GAUSS' LAW E. A. surface

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1 Prf. Dr. I. M. A. Nasser GAUSS' LAW GAUSS' LAW Intrductin: The electric field f a given charge distributin can in principle be calculated using Culmb's law. The examples discussed in electric field shwed hwever, that the actual calculatins can becme quit cmplicated. The electric flux (t flw) (Φ, [Φ] = N.m²/C) is represented by the number f electric field lines that penetrate a surface. If the electric field, E, is unifrm and makes an angle θ with the nrmal t the surface area A, the electric flux thrugh the surface is θ E A n In general case E.A EA cs E. d A surface Ex.- A unifrm electric field E = a i + b j intersects a surface f area A. Calculate the flux thrugh this area if the surface lies: a) in the XZ-plane. [Answer: ΦXZ = (a i + b j).(a j) = ba.] b) in the YZ-plane. [Answer: ΦYZ= (a i + b j).(a i) = aa.] c) in the XY-plane. [Answer: ΦXY= (a i + b j).(a j) = 0.] ==================================================== Gauss' law is a very pwerful therem which relates any charge distributin, having a high degree f symmetry, t the resulting electric field at any pint in the vicinity f the charge. It states that "the net electric flux, Φc, thrugh any clsed surface* is equal t the net charge inside the surface, qin, divided by ɛ". In symbls q E. A surface d in 1, 9 10 Nm, C *usually called Gaussian surface, which has the exact symmetry as the charge distributin. If the charge has symmetry, such as spherical, cylindrical, etc. ne can use E. A E d EA surface E 4 r Spherical case 4 rh Cylinderical case 1

2 Prf. Dr. I. M. A. Nasser GAUSS' LAW Example 1: Field f pint charge. Calculate the electric field due t a pint charge q. The field generated by a pint charge q is spherical symmetric, and its magnitude will depend nly n the distance r frm the pint charge. The directin f the field is alng the directin (see Figure 4.). Cnsider a spherical surface centered arund the pint charge q (see Figure). The directin f the electric field at any pint n its surface is perpendicular t the surface and its magnitude is cnstant. Using Gauss's law we btain the fllwing expressin which is Culmb's law. q q q E. d A surface E (4 r ) E k 4 r r Fr a cntinuus charge distributin, the electric field is given by q dq E = k r k r r i i i i r and dq culd be calculated using the charge density as fllws: charge density Vlume Surface Linear dq dq dq = = = dv da dl Ex. - Calculate the electric field due t an insulating sphere f radius R. The sphere has a Q unifrm charge density ρ and ttal charge Q, where 4 3 r ( r ) 3 Q k k r 3 r R r R qin E r 4 3 ( R ) 3 Q k k r R r r Ex. 3- Calculate the electric field due t a nncnducting plane sheet f charge Q and has a Q surface area A,. A

3 Prf. Dr. I. M. A. Nasser GAUSS' LAW Q E. d A E ( A) E surface Cnductrs and Insulatrs: 1. Cnductrs are material in which electric charges mve quite freely. Cnductrs are essentially metals, e.g. cpper, aluminum, they cntain free electrns which mve freely in the cnductr and can transfer the electric charge.. Insulatrs are materials that d nt readily transprt charge, e.g. glass, rubber, wd. Their electrns are tightly bund t their atms and can nt mve freely. Prperties f a cnductr in electrstatic equilibrium: 1. The excess charge resides entirely n its surface.. The electric field is zer everywhere inside it. 3. The electric field just utside it is perpendicular t its surface and has a magnitude ( where σ is the charge per unit area at that pint. 4. On an irregularly shaped cnductr, the cncentratin f charge n it is greatest where the surface is mst sharply curved (radius f curvature f the surface is the smallest). EXAMPLES Example 1. The electric field everywhere n the surface f a hllw sphere f radius r =0.7 m is measured t be equal t ² (N/C) and pints radially tward the center f the sphere. Q a- What is the net charge within the sphere's surface? [ E k ] r Since the electric field pints radially inward twards the center f the sphere, then ne can expect a negative charge inside. Using Gauss' law, we can have Q (9.810 )(0.7) 8 Ans: E k Q C 9 r (9.010 ) b- What can yu cnclude abut the nature and distributin f the charge inside the sphere? Negative charge has spherically symmetric distributin. Example.The electric field everywhere n the surface f a hllw sphere f radius r =11 cm is measured t be equal t ⁴ N/C and pints radially inward twards the center f the sphere. Hw much charge is enclsed by this surface? ), = EA = E(4πr²) = ⁴ (4π) (0.11)² = ³ N.m C 3

