar there is also a
suengldownward electric ﬁeld that is uniform close
fl toddlefﬁlanem surface. A ball of mass m carrying'a thrown upward at a speed 0 and hits the after an interval t. What is the potential dif;
ference betWeen the starting point and the top point? of the trajectory? ' ‘ '  .U‘ " @ L ' .A ~paint charge 4 or'_ mass m is. injected at inﬁnity with.
Inmal velocity val, towards the center of a uniformly charged
sphere of radius RaT'ne total charge on the sphere Q is the H / same sign a q. . (a).What is the minimum initial velocity necessary for the
point charge to collide with the sphere? '1 x ' " (b) if the inidal velocity is half aisle result in (study close I does the charge get to adhere? @ A spherical shell qf'radius Jiglass net chargel Q
spread uniformly over its surface. A point charge  q. mass "m is initially at a distance D'from the surface of the:
shell. Thispoin't charge is released'from rest and attracted
'to the shell. It turns out that the spot where the particle:
would hit the shell has a small hole in it’, so that the charged
particle enters the shell and hits the inner surface en the far
' side of the shell. With what speed does the particle hit the
shell?.". '.' .l ‘ 26 'Four identical particles each hare charge q and mass
'._,m. They are released from rest at the vertices of a
square of side L. Hew‘fast is each charge moving
When their distance fcgrg the center 9f the square
jdcvblss? . ‘ (QMWW ' Q ((1)4.58ax — 0.156y + 5.516Z
(b) —6.89 or 22.11* a" 159.73,, + 27.4% — 49.44
<a>(x+1) = 0.56 [(x+1)2+(y1)2+(zl—3)2]1'5
(b) 1.69 or 0.31 (a (a) —1.63 uc
(b) —30.llax — 180.6321,  150.53% (c) 183.12ap — 150.536Z (d) —237.1
(5—) (a) 82.1 pC (b) 4.24m (11) 57.56y  28.86z V/m (b) 23%  4632 (a) Dr(r < 2) = 0, Dr(r = 3) = 3.9 x
109 C/mz; D,(r = 5) = 6.4 x 1010 (2/1112
(12) pm = —(4/9) x 10—9 C/mz @ (a) [(87tL)/3][p% — 109] MC where p1
is in' meters (b) 40611“ — 10'9)/(3p1);/.C/m2
where p1 is in meters
(0) 0,,(03 mm) = o; 13,,(16 mm) = 3.6 x
106MC/m2; 1),,(24 mm) = 3.9 x 106 MC/mz @ (a) 1.20 mm? (b) o (c) 426sz
® (a) 3.47 c (b) 3.47 c 19V: 0‘2,” ’6“
C) V‘Q‘ ' 2 z
00 4— Q r
2056166) @ —68.4V (a) 4.626 v (bi 9.678__ v; ‘C
(a) VP = 279.9 v, E1, = —179.9ap — 75.0% mi; Dp = 4.593,, . ad, nC/mz, pvp = ,_
—443 pC/m3 (b) —5.56 nC  L . \l
'D 2 _ ‘ . . 
(3 Mac) 091} “[(Wﬁgx 61:2:va ’2 ...
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This note was uploaded on 05/26/2008 for the course ECE 302 taught by Professor Ferguson during the Summer '08 term at Cal Poly Pomona.
 Summer '08
 Ferguson
 Electromagnet

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