Lecture 6
Ch21. Coulomb’s Law
University Physics: Waves and Electricity
Dr.-Ing. Erwin Sitompul
http://zitompul.wordpress.com
Homework V: Ambulance Siren
An ambulance with a siren emitting a whine at 1600 Hz
overtakes and passes a cyclist pedaling a bike at 8 m/s. After
being passed, the cyclist hears a frequency of 1590 Hz.
(a) How fast is the ambulance moving?
(b) What frequency did the cyclist hear before being overtaken
by the ambulance?
January–April 2010
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Solution of Homework V: Ambulance Siren
(a) f  1600 Hz
f   1590 Hz
vD  8 m s, toward S
vS  ?, away from D
(b) f  1600 Hz
f?
vD  8 m s, away from S
vS  10.21 m s, toward D
v  vD
v  vS
343  8
1590  1600
343  vS
1590(343  vS )  1600(343  8)
545370  1590vS  561600
vS  10.21 m s
f f
v  vD
f f
v  vS
343  8
 1600
343  10.21
 1610.63 Hz
January–April 2010
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Electric Charge
 Static cling, an electrical phenomenon
that accompanies dry weather, causes
the pieces of paper to stick to one
another.. This is an example that
reveals the existence of electric charge.
 In fact, every object contains a vast
amount of electric charge.
 Electric charge is an intrinsic characteristic of the
fundamental particles making up those objects.
 The vast amount of charge in an everyday object is usually
hidden because the object contains equal amounts of the two
kinds of charge: positive charge and negative charge.
 With such a balance of charge, the object is said to be
electrically neutral (contains no net charge).
 If the two types of charge are not in balance, we say that an
object is charged, it has a net charge.
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Electric Charge
 Charged objects interact by exerting forces
on one another.
 Charges with the same electrical sign repel
each other, while charge with opposite
electrical signs attract each other.
 This rule will be described quantitatively as
Coulomb’s law of electrostatic force between
charges. (The term electrostatic is used to
emphasize that the charges are stationary
relative to each other.)
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Coulomb’s Law
 If two charged particles are brought near each other, they
each exert a force on the other.
 If the particles have the same sign of charge, they repel each
other.  The force on each particle is directed away from the
other particle.
 If the particles have opposite
signs of charge, they attract
each other.  The force on
each particle is directed
toward the other particle.
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Coulomb’s Law
 This force of repulsion or attraction due to the charge
properties of objects is called an electrostatic force.
 The equation giving the force for charged particles is called
Coulomb’s law, named after Charles-Augustin de Coulomb,
who did the experiments in 1785.
 If particle 1 has charge q1 and particle 2 has charge q2, the
force on particle 1 is:
F2  k
q1q2
rˆ
2 12
r12
 The term rˆ12 is a unit vector to the direction from position of q1
to position of q2. The term k is a constant.
k
1
 8.99  109 N  m2 C2
4 0
 0  8.854 1012 C2 (N  m2 )
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Coulomb’s Law
 ε0 is a constant denoted as permittivity in vacuum, a
measure of how the vacuum medium is affected by an electric
field.
 As can be deducted from the constants, the SI unit of charge
is the coulomb (C).
y
F2  k
q1
+
r→
12
→
r1
→
r2
+
q2
→
F2
x
January–April 2010
q1q2
r12
rˆ
2 12
r12  r2  r1
r2  r1
r12

rˆ12 
r2  r1
r12
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Some Examples on Vectors
 Example: If r12  4iˆ  2jˆ , find r12 and rˆ12 .
r12  (4)2  (2) 2  20  4.472
r12 4iˆ  2jˆ

