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Hmwk 4 Questions

# Hmwk 4 Questions - B.V/ft/lw i cadmium sure...

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Unformatted text preview: B .V/ft/lw i, cadmium sure FOLYTECHMC'UNNERSETL FOMONA I L. Ferguson . ESE , 3&2 . - - ~ , HCﬂE‘WOﬁK @4- 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 Else-meal 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 the-direction 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-+,35-7,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°)and-B(r=3cm,6}=45°,¢v= 90°). V V ‘_ ) 1-2:) \/ 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) Find-E 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 .. positive-charge, 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 " the-central perpendicular axis If this electron is re. leased'ﬂ-om 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 Jig-lass net charge-l- 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+(y-1)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 10-9 C/mz; D,(r = 5) = 6.4 x 10-10 (2/1112 (12) pm = —(4/9) x 10—9 C/mz @ (a) [(87tL)/3][p% — 10-9] 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 10-6MC/m2; 1),,(24 mm) = 3.9 x 10-6 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 2056-166) @ —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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Hmwk 4 Questions - B.V/ft/lw i cadmium sure...

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