Assisnment 44

Gauss's Law

Select the conect ansryer

1. Which one of the following is not an expression for electric charge?

, loon o) !o-a.a {'t .[ tdt (d) f n'aa @) eo

vohEe area line aH

2. Which of the following cannot be a statement of Gauss's Law for some physioal situation?

(a) 4re"rzE=g (b) 2tre,rlE=Q g e"fiu-de=leav (a) s,fiE'dA= p

(e) ZI"EA= [odA

3. Electric field lines:(a) Intersect ot point afequal values'(c) Are dense at rcgions of small E.

4. Figure (1) shows the electric field lines in a region of space containing two

small charged spheres CY and Z)' Then:

(a) Y is negdive andZ is Positive'|ry *" *o!"ilude of theit "t'XS'eU

isthe swne everyohere' (4 the etic*ic 6eti is strongest midwoy between Y and Z(d)Y is positive and Z is negalive(e)Y and Z must hwe the same sign

[s at

(b) Can\ iiltersed at all@1ere dense at Points of constant E

r. - .,--h-. ii' lil

., ll-/,i.,- r ,.. : ., -lilat--. --6\:-}V.",*-Y.--

,/ 'i ... .,' t '-.

" tr

,'''

Fig'(1)

5. The flux of the electric field E =24i+3}i+16/t(I'i/C) through a

(a) i2 N'm2 /c (b) s4 N'rrf /c @) 42 N'ri /c

6. Figure (2) shows the electlic field lines for two charges Qt and ql2 separated by a

small distance.i. The rutio of q1lq2 is equal to

(a) 3/2 O) 1/3 (c) 2/3 (d) 3

ii. The sign of gy is(a) negative (b) positive

2.A t* portion of the xY Plane is:

(d) 4s I{.r# /c (4 6A N'd /c

(d) s/4

\;,r\ /,.8--r\\ i,/---<\\t t/,/ -^*-\\\/.lr*

a--l'\ -----\

--_\9.r///s,\'\//t 1\',

/1t,,,ii\\ LC-/ I\--l t--'/

7. Two concentric imaginary spherical surfaces of radius R and 2R respectively- Fig. (2)

surrounded a point n-egative ch trge glocated at the center of the surfaces' The

electric flux p1 through the surface of-radius R and the electric flux rp2through the surface of radius 2R

are related with:I

(a) gz= 4gt

I(b) Qz=i9, (c) 9z= Pt (d) gr=2A (e) g, = 49,

8" Figure (3) shows a point charge Q located at the center of the flat surface

of a hemisphere of radius R.i. The elictric flux through the curved surface of the hemisphere is

(a) Zero (b) Q/2e, (c) Q/ d @) Q/e,

@) Q/ ar*iii. The electric flux through the flat face of the hemisphere is

(a) zero @ A/nrt G) Q/za @) Q/e" (Q Q/ ar{

Point charge(Q)

g. The electric field just outside the surface of a hollow conducting sphere of radius 20 cm has a magnitude

of 500 N/C and directed outlvard, An unknown charge Q is introduced into the center of the sphere and it

is noted that the il;il n.td'ir still directed outward but has decreased to 100 N/C. The magnitude of the

charge Q is:(a) 1.5 nC (b) 1.8 nC (c) 1.3 nC (d) 1.1 nC (e) 2.7 nC

10. A hemispherical surface (ralf of a spherical surface) of radius R is located in a uniform electric field of

magnituie Ethatis paraliel to the axis of the hemisphere. what is the magnitude of the electric flux

Fig. (3)

through the hemisphere surface?

(4 r#e @) 4re2w/3 @) 2rNE/3 @) tNE/2 @) tNE/3

1i. A long nonconducting cylinder of radius 6 mm has a nonunlfoqnvolgg-1@ge density given b-y d,where a= 6.2mClms"and r is the distance from the axis of the cyiinaer. The magnitude of the electric

14. Figure (4) shows four regions separate three parallel infinite sheets of

"tirg. with different surface charge densities as indisated in the figure.