4 Prf. Dr. I. M. A. Nasser GAUSS' LAW then Qin C It is a negative charge inside because the electric field pints radially inward twards the center f the sphere. Questin 1 A unifrm electric field E a ˆ b ˆ i j N/C intersects a surface f area A. If the surface f the area A lies in YZ-plane, the flux thrugh the area will be: (a) b A. (b) Zer. (c)@ a A. (d) a b. (e) A A. Questin When a piece f paper is held with ne face perpendicular t a unifrm electric field, the electric flux is 48 N m /C. When the plane f the paper makes 30 degrees with the directin f the electric field the electric flux thrugh it is: (a)@ 4 N m /C (b) 44 N m /C (c) 1 N m /C (d) 48 N m /C (e) 3 N m /C Questin 3 Calculate the electric flux (phi) thrugh the curved surface f a cne f base radius R and height h. The electric field E is unifrm and perpendicular t the base f the cne, and the field lines enter thrugh the base. The cne has n charge enclsed in it, as seen in figure (). (a) (b) (c) π R h E. - π R E. π R E. 4

5 Prf. Dr. I. M. A. Nasser GAUSS' LAW (d) - π (R ) E. (e)@ π (R ) E. Questin 4 Tw cncentric shells, ne with radius R and the ther with radius R, surrund an islated pint charge. The rati f the number f field lines thrugh the larger shell t the number f field lines thrugh the smaller is: (a) 1/4. (b)@ 1. (c) 1/. (d) 4. (e). Questin 5 Fr the electric field: E 4 ˆi 30 ˆj 16 kˆ N/C where,, and are the unit vectrs in the directins f x, y, and z, respectively, the electric flux thrugh a.0 m prtin f the yz-plane is: (a) 60 N m /C. (b) 9 N m /C. (c)@ 48 N m /C. (d) 3 N m /C. (e) 80 N m /C. Questin % A clsed cylinder whse main axis is alng the x-axis is shwn in figure. It is placed in a unifrm electric field f magnitude 00 N/C pinting in the negative x-axis. The cylinder has a crss sectinal area f 1.5 cm and a length f 6.0 cm. The fluxes thrugh faces I, II and III are respectively: î ĵ ˆk (a) zer, 0.5, zer N m / C 4 I EA N m / C (b) zer, -0.5, zer N m / C II 0, since E A (c) - 0.5, zer, 0.5 N m / C 4 (d) - 0.5, 0.5, N m / C III EA N m / C (e)@ 0.5, zer, N m / C Questin % A cube, as in figure (6), has an edge length f 3.00 m in a regin f a E 5 ˆj 6 kˆ N/C, unifrm electric field given by the equatin: where i, j, and k are the unit vectrs in the directins f x, y, and z respectively. Find the electric flux thrugh the tp face (shaded). (a) (b) Zer. 45 N m /C. ˆ ˆ ˆ EA. 5 j 6 k.9 j Nm / C 5

