Supplementary Problems
1.16.
Given A = 4ay + lOa,
1.17.
Given A = (l~/l/i)(a, + a,) and B = 3(a,
the direction of A.
Ans. 1.50(ax +- a,)
1.18.
and B = 2 q + 3a,.
Find the angle between A = 10a,
Am. 161.5'
+ 2a,
find the projection of A on B.
+ a,),
and
Am.
12/fi
express the projection of B on A as a vector in
B = -4q
+ 0.5a, using both the
dot product and the
cross product.
1.19.
Find the angle between A = .5.8aY+ 1.55a2 and B = -6.93ay
and the cross product.
Am. 135"
i4. Oa,
using both the dot product
+
1.B. Given the plane 4.r 3y + 22 = 12, find t h e unit vector normal to the surface in the direction away
fromtheorigin.
Ans. (4a,+3ay+2a,)/V'%
L.21.
Find the relationship which the cartesian components of A and B must satisfy if the vector fields are
everywhere parallel.
1.22.
Express the unit vector directed toward the origin from an arbitrary point on the line described
by x = O , y = 3 .
1.23.
Express the unit vector directed toward the point
plane y = -5.
Am,
a=
(1,-x)B,+
(y, +5)a,
(A,,
y,, z,) from an arbitrary point in the
+(z,-:)a,
G I+ (?',
)+~
5)2 + (I] :I2
-
1.24.
Express the unit vector directed toward the paint (0, 0, h ) from an arbitrary point in the plane
1.25.
Given A = 5 1 and B = 4% + B,a,. find B, such that the angle between A and B is 45". I f B also
Ans. 3,= k4,
3: = 4
has a term Bra,, what relationshkp musl exist between 8,and Bi?
4Show that the absolute value of A . B
1.26.
mi.
X
(Hint: First sbow that the hase has area
1.27.
Given
C is the volume of the parallelepiped with edges A, B, and C.
(BX CI .)
B = 3 u , + a Y , and C=-Za,+ha,-4a,,
A=2a,-a,,
1.28. Given A = a , - % ,
B = 2 a , , and C = - a , + 3 a , ,
this scalar triple product.
Am. - 4. f4
showthatcis 1 tobothAandB.
find A - B X C . Examine other variations of
1.29.
Using the vectors of Problem 1.23 find (A x B] X C.
1.30.
Find the unit vector directed from (2, -5, -2) toward (14, - 5,3).
Am.
12
z = -2.
Am.
-8a,
5
a=-a,+-a,
13
13
1.31.
Find the vector directed from 1 1 O , 3 ~ / 4 ,n/61 to (5, .-r/4,
n),where the endpoints are given in spherical
coordinates.
Am. -Y.Ma, - 3.54a, + 10.61a,
1.32.
Find the distance between (2, n/6, 0) and (1, x . 21, where the points are given in cylindrical
coordinates.
Am. 3.53
1.33.
Find the distance between ( 1 , n14,0 ) and (1,3x/4, n), where the p i n t s are given in spherical
ct~ordinates. Am. 2.U
1
1.34.
Use spherical coordinates and integrate Lo find the area of the region 0 5 @ c LX on the spherical shell
of radius a. What is the result when a = 2n?
Ans. 2aa1, A -=47ra'
1.35.
Use cyIindrical coordinates to find the area of the curved surface of a right circular cylinder of radius a
and height h.
Atu. 2nok
12
[CHAP. 1
VECTOR ANALYSIS
1.36.
Use cylindrical wrdinates and integrate to obtain the volume of the nght circular cylinder of Problem
1.35.
.4n5. na2h
1.37.
Use spherical coordinates to write the diffrrential surface areas dS, and dS, and then integrate to obtain
the areas of the surfaces marked 1 and 2 in Fig. 1-14.
Am. ~ 1 4~, 1 6
Fig. 1-14
1.38.
Use spherical coordinates to find the volume of a hemispherical shell of inner radius 2.00 m and outer
radius 2.02 m.
Am. 0.162~m3
1.39.
