Homework (lecture 1):
● Prove the shell theorem for the electrostatic force
● 1, 3, 6, 9, 12, 20, 26, 29 (page 574-576)
1. Two equally charged particles are held 3.2x10-3 m apart and
then released from rest. The initial acceleration of the first
particle is observed to be 7.0 m/s2 and that of the second to be
9.0 m/s2. If the mass of the first particle is 6.3x10-7 kg, what
are (a) the mass of the second particle and (b) the magnitude of
the charge of each particle.
F12  m2a2
F21  m1a1
F12  F21 : m2  a1 m1
F12  F21  k
2
q1q2
r
2
a2
 m1a1
7

m1a1r
6.3 10  7 3.2 10
q

9
k
8.99 10
q  71 (pC)

3 2
 7.110
11
(C )
3. What must be the distance between point charge q1 = 26.0 C
and point charge q2 = -47 C for the electrostatic force
between them to have a magnitude of 5.70 N?
F k
q1q2
r
1C  10
r
6
2
q1 q2
r k
F
C
9
6
8.99 10 (26 10
5.7
)(47 10
6
)
 1.39 (m)
6. In the figure as shown, four particles form a square. The
charges are q1 = q4 = Q and q2 = q3 = q. (a) What is Q/q if the
net electrostatic force on particles 1 and 4 is zero? (b) Is there
any value of q that makes the net electrostatic force on each of
the four particles zero? explain.
(a) q1 & q4 have the same sign,
all three forces act on q1
as shown 


F41  F21  F31
F41  2F21
Q2
qQ
Q
k
 2k

 23 / 2  2.83
2
2
q
a 
a 2


q
(b) if the net force acting on particle 3 is also zero:
 2.83
 this is inconsistent with (a), so the answer is NO Q
9. Two identical conducting spheres, fixed in place, attract each
other with an electrostatic force of 0.108 N when their centerto-center separation is 50.0 cm. The spheres are then connected
by a thin conducting wire. When the wire is removed, the spheres
repel each other with an electrostatic force of 0.036 N. Of the
initial charges on the spheres, with a positive net charge, what
was (a) the negative charge on one of them and (b) the positive
charge on the other?
Using the shell theorem:
Before:
F1  k
q1q2
r
2
 q1q2  3.0 1012
(C )
After: the net charge is positive
2
 q1  q2 


2
q
2 

F2  k
k
 q1  q2  2.0 10 6
2
2
r
r
(C )
2
6
12
q1  2.0 10 q1  3.0 10
0
The solutions:
1st:
2nd:
q1  3.0 10
6
q1  1.0 10
(C )  q2  1.0 10
6
6
(C )  q2  3.0 10
6
(C )
(C )
Note: if the net charge is negative
q1  q2  2.0 10
6
(C )
the solutions should be:
q1(or q2 )  1.0 10
6
(C )  q2 (or q1)  3.0 10
6
(C )
12. See the figure as shown, particle 1 (of charge q1) and
particle 2 (of charge q2) are fixed in place on an x axis, 8.0 cm
apart. Particle 3 (of charge q3 = +8.0x10-19 C) is to be placed on
the line between particles 1 and 2 so that they produce a net
electrostatic force F3,net on it. The diagram gives the x component
of that force versus the coordinate x at which particle 3 is
placed. What are (a) the sign of charge q1 and (b) the ratio
q2/q1?
● at 2 cm: Fnet=0 so 1 and 2 must have
the same sign.
● when q3 approaches q1,F13
increases in magnitude. Fnet
increases in the positive x direction,
so F13 is a repulsive force,
q1 > 0
qq
q q
k 1 3 k 2 3
r 213
r 2 23
q2
q1
2
2
 r23 
6

     9
 
2
 r13 
20. See the figure, particles 1 and 2 of charge q1 = q2 =
+3.2x10-19 C are on a y axis at distance d = 17.0 cm from the
origin. Particle 3 of charge q3 = +6.4x10-19 C is moved gradually
along the x axis from x = 0 to x = +5.0 m. At what values of x
will the magnitude of the electrostatic force on the third particle
from the other two particles be (a) minimum and (b) maximum?
what are the (c) minimum and (d) maximum magnitudes?
The net force acting on particle 3:
qq3
Fnet  2  k
r
r
2
 cos 
x
x  d ; cos  
r
2
2
Fnet  2kqq3
x
x
2
d