6 Prf. Dr. I. M. A. Nasser GAUSS' LAW (c) (d) - 30 N m /C. 30 N m /C N m /C. 4-3 flux f an Electric field Questin 8 An infinitely lng line has a charge density f 7.6 nan-c/m. Calculate the electric flux thrugh a spherical surface f radius R = 7.7 cm whse center, C, lies n the line charge as shwn in Figure 3. (a) zer (b)@ 13 N m /C (c) 9.0 N m /C (d) 415 N m /C (e) 610 N m /C Questin 9 A ttal charge f (-6) C is unifrmly distributed inside an irregular insulatr. The vlume f the insulatr is.50 m 3. Nw, imagine a cube f vlume 0.50 m 3 inside the insulatr. What is the ttal electric flux thrugh the surface f the cube? (a) N m /C. (b) Zer. (c)@ N m /C. (d) N m /C. (e) N m /C. Questin 10 A pint charge f.0 micr-c is placed at the center f a cube 50 cm n edge. What is the flux thrugh the bttm surface? (a) N m /C. (b) N m /C. (c) N m /C. (d) N m /C. (e)@ N m /C. Questin % The cube in figure 3 has edge lengths f.00 m and is riented as shwn in a regin in which a E 5.00 ˆi 8.00 kˆ N/C, where i unifrm electric field exists. The electric field is given by: 6

7 Prf. Dr. I. M. A. Nasser GAUSS' LAW and k are unit vectrs parallel t the x-axis and z-axis respectively. Find the electric flux thrugh the right face (shaded) f the bx. (a)@ zer (b) +6.0 N m /C (c) N m /C (d) N m /C (e) N m /C Questin % A spherical cnducting shell has a radius f 0 cm. Pint A is a distance f 30 cm frm the center f the sphere. The electric field at pint A is 500 N/C and is directed radially utward. An additinal charge Q is intrduced at the center f the shell. The electric field at pint A decreases t 100 N/C. What is Q? (a) - 5 nan-culmbs (b) + 5 nan-culmbs (c)@ - 4 nan-culmbs (d) + 1 nan-culmbs (e) - 1 nan-culmbs majr-033-q Questin % The electric field in the regin f space shwn in figure 3 is given by: E = (8.0 i + y j) (N/C), where y is in meters. What is the magnitude f the electric flux, in units f N m /C, thrugh the tp face f the cube? (i and j are the unit vectrs in the x and y directins, respectively) (a) 1 (b) 6.0 (c) 90 (d) 130 (e)@ 54 final-033-q Questin % 7

8 Prf. Dr. I. M. A. Nasser GAUSS' LAW A charged cnducting spherical shell with an uter radius f.0 m has a pint charge f +3.0 micr-culmb at its center. The electric field at a distance f 3.0 frm the center has a magnitude f N/C and is radially utward. What is the charge n the uter surface f the shell? (a)@ +1 micr-culmb (b) -1 micr-culmb (c) +6 micr-culmb (d) -3 micr-culmb (e) +3 micr-culmb final-04-q Questin % If a rectangular area is turned in a unifrm electric field frm a psitin where the maximum electric flux ges thrugh it t a psitin where nly half the maximum flux ges thrugh it, what is the turned angle? (a) 45 degrees (b) 30 degrees (c) 90 degrees (d) 3 degrees (e)@ 60 degrees 4-4 Gauss' Law majr-993-q Questin % A pint charge Q= 6 micr-c is placed at the center a rectangular bx with dimensins a=b= 0.4 m and c= 0.6 m. Find the ttal electric flux thrugh the surface f the bx. (a)@ N m/c (b) N m/c (c) N m/c (d) Zer (e) N m/c final-993-q Questin % Charges q and Q are placed n the x axis at x= 0 and x=.0 m, respectively. If q= -40 pic-c and Q= +30 pic-c, determine the net electric flux thrugh a spherical surface f radius 1.0 m centered n the rigin. (a) -8.5 N m/c (b) -1.1 N m/c (c)@ -4.5 N m/c (d) -9.6 N m/c (e) -6.8 N m/c final-001-q Questin 18 In figure (1), what is the rati f the electric flux that penetrates surface S1 t that penetrates surface S? (Nte that S1 and S are clsed surfaces and q is a charge.) 8