Using spherical coordinates to express the differential volume, integrate to obtain the volume defined
77d
by l 5 r 5 2 m , O ~ B s x / 2 , and 05#51r/7. Am. -m3
6
Supplementary Problems
Two point charges, Q , = 250 pC and Q,= -300 pC, are located at (5,0,0) m and ( O , O , - 5 ) m ,
respectively. Find the force on Q,
Two point charges, Q,= 30 pC and
respectively. Find ths brce on
a +a
F: = ( 1 3 . 5 ) ( ~ ) N
AN.
el.
In Problem 2.26 find the force on Q2.
QZ
= -100 P C ,
Am.
Ans.
are located at (2.0,5)rn and (-1,O. -2)
-3a - 7 a
F, = ( 0 . 4 6 5 ) ( d 2 )
fi
rn,
N
/
-F,
Four point charges, each 20 PC, are on the x and y axes
charge at (O,0, 3) m.
Am. 1.73a, N
at
i d m.
Find the force on a 100-PCpoint
Ten identical charges of 500 pC each are spaced equally around a circle of radius 2 m. Find the force
Am. (79.5)(-a,) N
on a charge of -20 pC located on the axis, 2 m from the plane of the circle.
Determine the force on a point charge of 50 pC at (O,0,5) m due to a p i n t charge of 5 0 0 p~ C at the
origin. Compare the answer with Problems 2.4 and 2.5, where this same total charge is distributed over
a circular disk.
Am. 2 8 . 3 N
~~
Find the force on a point charge of 30 pC at (0.0,s) m due to a 4 m square in the z = 0
between x = *l m and y = i2 m with a total charge of 500 P C , distributed uniformly.
Am. 4.Mm, N
plane
Two identical point charges of Q ( C ) each are separated by a distance d ( m ) . Express the electric field E
for points along the line joining the two charges.
Ans. If the charges are at r = O and x = d, then, for 0 < x < d ,
Identical charges of Q(C)are located at the eight corners of a cube with a side f (m). Show that t h e
coulomb force on each charge has magnitude (3.29Qz/4nr,P) N.
Show that the electric field E outside a spherical shell of uniform charge density p, is the same as E due
to the total charge on the shell located at the center.
Develop the expression in cartesian coordinates for E due to an infinitely long, straight charge
configuration of uniform density p r .
PI xa,+ya,
E =--
Am.
2x6, x Z + y 2
TWOuniform line charges of p, = 4 nCim each are parallel to the z axis at x = 0, y = f4 m.
Determine the electric field E at (f4 , 0,z ) m.
AW. f 18a, V/m
Two uniform line charges of p , = 5 nC/m each are parallel to the x axis, one at r = 0 , y = -2 rn
and the other at z = 0, y = 4 m. Find E at (4, 1,3)m.
Am. 30a, V / m
Determine E at the origin due to a uniform line charge distribution with p, = 3.30 nC/m located
at x = 3 m , y = 4 m .
Am. -7.13~~-9.SOa,V/m
Referring to Problem 2.38, a t what other points will the value of E be the same?
Am.
(0, 0, r )
Two meters from the z axis, [Eldue to a uniform line charge along the z axis is known to he
1-80x 10' Vim. Find the uniform charge density p,.
Ans. 2.0 pC/m
The plane -x + 3y - 6x = 6 m contains a uniform charge distribution p, = 0.53 nC/m2. Find E on
the side containing the origin.
Am.
30(
a.
- 3ay + b,
v%
-)v/m
Two infinite sheets of uniform charge density p, = (10-'/tin) C/m7 are located at z =
-5 m and y = -5 m. Determine the uniform line charge density p, necessary to produce the same
value of E at ( 4 , 2 , 2) m, if the line charge is located at z = 0, p = 0 .
A m . 0.667 nClm
Two uniform charge distributions are as follows: a sheet of uniform charge density p, =
-SOnC/rn2 at y = 2 m a n d a u n i f o r m l i n e o f p,=O.ZpC/m at z = 2 m , y = - l m . At what
Am. (x, -2.273, 2 . 0 ) m
points in the region will E b?zero?