The electric field intensity in region (I) is equal to

field at a point 2 mm from the axis is:(a) 1.4 N/C (b) l.s N/C (c) 2.2 N/C @ s.6 N/C (Q 5.a N/C

12. Theelectric field intensity just outside a charged conductor of surface charge density o is:

(a) perpendiculat to its surface and has a magnitude dZen

@) peryendicular to i8 surface and has a magnitwde de,(c) parallel to its swface andhas amagnitude d2e.(d) paraltel to its swface and has a magnitude de;(e) Zero

13. Charge of uniform surface density (0.195 nc/m2) is distributed over the entire ry plane. Detennine the

*ugnitude of the electric field at any point havilq-z:2'0 m'

(a) 17N/c (b) I|N/C (c) B NIC @) 2S N/C @) 40 N/c

6tu)-r" 1eouoot Yi G)

to

5o. o.-

I (ti) --_-r2€" 2€o I

e) none ofthe aboveFie.(4)

15.Figure (5) shows charges distributed uniformly over a volume of hollow

"ytira"r of inner radius 0.02 m and outer radius 0.04 m. The distributions is

in{inite in extent with Z-axis with volume charge density p: 50 pC/m3. The

radial variation of the electric field intensity inside the hollow clyinder (0.02

<r < 0.04) is given by:

(a) Zero

(c) E=plrT-(0'02)2)

(e) E-

2r€o

pnlrz - (0.02)1

p= 5o YC/m3

(b) E= P',1cLOO

plrz +(0.02)21(o.1 a =

2r€"

A uniform line charge of x. : 3 pcim lies along the Z-axis and is

sunounded by a concintric cylindrical sheet of charge of radius 0'2 m as

shown in figure (6). The cylinder has surface charge density ofo = (-1.5/4n) pcln*. Both distributions are infinite in extent with

Z-axis. Using Gauss' law, find the electric field intensity at r: 0.1m and

r = 0.5m [eo: 8.85x10 t2 F/m]

A non uniform, but spherically symmetric, distribution of charge has a

charge density p(r) given as follows:

p(r)=r,(r-*)P(r)=0

t-@ Fig.(s)

_-_ /.

2.

4.

2r€o

FroblemsA 4g.0-cmdiameter loop is rotated in a uniform electric field until the po_sition of maximum electric flux

is found. The flux in this position is measured to be 5.20 x lOs N 'mzlc. what is the magnitude of the

electric field?Consider a closed triangular box resting within a horizontal electric

field of magnitude E:7.80 x 104 N/c as shorvn in Figure shown.

Calculate ttre electric flux through (a) the vertical rectangular surface, ;ft) the slanted surface, and (c) the entire surface of the box'

,;,T.,;

J.

,=-E rg1^'4r

for r<R

for r>Rwhere po is apositive constant. a) Find the total charge contained in the

I'-: Fig'(o

charge distribution. b) Obtain an expression of the electric field in the

region r > .R . c) Obtain an expression of the electric field in the region r S ,R, d) Graph the electric field

magnitude as a function of r. "1

rira the value of r at which the eiectric field is maximurn, and find the

value of maximum field.

H

ZI

I

-5. A closed surface with dimensions a:b:0.40m and c:0'60m is

Iocated as in Fig.(7). The left edge of the closed surface is located at

position x: a. The electric field throughout the region is nonuniform

and given by E = (3.0+ 2.0x:1l N/C, where x is in meters. Calculate

the net electric flux leaving the closed surface. What net charge is

enclosed by the surface?

A solid, insulating sphere of radius a has a uniform charge density p and

a total charge Q. -oncentric with this sphere is an uncharged, conducting

hollow sphere whose inner and outer radii are b and c, as shown in

Figure tho*r, (a) Find the magnitude of the electric field in the regions r.2, o < r I b, b < r <c, and r > c. O) Determine the induced charge per

unit area on the inner and outer surfaces of the hollow sphere.t,.

I\

) ca= \dt 1O=f^ lt=0 .lrtob C hq-rtt,fU'

^r I ,Pr,(v ?tlo t1

t) )*at?.) /^dt )d) f t" clA

ne) €) e "d..A

Q=

q

a:

f"lV

=*Cl

AL

Ass+e dq,. fdu gQ

)dv.*a{laQ=l//u=f*"u

{=

Jt=

td

I?.,)

en

3

s)

t)

5) r_L-

a|.L^

: EA: (te) (e) :

is t4 ZE//

c

^( "11

- 'L/<

1

-j

+kql ,t

Kqa?