9 Prf. Dr. I. M. A. Nasser GAUSS' LAW (a). (b) 1.3. (c) (e) 1.0. majr-03-q Questin % A pint charge, q1 = (-6) C, is placed inside a cube f side 5.0 cm, and anther pint charge q = (-6) C is placed utside the cube. Find the net electric flux thrugh the surfaces f the cube. (a) N m /C (b)@ N m /C (c) N m /C (d) N m /C (e) N m /C cube q N m /C majr-03-q Questin % A ttal charge f C is unifrmly distributed inside an irregularly-shaped insulatr. The vlume f the insulatr is 3.0 m 3. Nw, imagine a cube f vlume 0.5 m 3 inside the insulatr. What is the ttal electric flux thrugh the surfaces f the cube? (a) N m /C. (b) Zer. (c) N m /C. (d) N m /C. (e)@ N m /C. 6 Q qin Vin V 3 6 qin N m /C C, majr-033-q Questin % Charge is unifrmly distributed ver the entire xy plane with a surface charge density f 0 micr-c/m. A sphere has a radius f 1.0 m, and is centered at the rigin. What is the net electric flux thrugh the surface f the sphere? 9

10 Prf. Dr. I. M. A. Nasser GAUSS' LAW (a) (b) N m /C (c) zer (d)@ (e) majr-041-q Questin % An imaginary clsed spherical surface S f radius R is centered n the rigin. A psitive charge is riginally at the rigin, and the flux thrugh the surface is Phi. The psitive charge is slwly mved frm the rigin t a pint R away frm the rigin. In ding s the flux thrugh S (a) decreases t Phi/4. (b) remains the same Phi. (c) increases t Phi. (d)@ decreases t zer. (e) increases t 4 Phi. majr-041-q Questin % Figure 1 shws three situatins in which a Gaussian cube sits in an electric field. The arrws and the values indicates the directins (in N m /C)f the flux thrugh the six sides f each cube. In which situatins des the cube enclse, a psitive net charge, a negative net charges and zer net charge? respectively. Nm /C Nm /C Nm /C (a) 1,3 and. (b),1 and 3. (c)@,3 and 1. (d) 3, and 1. (e) 1, and 3. majr-04-q Questin % The net electric flux passing thrugh a clsed surface is N m/c. What is net electric charge cntained inside the surface if the surface is a cylinder f height 3.5 cm and radius 1.1 cm. (a)@ (-9) C. (b) (-) C. (c) (-) C. (d) zer. () : (1) : (3) :

11 Prf. Dr. I. M. A. Nasser GAUSS' LAW (e) (-9) C. 4-5 Gauss' Law and Culmb's Law final-961-q Questin 5 The electric field everywhere n the surface f a hllw sphere f radius 11 cm is measured t be equal N/C and pints radially inward twards the center f the sphere. Hw much charge is enclsed by this surface? (a) C. (b)@ C. (c) C. (d) C. (e) C. majr-04-q Questin % A psitive pint charge q sits at the center f a hllw spherical shell. The shell, with radius R and negligible thickness, has net charge -q. The electric field strength utside the spherical shell (at r>r) will be: (a) 3k q/r radially inwards. (b)@ k q/r radially inwards. (c) 3k q/r radially utwards. (d) k q/r radially utwards. (e) zer. 4-6 A Charged Islated Cnductr majr-99-q Questin 7 A spherical cnducting shell f inner radius r1 and uter radius r has a net charge f C. If a pint charge f -4.0 charge n the uter surface f the spherical shell? (a) -4.0 (b) +4.0 (c)@ -.0 (d) zer (e) +. C majr-001-q Questin 8 An islated cnducting spherical shell has an inner radius f 4.0 cm and uter radius f 5.0 cm. A charge C is puted n the shell. What is the rati f the charge n the inner surface f the shell t the charge n the uter surface? (a) 7/10. (b) 5/4. (c) 8/5. (d)@ Zer. C C C C is placed at the gemetrical center f the spherical shell, what is the 11