A uniform sheet of charge with p, = (-113x1nC/m2 is located at z = 5 m and a uniform line of
charge with p, = (--2519)nC/m is located at z = -3 m, y = 3 m. Find the electric field E at
(0,-1,O)m.
Am. 8 j V l m
CHAP. 21
2.45.
31
COULOMB FORCES AND ELECTRIC FIELD MTENSITY
A uniform line charge of p, = ( ~x 210-'/6) Clm lies along the x axa and a uniform sheet of charge
is located a1 y = 5 m. Along the Line v = 3 m, r = 3 m the eleclric field E has only a z
component. What is p, for the sheet?
Ans. 125 pC/nll
2.46. A uniform line charge of pp = 3.30 nC/m i s located at x = 3 m, y = 4 m . A point charge Q is 2 m
from the origin. Find the charge Q and its location such that the electric field is zero at the
m
orig$n.
Arts. 5 -28nC at (- 1.2, - 1 -6,O)
2-47.
A circular ring of charge with radius 2 m lies in the 2 = 0 plane, with center at the origin. If rhe
uniform charge density i s p, = 10 nClm, find the point charge Q at the origin which would produce
the same electric field E at (0, il. 5) m.
Atn. 100.5nC
2.48.
The circular disk r 5 2 m in the
2.49.
,Examine the result in Problem 2.48 as k becomes much greater than 2 rn and compare it to the field at h
which results when the total charge on the disk is concentrated at the origin.
2.50.
A finite sheet of charge, of density p, = k ( x 2 + ,v2 + 4)m (C/m2), hes in the
O ~ x 5 2 mand U s y S Z r n . Detenn~neEat(O,O,Z)m.
plane has a charge density p3= 1 O - v ~ (C/lm2).
1.13 X 10"
Determine the electric field E for the point (0. @, A ) .
Am.
-a, (Vim)
h
m
Am.
(18 x
lov)(-
16
- a,
3
2
=O
i
- 4a, + 8.. V/rn = 18
i :"
--a,
- 4a,
+ .8.
z = 0 plane for
1
CVIm
-
2.51.
Determine the electnc field E at (8,0,O j m duc to a cl~argeOF 10nC distributed uniformly along the x
axis between x = -5 m and x 5 m. Repear for the same total charge distributed between
x = - l r n and x = l m .
Am. 2.31a,Vlm,1.43iixV!m
2.52.
The circular disk r 5 1 rn, z = 0 has
{ 0 , 0 , 5 ) m.
Ans. 5.66axGV/m
2.53.
Show that the electric field is zero everywhere Inside a uniformly charged spherical shell.
2.54.
Charge is distributed with constant density p thraughout a spherical volumz of radius a.
results of Problems 2.34 and 2 . 5 3 , show that
a
charge density p, = 2(r2 + 2 5 ) ' R e - ' " ( ~ / m ' ) . Find E at
where r is the distance from the tenter of the sphere.
By using the
ELECTRIC FLUX AND GAUSS' LAW
Fig. 3-18
Supplementary Problems
3.18.
Find the net charge enclosed in a cube 2 rn on an edge. parallel to the axes and centered at the origin, if
the charge density i s
3.19.
Find the charge enclosed in the voiume I 5 r -= 3 rn, 0 < @ s n / 3 , 0 5 2
dens~ty p = 22 sin' # (Cirri').
Am. 4.91 C
3.20.
Given a c h ~ r g edensity in spherical coordinatts
PI closed surface S
contains a finite line charge distributinn, O 5 t
P
pt = - p,, sin -
2
What net flux crosses the surface S?
3.22.
Am
-2p,,
I7r
2m
given thc cliaige
r = r d , r = St,, and r = =.
find the amounts of charge in the spherical volumes enclosed by
.4ns. 3.97p,,r:, b.24pl,q',, 6.2Hp,d
3.21.
5
m, with charge density
(Clml
(C)
Charge is distr~buiedIn the spherical region r i
2 rn with density
P=
-200
7
(pC/m")
What net flux crosses the surfaces r = lm, r = 4 m ,
Ans. -WOn fiC. -Ir3Mx PC, -1600,~ pC
and
r =5mm?