€ "1A

n oL ia6 q'sk(\"

+w(zd,

Q = € reRq

a

= +Tt€, Er"= € A E"

=€[ E -CA

4

6)

-TI-

i)

tL)

z+t f sos + t6k

31-N,^,/ c{i,<t(on

V. Ua- tk

-\Aq s<4_L<

liLLl Eal ntn at\a"

1b

nk qlri c

7)

8)

il

@,: Q"C

fr,-^ctc1h t h<nisph<y \a -_: % ' ae- zE,

b

Z<,r ati) 1),",*1 /, r Fla t

c\))

cq't<r sy h"t* :

5c -ica a 4oc

/ir kC

?oa = -{-:{-( o. ,1"

)D\ i ,v le,l =(e,l

Q) ,,= rA: E rr[<za

t'tl

i

<<---- P\"--' (Xrs /l'

s,y nn t 11

E, --"r

hA

lt4

Q=b

1^/" b .,c

II

"..--"-."*--*:-.L-2.. rr, , , :. i, -| -' t I

=7< n hlt't I't

rr)t e=+9,n =) / clv , - f'. 1''zrrl2

'V= TfrhJv=tvrh d,

(p.- j e 14

=' tuot zrr l,)

-\

Etqnrl)= c(

f = >< 13

----=---=-:-t'b.

rrh f+T*

a.fil{ (o ''L x8.5D{6='? -

(='L n'\

F. --(,

ct(

6.t x J)*iu! I "+ *l e

oH.6"e.

r/)s)

,nrrc1nrt.*'/ll

V,

r.) b ^^,1

L_t"

?6-- ?€.

=L€------_--_------ ,

_g.Eo- 'iLc',

Ir)

J.i \'j ,1.

aQ:

Il5)

Q= €A = € LTrrA

%a= f Vz / ( ffr uh tr a.uz'L )

EzTr_*= f (/ / Li

I t rL o,ozJC f {t O'tt-L,

-'

C ZrL-

' P roblens,r\ )

€ = ,n 'Lo

,'*

e1 E A= € TTrZ

5.2 N ro5 -- E rr (0, z) ,

E = +l (/06

I

i

!

N.a,/ <

*, lc

L)a) a.: €"4

E A ces&7, k{tcr+ * 3oc:

'23+Kto, .A),e'AE A C",.s p7,Ei( /ol &

+ cos ik0N.*,n ,l C

,rzftb)

{13 or (A

c) ri zw,.-Y tot.t(

3)

p

e

a6 /^ =d.l

= E (*rrh)

\ ,3 )r/*

4,L *e- ra-rh

C"

E (''r'x): 4t*---_'zrf (E"

.3xlo-6z rr 0.t ,t giI)-,,

€-

6 -qJ

^(

/' -+."/

E--

e=q(.9,, n ,

€"

e"

r:

€.

(p", e (znrl)

@{"

3 x /b 6 x ($7--) x,o-6.t rrxo.yz tT \ o.s) 4. vx rb-i?

[- (zurfi =

c = -\ luF-r..oR

+) \-T'cx - ,P JuV: + rrf 3

otv = '4 rT t"' 4n

rA

\c)

C)(- I

f t( ,-^S) ( +7Tr?dr)

f+rr f f' C dr/;' ) ?-o

,t ni f fr --P*-rL+- -#rl

€n4r,o .

r:

(

aa

q, "-IlrL a ): (t- .3i) 4rrzr"vf5 ^r|- /l I

='t'/ n IaJc,

J4

4\flsn/

= '/<zx

a(. o

= lxro

- En= G (+u")

e @,rr') : 3 €,

E

€ *oo

[,,*n

r,F lclr

l-

--a-..t :,'

(

2RrzG,

) Qr/i

+:orzr I

AI

r-_ R2

R7-flF

E r

3lz(o*.) '(oL)3 te(r)"(0.*)(a.()

(s +?x?) e

=b= 6" t: O,6

cL

C.

e A casl&,b

G * z d) (,"b) (-,J(s *iod) (0,*lto.e)

^3

4

I

IQ",

Q,r[t -^

0€"6)

6.@a2

o--=G,u,0

f'rf6-c'