2 3/ 2
r
 x  0 : minimum
F net  0  d  2 x  0  x  d / 2 : maximum
'
2
2
http://www.function-grapher.com/index.php
26. Calculate the number of coulombs of positive charge in 250
cm3 of (neutral) water. (Hint: A hydrogen atom contains one
proton; an oxygen atom contains eight protons)
The mass of the sample:
m  V  1 250  250( g )
The number of moles:
m
250
n

 13.9
M molar
18
The positive charge:
Q  nN Aq
23
 13.9  6.023 10 10 1.6 10
19
7
 1.34 10 (C )
29. In crystals of the salt cesium chloride, cesium ions Cs+ form
the eight corners of a cube and a chlorine ion Cl- is at the cube’s
center. The edge length of the cube is 0.4 nm. The Cs+ ions are
each deficient by one electron (and thus each has a charge of
+e), and the Cl- ion has one excess electron (and thus has a
charge of –e). (a) What is the magnitude of the net electrostatic
force exerted on the Cl- ion by the eight Cs+ ions? (b) If one of
the Cs+ ions is missing, the crystal is said to have a defect, what
is the magnitude of the net force exerted on the Cl- ion by the
seven remaining Cs+ ions?
(a) The net force is zero due to
the symmetric distribution of
the eight Cs+ around Cl2
e
(b)
F7Cs   F1Cs   k
2
r
2
2
r
a  (a 2 )
; a  0.4nm
2
2
F7Cs   k
e
(3 / 4)a 2
Homework (lecture 2): 1, 3, 5, 8, 12, 19, 22, 24, 27, 31,
34, 46, 50, 53 (page 598-601)
1. Sketch qualitatively the electric field lines both between and
outside two concentric conducting spherical shells when a uniform
positive charge q1 is on the inner shell and a uniform negative
charge –q2 is on the outer. Consider the cases q1 > q2, q1 = q2,
and q1 < q2.
E=0
q1 > q 2
E=0
q1 = q 2
q1 < q 2
3. Sketch qualitatively the electric field lines for a thin,
circular, uniformly charged disk of radius R. (Hint: Consider as
limiting cases points very close to the disk, where the electric
field is directed perpendicular to the surface, and points very
far from it, where the electric field is like that of a point
charge)
Edge-on view
5. What is the magnitude of a point charge whose electric field
50 cm away has the magnitude 2.0 N/C?
E
kq
2
r
Er
q 
k
2

2  0.5
2
9
8.99 10
 5.6 10
11
(C )
8. See the figure, particle 1 of charge q1 = -5.0q and particle
2 of charge q2 = +2.0q are fixed to an x axis. (a) As a multiple
of distance L, at what coordinate on the axis is the net electric
field of the particles zero? (b) Sketch the net electric field
lines.
+
E1 
k q1
x
2
x  L 
2
x2
 E2 
E2
E1
k q2
x  L 
2
2

 x  2.72 L
5
12. The three particles are fixed in place and have charges q1 =
q2 = +e and q3 = +2e. Distance a = 6.0 m. What are the (a)
magnitude and (b) direction of the net electric field at point P
due to the particles?


E1  E2  0


Enet  E3

E3
2e
a 2
E3  k
; OP 
2
2
OP
9
E3  8.99 10
4 1.6 10
19
6.02 1012

0
E3  160( N / C ); ( E3 , Ox)  45
19. The figure shows an electric dipole. What are the (a)
magnitude and (b) direction (relative to the positive direction of
the x axis) of the dipole’s electric field at point P, located at
distance r >> d?
Enet  2k
q
r
2
cos  
kqd
r
3
p  qd : electric dipole moment
Enet 
•
•
If x>>d:
kp
 d 