3.23.
A point charge 0 is at the ctrigin of spherical coordinates mil a spheric*] \hell chargc distribution
at r = a has a total charge of Q' - Q, uniformly distributed. What flux crosses the surfaces r =
k for k < a and k :>a?
Ans. Q , Q'
3.24.
A uniform I~necharge with pr 3 pC/m l ~ c sa1011gthe 1 axis. What flux crosses a spherical surface
centered at the origin with r = 3 tn?
Am. 14 pC
3.25.
If a point charge Q is at the origin, find an expression for the flux which crosses the portion
-
cznrered at the origin, described by
a 5 #J 4 15.
Ans.
P2n
df
a sphere.
3.26.
A poin! charge of Q ( C ) is at the center of a spherical coordinate system. Find the flux Y which crosses
an area of 4rc mZ on a concentric spherical shell of radius 3 m.
Ans. Q/9 (C)
3.27. An area of 40.2m2 on the surface of a spherical shell of radius 4 rn is crossed by 10pC of flux in an
inward direction. What point charge at the orrgln is indicated?
An.. -50 pC
3.28.
A uniform line charge pp lies along the x axls. What percent of the flux from the line crosses the strip
of the y = 6 plane having - 1 5 z 5 I?
AIM. 5.26%
3.29.
A p a n t charge, Q = 3 nC, is located at the origin of a cartesian coordinate system. What flux W
crosses the portion of the z = 2 m plane for which -4 5 x 5 4 m and - 4 5 y s 4 m?
Ans. O 5 nC
3.30.
A un~fclrmline charge with
2ay + a,
A .
)
(0.356)j
p,
= 5 pC/m
lies along the x axis. Find D a! (3.2.1) m
rc/m2
3.31.
A point charge of + Q is at the ongin of a spherical coordinate system, surrounded by a concentric
uniform distribution of charge on a spherical shell at r = u for which the total charge is -Q. Find
the flux Y crossing spherical surfaces at r < a and r > a. Obtain D in all regions.
3.32.
Given that
D = 500e-0,'"a, ((tC/m2), find the flux Y crossing surfaces of area 1 rn' normal to the x
u i s a n d l o c a t e d a t x = l r n , x = 5 m , and r = l O m .
.4m. 452pC,3U3pC,184pC
3-33. Given that D = 5x2a, + 10za, (C/rn2), find the net outward flux crossing the surface of a cube 2 m
on an edge centered at the origin. The edges of the cube are parallel to the axes.
Am. SO C
3.34.
Given that
in cylindrical coordinates, find the outward Aux crossing the right circular cylinder described
by r = 2 b , r = 0 , and z = Sh (m).
Am. 129b2(C)
3.35. G ~ v c nthat
in cylindrical coordinates, find the flux crossing the portion of the z = 0 plane defined by r 5 a,
0 i # 5 n/2. Repeat for 3 r / Z 5 @ 5 2n. Assume flux positive in the a, direction.
Ans.
a u
--
-
3'3
3.36.
In cyclindrical coordinates, the disk r s a , r = I ) carries charge with nonuniform density
r
$1. Use appropriate special gaussian surfaces to find approximate values of D on the z axis ( u )
very close to the disk (O < r e a ) , ( b ) very far irutn the disk ( r + a ) .
P (0.
Q
ART. ( a ) - - .
, b )
where
p.(r. 6 ) r d r d $
3.37.
A point charge, Q =20W pC. i s at the orign of sphcncal coordinates. A concentric spherical
distribution of charge at t = 1m has a charge density p, = 40n pC/m'. What surface charge
density on a concentnc shell at r = 2 m would tesult in D = 0 for r > 2 m?
&hcad
Am.
-71.2pC/m'
3.38.
Given a charge distribution with density p = 5r (Clm')
find D.
Am. (5r1/4)a,lC/mz)
3.39.