 0.894iˆ  0.447ˆj
rˆ12 
4.472
r12
 Example: If r1  2iˆ  3jˆ and r2  4iˆ  2jˆ , find r12 and rˆ21 .
ˆ  (2iˆ  3j)
ˆ  2iˆ  5jˆ
r12  r2  r1  (4iˆ  2j)
ˆ  (4iˆ  2j)
ˆ  2iˆ  5jˆ
r21  r1  r2  (2iˆ  3j)
r12  r21
r12  r21
January–April 2010
 Both vectors are of
opposite direction, but
have the same magnitude
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Coulomb’s Law
 If we have n charged particles, they interact independently in
pairs, and the force on any one of them, is given by the vector
sum.
 Let us say, we have n particles, then the force on particle 1 is
given by:
F1,net  F12  F13  F14 
 F1n
y
+
q1
q2
→
F21
+
q3
–
→
F23
→
F2,net
x
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Example 1: Coulomb’s Law
The figure below shows two positively charged particles fixed in
place on an x axis. The charges are q1 = 1.610–19 C and q2 =
3.210–19 C. The q1 is located on the origin, while the
separation is R = 0.02 m.
What are the magnitude and direction of the electrostatic force
→
F12 on particle 1 from particle 2?
r1  0
r2  0.02iˆ m
r21  r1  r2  0.02iˆ
r21  0.02
rˆ21 
F12  k
q1q2
r21
rˆ
2 21
(1.6  1019 )(3.2  1019 ) ˆ
 8.99  10
(i)
2
(0.02)
 1.151  1024 ˆi N
9
r21
 ˆi
r21
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Example 2: Coulomb’s Law
Now, particle 3 lies on the x axis between particles 1 and 2.
Particle 3 has charge q3 = –3.210–19 C and is at a distance
3/ R from particle 1.
4
→
What is the net electrostatic force F1,net on particle 1 due to
particles 2 and 3?
r1  0
r3  0.015iˆ m
r31  r1  r3  0.015iˆ
r31  0.015
r
ˆr31  31  ˆi
r31
F13  k
q1q3
r31
rˆ
2 31
19
19
(1.6

10
)(

3.2

10
) ˆ
9
 8.99  10
(i)
2
(0.015)
 2.046  1024 ˆi N
F1,net  F12  F13
 1.1511024 ˆi  2.046 1024 ˆi
 8.95  1025 ˆi N
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Example 3: Coulomb’s Law
Particle 3 from previous example is now replaced by particle 4.
Particle 4 has charge q4 = –3.210–19 C, is at a distance 3/4R
from particle 1, and lies on a line that makes an angle θ = 60°
with the x axis.
→
What is the net electrostatic force F1,net on particle 1 due to
particles 2 and 4?
r1  0
r4  (0.015)cos60ˆi
 (0.015)sin 60ˆj
 0.0075iˆ  0.013jˆ
r41  r1  r4
 0.0075iˆ  0.013jˆ
r41  (0.0075)2  (0.013) 2
 0.015, equals 34 R
January–April 2010
rˆ41 
r41
r41
0.0075iˆ  0.013jˆ

0.015
 0.5iˆ  0.867jˆ
University Physics: Wave and Electricity
6/13
Example 3: Coulomb’s Law
F14  k
q1q4
r41
rˆ
2 41
19
19
(1.6

10
)(

3.2

10
)
ˆ  0.867ˆj)
 8.99  109
(

0.5i
(0.015)2
 1.023 1024 ˆi  1.774 1024 ˆj N
F1,net  F12  F14
 1.151  1024 ˆi
 (1.023  1024 ˆi  1.774  1024 ˆj)
 (0.128 ˆi  1.774 ˆj)  1024 N
January–April 2010
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Checkpoint
The figure below shows three arrangements of one electron (e)
and two protons (p).
(a) Rank the arrangements according to the magnitude of the
net electrostatic force on the electron due to the protons,
largest first
a, c, b
(b) In situation c, is the angle between the net force on the
electron and the line labeled d less than or more than 45 °?
Less than 45°
p  1.602 1019 C
e  1.602  1019 C
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Example 4: Coulomb’s Law
Two particles are fixed in place: a
particle of charge q1 = +8q at the
origin and a particle of charge q2 =
–2q at x = L.
At what point can a particle of
charge q3 = +4q be placed so that it
is in equilibrium (the net force on q3
is zero)?
→
→
F32
F31
: impossible to place
q3 on the left of q1 or
in the middle
between q1 and q2
: the only possibility
is to place q3 to the
right-hand side of q2
January–April 2010
→
F32
→
F31
→
F32
→
F31
University Physics: Wave and Electricity
6/16
Example 4: Coulomb’s Law
F3,net  F31  F32  0
q1q3
q2q3
k
rˆ  k
rˆ  0
2 13
2 23
r13
r23
qq
qq
k 1 32 ˆi  k 2 32 ˆi  0
r13
r23
q1
q2

0
2
2
r13
r23
8q
2q

0
2
2
( L  x)
x
8
2

( L  x) 2 x 2
January–April 2010
8x2  2( L  x)2
8x2  2( L2  2 xL  x2 )
6 x 2 4 L x 2 L2  0
c
a
b
b  b2  4ac
x1,2 
2a
x1   13 L • q3 between q1 and q2
x2  L
• q3 to the right of q2
Charge q3 must be
placed on the x
axis, at distance L
to the right of q3, or,
at point (2L,0)
→
F32
L
x
University Physics: Wave and Electricity
→
F31
6/17