12 Prf. Dr. I. M. A. Nasser GAUSS' LAW (e) 1. majr-011-q Questin % A pint charge f qcenter 50e lies at the center f a hllw spherical metal shell that has a net charge f qnet 100e, as seen in figure (4). Calculate the charge n the (a) shell's inner surface, and (b) n its uter surface. [e is the magnitude f the charge n the electrn.] q q 50 e, inner center q q q 50 e ( 100 e) 150e ut inner net (a)@ (a) 50e (b) -150e. (b) (a) 50e (b) -100e. (c) (a) -50e (b) -100e. (d) (a) Zer (b) -150e. (e) (a) -50e (b) 150e. majr-031-q Questin % A pint charge f +4.0 micr-c lies at the center f a hllw spherical cnducting shell that has a net charge f micr-c. If the inner radius f the shell is.0 cm and the uter radius is 3.0 cm, then the rati between the charge density n the inner surface t the charge density n the uter surface is: (a) 4 : 1. (b) 1 :. (c) -1 : 1. (d) -1 :. (e)@ 1 : 1. majr-033-q Questin % A +0 micr-culmb pint charge is at the center f a cnducting spherical shell that has an uter radius f 1.0 m and an inner radius f 0.50 m. The net charge f the spherical shell is zer. What is the surface charge density n the uter surface f the shell? (a) +6.4 C/m (b) -6.4 C/m (c)@ +1.6 C/m (d) -1.6 C/m (e) zer 4-7 Applying Gauss' Law: Cylindrical Symmetry majr-99-q

13 Prf. Dr. I. M. A. Nasser GAUSS' LAW Questin 3 Cnsider an infinitely lng line f charge density.0 micr-c/m lying alng the x-axis as shwn in Figure 4. What is the rati f electric field strength at pint A t that at pint B? (a) 0.5 (b) 1.00 (c).00 (d)@ 0.50 (e) 4.00 final-011-q Questin % An infinite line f charge prduces an electric field f N/C at a perpendicular distance f.5 m frm its axis. Calculate the linear charge density. (a) (-6) C/m. (b) (-6) C/m. (c)@ (-6) C/m. (d) (-6) C/m. (e) )-6) C/m. majr-01-q Questin % Tw lng, charged, cncentric cylindrical shells have radii 3.0 and 6.0 cm. The charge per unit length is (-6) C/m n the inner cylinder and (-6) C/m n the uter cylinder. Find the electric field at r = 4.0 cm, where r is the radial distance frm the cmmn central axis. (a) N/C radially utward (b)@ N/C radially inward (c) N/C radially inward (d) N/C radially inward (e) N/C radially utward majr-0-q Questin % Figure 3 shws tw infinitely lng rds carrying unifrm linear charge densities and. If 1 the net electric field at pint A is zer, then the rati / is: 1 13

14 Prf. Dr. I. M. A. Nasser GAUSS' LAW The tw electric fields must be ppsite at A, then (b) r 7 E A k k (c) 0.71 r1 r 1 r1 5 (d).7 (e).3 majr-033-q Questin % Figure 1 shws tw infinitely lng lines f charge with unifrm linear charges densities: lambda1 = nan-culmb/m and lambda = nan-culmb/m. The separatin between the tw lines f charge is d = 1.00 m. What is the net electric field at pint P? (a)@ 7 N/C t the left (b) 7 N/C t the left (c) 7 N/C t the right (d) 135 N/C t the left (e) 135 N/C t the right majr-041-q Questin % In figure, the magnitude f the electric field at pint A, due t an infinite line charge density f C/m, is N/C. If the pint A is at a distance R frm the line charge, what is R? (a) 1. m. E k R k.3 m A R EA 7.10

15 Prf. Dr. I. M. A. Nasser GAUSS' LAW (b) 0.3 m. (c) 3.4 m. (d) 5 m. (e)@.3 m. majr-04-q Questin % A very lng unifrm line f charge having a linear charge density f 6.8 micr-c/m lies alng x- axis. A secnd line f charge has a linear charge density f micr-c/m and is parallel t x- axis at y = 0.5 m. What is the net electric field at pint where y= 0.5 m n y-axis? (a) N/C alng -y-axis. (b)@ N/C alng +y-axis. (c) N/C alng -y-axis. (d) N/C alng +y-axis. (e) N/C alng +y-axis. 4-8 Applying Gauss' Law: Planar Symmetry final-99-q Questin 39 Tw infinite nn-cnducting parallel surfaces carry unifrm charge densities f 0.0 nan- C/m and nan-c/m. What is the magnitude f the electric field at a pint between the tw surfaces? (a) 17 N/C (b) 34 N/C (c) 90 N/C (d)@ 45 N/C (e) 3 N/C majr-00-q Questin 40 Fig. 7 shws tw parallel plates, infinite and nn-cnducting, with surface charge densities f (-4)C/m and (-4)C/m. B, a ball with negligible mass, carries a psitive charge f (-8) C and is attached t pint A with a nn-cnducting string f length 10 cm. At equilibrium, the tensin in the string is: (a) 3.0 N. (b) Zer. (c)@ 6.0 N. (d) 1.5 N. (e) 0.3 N. majr-011-q