A uniform charge density of 2 C/rn"exists in the volume 2 5 x 5 4 m (cartesian coordina~es). Use
AM. -2a, C/mz, 2(x - 3)ax (C/rn2), 2 9 C / m 2
Gauss' l a w to find D in all regions.
3.40.
Use Gauss' law to find D and E in the region between the concentric conductors of a cylindrical
Am. p,,(a/r), p , , ( a / ~ ~ r )
capacitor. The inner cylinder is of radius a. Neglect fringing.
3.41.
A conductor of substantial thickness has a surface charge of density p,. Assuming that V = 0 wthin
the conductor, show that D = k p , just outside t h e conductor, by constructing a small special
gaussian surface.
in spherical coordinates. use Gauss' law
to
DIVERGENCE AND THE DlVERGENCE THEOREM
CHAP. 4)
For the right side of the divergence theorem:
jC ( T . D ) d u = p ~ ~ ~ ~ ~ r ) r ' r i n B d r d ~ d @ = S b 9 ~ C
and
Supplementary Problems
4.20,
Develop the d~vergence In spherical coord~nates. Usc the delta-volume with edges Ar, r A@, and
r sln 0 A+.
4.21,
Show that P . E is zero for the held of a uniform sheet chargt
4.22.
The field of an electric dipole with the charges at fd / 2 on the z axis is
E;-
Vd
(2cos ba, +sin 8s.)
ln~,,r~
Show that the divergence nf th~sfield is zero.
Given
A=e5"a,+2cosya,+2sinza,,
firiilV-Aattheoripin.
findV.A.
Given A=(3x+y2)a,+(x-!')a,,
Am.
A
= 4rya,
- xy'a, + 5 sin za, ,
Given A = 2 r cob2 @ar+ 3r'sin ,ac
G ~ v e n A = (lD/r')a,
Ciiven
+ 5e-"a,,
-
A = 5 cos ra, + (3:e-3,'r)a, , find '? A
Given A = IDa,
+ 5 sin ee, ,
Cilvcn
A=ra,-r2cotBaR,
Given
A = [ ( l U sin' P)/rja,
= r'rin
0..
find C . A.
find C - A at ( 2 , I$, 1).
at
(n.@,
-6.0
Ans.
find V A at (2,z.O).
+ 4z sin2 $ at .
Am.
Atis
-2.6CI
Am.
2).
Am.
( 2 + cos B ) ( l O / r )
find'7-A.
Ans.
3-r
V
(Nlm), find G . A
+ 13$a, + 2 r q ,
5.0
Anr. 4.11
.A .
find
- 1.59
at (2 m, n/4 rad, z/2 rad).
find V . A .
4.33.
Giwn A
4.34.
Show that tbc divergence of E is zero if
4.35.
In the region a 5 r 5 b (cylindrical coordinates),
E = (10U/r)a,
AM.
7.0
3-2y
Given A = Z r y a , + ~ ~ , + ~ z ' m ,f ,i n d C v A a t ( l , - 1 , 3 ) .
Given
Am.
4r sin 8
+ 40a,.
Ans.
13#
+ (T)
cot t)
1.25 N/mZ.
DIVERGENCE AND THE DIVERGENCE THEOREM
and for
For
[CHAP. 4
r > b,
Am. O,p,,l)
D=O. Findpinallthrteregions.
r<a,
1.36.
In the region 0 < r 5 2 (cylindrical coordinates). D = (4r-' + ~ c - O" + 4 r - I ~
2, D = (2.057/r)a,. Find p in both regions.
Ans. - c - O ", o
)a,,
4.37.
In the region r 5 2 (cylindrical coordinates), D = [ l o r
13/(128r)]a, Find p in both regions.
Atas. 20 + r . 0
+ ( r 2 / 3 ) ] a , , and
for
4.38.
Given D = 10s1n Ba,
4.39.
Given
.
+ 2 cos #aA ,
find the charge density.
in spherical coordinates, find the charge density.
4.40.
Ans.
r > 2,
D=
sin 8
5
-(18 + 2 cot2 0)
+ 3 ) / ( r 2+ I
) ~
Given
10
D = 7 [I - e "(1
r-
in spherical coordinates. find the chargc density.