2
   x 
 2 


Enet points
2

3/ 2
Enet 
kp
x
3
in the negative direction of the y axis


Enet
22. The figure shows two parallel nonconducting rings with their
central axes along a common line. Ring 1 has uniform charge q1
and radius R; ring 2 has uniform charge q2 and the same radius
R. The rings are separated by distance d = 3.0R. The net
electric field at point P on the common line, at distance R from
ring 1, is zero. What is the ratio q1/q2?
E
qz
4 0 ( z  R )
2
kq1R
2
2 3/ 2
2 3/ 2

kq2 2 R
2
2 3/ 2
(R  R )
(4 R  R )
q1 4 2

 0.51
q2 5 5
Note: q1 and q2 must have the same sign to produce a net
electric field equal to zero
24. Two curved plastic rods, one of charge +q and the other of
charge –q, form a circle of radius R = 8.5 cm in an xy plane. The x
axis passes through both of the connecting points, and the charge is
distributed uniformly on both rods. If q = 15.0 pC, what are the (a)
magnitude and (b) direction (relative to the positive direction of the
x axis) of the electric field produced at P, the center of the circle?
Consider a differential charge dq:
dq  ds   ( Rd )
(ds)
k
dE1rod  k
cos  
cos  dθ
2
For two rods:
dq
R
R
k
dE2rods  2
cos  dθ
R
k
E2rods  2 
R
E2rods 
90

4k 4kq
cos

dθ



2
R
R
90
4  8.99 109 15 1012
3.14  8.52 10 4

dE
 23.8( N / C )
27. A nonconducting rod of length L = 8.15 cm has charge –q =
-4.23 fC uniformly distributed along its length. (a) What is the
linear charge density of the rod? What are the (b) magnitude
and (c) direction (relative to the positive direction of the x axis)
of the electric field produced at point P, at a = 12.0 cm from
the rod? What is the electric field magnitude at a = 50 m by
(d) the rod and (e) a particle of charge –q = -4.23 fC that
replaces the rod?
(a)
 q  4.23 1015


 0,519 1013 (C / m)
L
8.15 10 2
(b)
dE x  k
dx
L  a  x 
2
q
3
E x  k
 1.6 10 ( N / C )
a L  a 
(c) the negative direction of the x axis
La :
E x  k
Ex  1.5 10
q
a
2
8
( N / C)
(d) for a distant point, the rod acts like a
point charge, so the electric field of the
point charge is the same as that of the rod:
Ex  1.5 10
8
( N / C)
31. At what distance along the central perpendicular axis of a
uniformly charged plastic disk of radius 0.6 m is the magnitude of
the electric field equal to one-half the magnitude of the field at
the center of the surface of the disk?

Ez 
2 0

1 




2
2 
z R 
z
At the center (very close to the center):
1
E z  Ec :
2

Ec 
2 0
1
z  R/
z
1

2
2
2
z R
3  0.35(m)
34. An alpha particle (the nucleus of a helium atom) has a mass
of 6.64 x 10-27 kg and a charge of +2e. What are the (a)
magnitude and (b) direction of the electric field that will balance
the gravitational force on the particle?

E
qE  mg

Fg
mg
E
2e
46. An electron is shot at an initial speed of v0=2.0x106 m/s, at
angle 0 = 400 from an x axis. It moves through a uniform
electric field E  (5.0 N/C) ˆj. A screen for detecting electrons is
positioned parallel to the y axis, at distance x = 3.0 m. In unit
vector notation, what is the velocity of the electron when it hits
the screen?
Fg  9 1030 ( N )
F  8 1019 ( N )
So, we ignore gravity
eE 1.6 1019  5.0
ay 

m
9.111031
 8.78 1011 (m / s 2 )
Time to

F
x
3
6
t



1
.
96

10
( s)
hit the screen:
v0 cos 0 2 106 cos 400
v y  v0 y  a yt  v0 sin   at  0.44 106 (m / s)

v  1.53 106 (m / s)iˆ  0.44 106 (m / s) ˆj
50. An electric dipole consists of charges +2e and -2e
separated by 0.78 nm. It is in an electric field of strength 3.4
x 106 N/C. Calculate the magnitude of the torque on the dipole
when the dipole moment is (a) parallel to, (b) perpendicular to,
and (c) antiparallel to the electric field.
 
  r F

  pE sin 
(a)
(b)
(c)
 net  0
(  0)
 net  2edE  2 1.6 1019  0.78 109  3.4 106
 net  8.5 1022 ( N .m)
0
 net  0 (  180 )
53. How much work is required to turn an electric dipole 1800 in
a uniform electric field of magnitude E = 46.0 N/C if p = 3.02
x 10-25 C.m and the initial angle is 640?
U   pE cos 
Wapplied  WE  U
Wapplied  U  0  U 0   pE cos(1800  640 )  pE cos 640
Wapplied  2 pE cos 640
Wapplied  2  3.02 10
25
 46.0  0.438
Wapplied  1.22 10
23
(J )