16 Prf. Dr. I. M. A. Nasser GAUSS' LAW Questin % As shwn in figure (3), a small, nn-cnducting ball f mass m = (-6) kg and charge q =.0 10(-8) C, distributed unifrmly thrugh its vlume, hangs frm an insulating thread that makes an angle theta = 0 degrees with a vertical, unifrmly charged nn-cnducting sheet (shwn in crss sectin). Cnsidering the weight f the ball and assuming that the sheet extends far vertically and int and ut f the page, calculate the surface charge density f the sheet. (a) (-9) C/m. (b) (-9) C/m. (c) (-9) C/m. (d).5 10(-9) C/m. (e)@ 3. 10(-9) C/m. majr-01-q Questin % Figure 4 shws crss-sectins thrugh tw large, parallel nn-cnducting sheets with identical distributins f negative charge. The surface charge density fr each sheet is (-15) C/m. What is the electric field at pint A? (a) (-4) N/C upward (b) (-4) N/C dwnward (c) (-4) N/C upward (d)@ (-4) N/C dwnward (e) 0 final-01-q Questin % A small insulating sphere f mass m = (-9) kg and charge q = nan-culmb is hanging at equilibrium abve a charged insulating sheet. What is the surface charge density f the sheet? (a) nan-culmb/m (b) nan-culmb/m (c)@ nan-culmb/m 16

17 Prf. Dr. I. M. A. Nasser GAUSS' LAW (d) +.50 nan-culmb/m (e) nan-culmb/m majr-0-q Questin % Figure 4 shws tw large, parallel, nn-cnducting sheets, each with fixed unifrm charge 6 6 density: C/m, C/m. The rati f the magnitude f the electric field at pint A t that at pint B is: 1 E ˆ A E 1 E ( 1 ) i 1 E ˆ B E 1 E ( 1) i E E A 1 B 3.1 (a) 1.5 (b) 0.6 (c) 4.4 (d). (e)@ 3.1 majr-031-q Questin % Fr the tw infinite dielectric sheets, see figure (5), find the magnitude f the electric field at a pint P. Cnsider that each sheet has a psitive surface charge density f 10 C/m. (a) N/C. (b) N/C. (c)@ N/C. (d) Zer. (e) N/C. majr-03-q Questin % Figure 7 shws prtins f tw large, parallel, nn-cnducting sheets, A and B. The surface charge densities are: 1 = -4.5 μc/m and = -6.5 μc /m. Find the electric field at any pint between the tw sheets. 17

18 Prf. Dr. I. M. A. Nasser GAUSS' LAW E 1, t the right because 1 6 ( ) N/C twards B (a) N/C twards A. (b)@ N/C twards B. (c) zer. (d) N/C twards A. (e) N/C twards B. majr-03-q Questin % A 40 N/C unifrm electric field pints perpendicularly tward a large neutral cnducting sheet, as shwn in figure 8. The surface charge densities (in C/m ) n the right, sigma-r and left, sigma-l, respectively are: E E C/m, S, the charges will be psitive n the left side and negative n the right side (a) ; (b) ; (c) ; (d) zer ; zer. (e)@ ; final-033-q Questin % Tw large flat nn-cnducting sheets have equal but ppsite surface charge densities. The distance between them is.0 cm. An electrn released frm rest frm the negative plate strikes the psitive plate after 15 nan-secnds. What is the magnitude f the surface charge density n each sheet? (a) 3.0 nan-culmb/m (b) 18 nan-culmb/m (c) 7.5 nan-culmb/m (d) 4.5 nan-culmb/m 18