4.41.
3(r2
Aw.
and for r >
In the region r
and for
5
I
+ 2r + 2r2))a,
Am.
40e-"
(spherical coordinates),
r > 1, D = [5/r63r2)]a, . Find the charge density in both regions.
Am.
4 - r2, 0
4.42.
The region r 5 2 rn (spherical coordinates) has a field E = ( S r x l ~ - " ~ , ) a , (Vlm). Find the net
charge enclosed by the shell r = 2 m.
Ans. 5.03 x
C
4.43.
Given that D = (5r2/4)8, in spherical coordinates, cvnluate both sides sf the divergence theorem f v r
the volume enclosed between r = 7 and r = 2.
Am. 75n
4.44.
Given that 0 = ( 10r3/4)a, in cylindrical coordinates, evaluate both sides of the divergence theorem
for the volume enclosed by r = 2 , z = 0, and z = 10.
Am. 8 0 0 ~
4-45.
Given that 1)= 111 sin Ba, + 2 cos Oa, , evaluate both sides of the divergence theorem for the volume
enclosed by the shell r = 2.
Am. Nx'
THE ELECTRDSTATJC FIELD: WORK, E N E R G Y . AND POTENTIAL
CHAP. 51
potential represents
Note that the total charge on the shell is, from Gauss' law,
while the puter~tial at thc shcll is
= K , . Thiis,
W , = ;@V. the familiar resuit for the energy
stored in a ca?acitor (in this case, a spherical capacitor with the other plate of infinite radius).
Supplementary Problems
5.29.
Find the work done in moving a point charge Q = -20 (LC from the origin to (4,2,O) m in 1he field
E =Z(x
along the path x2 = 8y.
5.24.
Am.
+4yjax
+ Ua,
(Vfrn)
1.61JmJ
Repeat Problenl 5.2using the direct rad~alpath.
-39.35 pJ (the nature of the singularity along the z axis makes the field nonconsewative)
Am.
5.21.
Repeat Problem 5.2 using the path shown in Fig 5-14.
Ans
-1 17.9pJ
F i g 5-14
5.22,
Find the work done in moving a point charge Q = 3 pC from (4 m, lr, 0) to (2 m.ni?. 2 m),
cylindrical coordinates, In [ l ~ field
c
E (li15/r)a, + 1U5ra, (Vtrn).
Arzs. -0.392 J
;
5.W.
Find the difference In the anlounts of work required to bring a point charge Q = 2 nC from infinity
to r = 2 m and Erum iufinity to r = 4 m, in thc field E = ( 1 0 5 / r ) a , {V/m).
Ans. 1 . 3 9 ~1U'J
5.24.
A uniform 1i1w chargc of dcnsity p, = 1 n C l m i s arranged in the form of a square 6 m on a side. as
showninfig. 5-15, Find the potential ar ( O . 0 , S ) m .
Am. 35.6V
5.25.
IkveIup an expression for the potential at a point d meters rad~allyoutward horn the midpoint of a
finite line charge L meters long and of uniform density p , (C/m). Apply this result to Problem 5 . 1 4 as
a check.
--L / l d d 2 + 1?/4
Am. - p2 I n - (V)
- --
--
216~~
+
d
THE ELECTROSTATIC FIELD: WORK, ENERGY. AND POTENTIAL
[CHAP. 5
X
Fig. 5-15
5.26.
Show that thr potential at the origin due to a unlform surface charge dens~ty p, over the
ring z = 0 , R c r 5 R + 1 is independent of R.
5.27.
A tola1 charge of lb(J nC is first separated Into four equal point charges spaced :it YOo intervals around a
circle of 3 m radius. Find the potential at a point on t h r axis, 5 m from the plane uf the
c~rcle. Separate the tnral charge into eight equal parts and repeat with the charges <it 45'
intervals. What would be [he answer in the limit p , = (tbCl/b..c) nCJm?
Ans. 247 V
5.28.