19 Prf. Dr. I. M. A. Nasser GAUSS' LAW nan-culmb/m majr-04-q Questin % A charged, islated, large nn-cnducting plate is placed n the XY-plane. At 1.5 m frm the plate, n Z-axis, the electric field measured was 10 4 N/C and directed int the plate. What is the charge density n the plate? (a) (-7) C/m. (b) 3. 10(-7) C/m. (c) (-7) C/m. (d)@ (-7) C/m. (e) zer. 4-9 Applying Gauss' Law: Spherical Symmetry majr-991-q Questin % Which ne f the graphs shwn in Figure represents the variatin f the magnitude f the electric field with the distance frm the center f a slid charged cnducting sphere f radius R in electrstatic equilibrium? (a)@ #5 (b) # (c) #3 (d) #1 (e) #4 majr-993-q Questin % Tw cnducting spheres are far apart. The smaller sphere carries a ttal charge f 4 micr-c, and the larger sphere carries a ttal charge f micr-c. The larger sphere has a radius that is twice that f the smaller sphere. After the tw spheres are cnnected by a thin cnducting wire, the charges n the smaller and larger spheres, respectively, are: (a) 3 micr-c and 3 micr-c (b)@ micr-c and 4 micr-c (c) - micr-c and 8 micr-c (d) 0 micr-c and 6 micr-c 19

20 Prf. Dr. I. M. A. Nasser GAUSS' LAW (e) -4 micr-c and 10 micr-c majr-001-q Questin 5 A slid insulating sphere has a charge f 0 micr-c unifrmly distributed thrughut its vlume. The magnitude f the electric fields inside the sphere at r = cm and utside the sphere at r = 10 cm, measured frm the center f the sphere, are equal. Find the vlume charge density f the sphere. (a) 48 milli-c/m 3. (b)@ 4 milli-c/m 3. (c) 1 milli-c/m 3. (d) 54 milli-c/m 3. (e) 0 milli-c/m 3. majr-031-q Questin % A lng nncducting cylinder (radius 1.0 cm) has a charge f unifrm density 5.0 nan-c/m 3 distributed thrugh its clumn. Determin the magnitude f the electric field 5.0 cm frm the axis f the cylinder. [See figure (3)]. (a)@ 14 N/C. (b) 4 N/C. (c) N/C. (d) 34 N/C. (e) 31 N/C. qin E. da, surface qin V i r h rh 9 r E ( rh ) E 14 N/C final-031-q Questin % A nncducting shell has a unifrm negative charge f magnitude (-5) C. Its inner and uter radii are 5.0 cm and 6.0 cm, respectively. The electric field at r = 3.0 cm, frm the center, is: (a) N/C, inward. (b) N/C, utward. (c) N/C, inward. (d) N/C, inward. (e)@ zer. majr-03-q Questin % A hllw metallic sphere, f radius.0 cm, is filled with a nn-cnducting material which carries a charge f C distributed unifrmly thrughut its vlume. What is the magnitude f the electric field 1.5 cm frm the center f the sphere? (a) Zer. (b) 17 N/C. (c) 90 N/C. (d)@ 84 N/C. (e) 68 N/C. q E k r R N/C

21 Prf. Dr. I. M. A. Nasser GAUSS' LAW majr-041-q Questin % A nn cnducting sphere, f radius 4.0 m, has a charge density f.0 micr-c/m 3. What is the electric field at a distance 1.7 m frm the center? (a) N/C. (b) N/C. (c)@ N/C. (d) N/C. (e) N/C. Q Q R 3 3 R 3 q E k r 3 R N/C majr-991-q Questin % A slid insulating sphere has a charge f 0 micr-c unifrmly distributed thrughut its vlume. The magnitude f the electric fields inside the sphere at r = cm and utside the sphere at r = 10 cm, measured frm the center f the sphere, are equal. Find the vlume charge density f the sphere. (a) 1 milli-c/m 3 (b)@ 4 milli-c/m 3 (c) 0 milli-c/m 3 (d) 54 milli-c/m 3 (e) 48 milli-c/m 3 1

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