In spherical coordinates, point A is at a radius 2 m while R is at 4 m . Given the field E =
(-161r2)a, (Vlm), find the potential of point A , zero reference ;it infinity. Repeat for point
3. Now express the potential difference V, - C:, and compare thc result with Problem 5.6.
.4m. v* = 1v* = -8 v
5.29.
[f thc zero p a t e n l l n l reference is at r = 10 m and a point chargc Q = 0.5 nC is at the o r i g n , find
the potentials at r = 5 rn and r = 15 m. At what radius is the potential the same in magnitude as
thatat r = 5 m butoppositeinsign?
,411.v. 0 . 4 5 V , - 0 . 1 5 V . x
5.30.
A pohnt charge Q = 0.4 nC i s located at ( 2 , 3 , 3 ) m in c a r t w a n coordinates. Find the potentla1
difference V',,, where point .4 IS ( 2 , 2 , 3 ) m and 3 is ( - 2 , 3 , 3 ) m.
Ans. 2.70 V
5.31.
Find the potential in sphericdl coordinates due lo two cqual hut opposite point charges on the y axis
at y = fd / 2 . Assume r d
Ans. (Qd sln ~ ) / ( 4 x ~ , , r ' )
5.32.
Repeat Prrjhlznl 5 . 3 1 with the charges nn the z axis.
5.33.
Find the charge densities on the conducturs in Problem 5.14
5.34.
A uniform h r ~ echarge p , = 2 nC/m lies in the .z = O plane parallel 10 the axis dt y = 3 m . Find
A m . - 18 4 V
the potentla1 difference V,, for the points d ( l nl. 0, 4 rn) and B(0,0. 0)
5.35.
A
5-36,
Given the cylindr~cal coordinate clcctric fields E = ( S / r ) a , (Vim) ftlr O s r ~ 2 r nand
2 . 5 ~ 1V/m
,
for r > 2 m, bind the putcntial difference V,, for .,I (1 nl, 0,O) and B(4 m,O,0).
*
uniform sheet of chargc, p, = ( l / h n ) nC/m2. i s at x = C) and a second sheet, p, =
(-l/hn)nC/m',
i s a t r = l O m . Find V , , , V,,., and
for A(lnm,O,O), B(Jm.(J.O), and
C(O,O,O). A m . - 3 6 V , - 2 4 V , - 6 0 V
Ans.
5.37.
Am. ( Q d cos 8 ) / ( 4 a ~ , r ' )
v,,
E=
8.47 V
A parallel-plate capacitor 0.5 rn by 1.0m, ha5
;I scp,iration dist;lnue of 2 crn and a voltage difference of
LOV. Findtliestor~denergy.assumingtha~r = c , , .
Ans. I l . l n J
CHAP. 57
5-38.
THE ELECTROSTATIC FIELD. WORK, ENERGY, AND POTENTIAL
75
The capacitor described in Problem 5.37 has an applied voltage of 201, V .
Find the stored energy.
( b ) Hold d, (Fig. 5-16) at 2cm and the voltage difference at 2 W V , while ~ncreasing dl to
2.2 cm. Find the final stored energy. [Hinr:
A W, = $(Ac)v*]
(a)
Ans.
( a ) 4.4pJ; ( b ) 4.2pJ
Fig. 5-16
5.39.
Find the e n e r a stored in a system of three equal point charges, Q = 2 nC, arranged in a line with
0.5m separation between them.
Am. 180 nJ
5.a.
Reprat Problem 5.39 if the charge in the center is -2 nC.
5.41.
Four equal point charges, Q = 2 nC, are to he placed at the corners of a square 4 m on a side, one at
a time. Find the energy in the system after each charge is positioned.
Any. 0, 108 nJ, 292 nJ, 585 nJ
5.42.
Given the etectric field E = -5e-""a, in cyl~ndricalcoordinates, find the energy stored in the volume
described by r 5 2a and 0 5 z 5 50.
Ara. 7.89 x 10-l"u'
5.43.
Given a potential V = 3x2 + 4y2 (V), find the energy stored In the volume described by 0 5 x
l m , O ~ y S l m , and O c z l l r n .
Ans. 137pJ
Am.
- 108
nJ
5