Download Coulomb`s law

Coulomb’s law
1
Dr. Loai Afana
Coulomb’s Law
Coulomb discovered in 1785 the fundamental law
of electrical force between two stationary charged particles.
An electric force has the following properties:
– Inversely proportional to the square of the separation, r, between the
particles, and is along a line joining them.
– Proportional to the product of the magnitudes of the charges |q1| and
|q2| on the two particles.
– Attractive if the charges are of opposite sign and repulsive if the charges
have the same sign.
1
F 2
r
F  q1 q 2
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Dr. Loai Afana
Coulomb’s Law (Mathematical Formulation)
F  q1 q 2

1
F 2
r
• ke
known as the Coulomb constant.
• Value of ke depends on the choice of units.
• SI units
– Force: the Newton (N)
– Distance: the meter (m).
– Charge: the coulomb ( C).

F  ke
Dr. Loai Afana
q1q2 
r
2
r
3
1
1
9
2
2
Ke 


9

10
Nm
/
C
4e
4  8.85 1012
where eo is known as the Permittivity constant of free space.
eo = 8.85 x 10-12 C2/N.m2
ke = 8.9875×109 Nm2/C2.
Reasonable approximate value: ke = 8.99×109 Nm2/C2.
Experimentally measurement:
ke = 9 ×10 9 Nm2/C2
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The Coulomb constant unit
:
Then the Coulomb constant unit is
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Dr. Loai Afana
Example:
The electron and proton of a hydrogen atom are separated (on the average) by a
distance about 5.3x10-11 m.
Find the magnitude of the electric force that each particle exerts on the other.
Solution:
q1 = - 1.60x10-19 C
q2 = +1.60x10-19 C
r = 5.3x10-11 m
Fe  k e
e
2
r2
 9 10
9 Nm 2
C2
1.6 10
 5.3 10
19
11

m
C
2
2
 8.2 108 N
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Dr. Loai Afana
Example:
Two protons in a molecule are separated by a distance of 3.8*10-10m. Find the
electric force exerted by one proton on the other.
Solution:
F  Ke
q1 q2
r2
9
9
9
1
.
6

10

1
.
6

10
14
.
4

1
.
6

10
9
 9 109


1
.
6

10
N
2
21
14.4 10
3.8 1010


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The electrostatic force
• The electrostatic force is often
called Coulomb force.
• It is a force (thus, a vector):
– a magnitude
– a direction.
F12  F21
F12  F21
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Dr. Loai Afana
Super position of Forces
FT  F10  F20  F30  ....
+Q1
+Q2
+Q3
r10
r20
F30
F20
+Q0
F10
r30
kq 0q1
kq 0q 2
kq 0q 3
ˆ
ˆ
FT 
r10 
r20 
rˆ30  ....
2
2
2
r10
r20
r30
N
 q1

q3
q2
qi
ˆ
ˆ
ˆ
FT  kq 0  2 r10  2 r20  2 r30  ....   kq 0  2 rˆi0
r20
r30
i 1 ri0
 r10

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Dr. Loai Afana
Gravitational versus Electrical Force
Newton's law of universal gravitation:
q1
m1
F
F
q2
m2
r
ke = 9 ×10 9 Nm2/C2
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Dr. Loai Afana
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Problem:
The electron and proton of a hydrogen atom are separated (on the average) by a distance
approximately : 5.3x10-11m
1- Find the magnitudes of the electric force and the gravitational force between the
two particles.
2- what is a fundamental difference between the two forces?
Particle
Charge ( C)
Mass (kg)
Electron
-1.60 *10-19
9.11 *10-31
Proton
+1.60 *10-19
1.67 *10-27
Neutron
0
1.67*10-27
ke = 9 ×10 9 Nm2/C2
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Dr. Loai Afana
Discussion
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Example:
1.3µC charge is located on the x-axis at x = - 0.5m,
3.2µC charge is located on the x-axis at x=1.5m, and
2.5µC charge is located at the origin.
Find the net force on the 2.5µC charge (origin)
+1.3µC
F23
1
+2.5µC
2
- 0.5m
F21
1.5m
+3.2µC
3
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Example:
Two fixed charges, 1µC and -3µC are separated by 10cm as shown in figure:
where may a third charge be located so that no ele.force acts on it?
1µC
1
-3µC
10cm
2
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Example
What’s the (x) coordinate of q3 if the resultant force acting on q3
equal to zero?
q1 = 15.0 mC
q2 = 6.0 mC
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Example:
Three point charges lie along the y-axis.:
A charge q1=-9µC is at y=6.0m, and
a charge q2=-8µC is at y=-4.0m.
Where must a third positive charge, q3, be placed such that the resultant
force on it is zero?
Example:
Tow charges q1 and q2, its sum is 5*10-5C.
The electric force Between them is F=1N and the distance is r = 2m.
Find q1 and q2
Example:
Three charges q1, q2 ,q3 are fixed on a straight line. The distance between q1, q2 is d,
and between q3, q2 also d.
The charge q3 lets free , but it doesn’t leave its position. (equilibrium)
Find the relation between q1, q2
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Discussion
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Example:
Consider three point charges at the corners of a triangle, as shown below.
Find the resultant force on q3 if:
y
q1 = 6.00 x 10-9 C
q2 = -2.00 x 10-9 C
q3 = 5.00 x 10-9 C
-
3m
q2
q1+
4m
F32
Ө’
F31
+
37o
q3
x
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Dr. Loai Afana
Example
Four charges are arranged in a square with sides of length 2.5 cm.
The two charges in the top right and bottom left corners are + 3.0 x 10-6 C.
The charges in the other two corners are - 3.0 x 10-6 C.
What is the net force exerted on the charge in the top right corner by the other three
charges?
2.5 cm
Solution:
2.5 cm
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Dr. Loai Afana
Example
A charge +Q is fixed at each of two opposite corners of a square as shown in figure.
A charge - q is placed at each of the other two corners.
If the resultant electrical force on Q is Zero,
how are Q and q related.
Solution:
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Dr. Loai Afana
Example:
In figure, two equal positive charges q =+2x10-6C interact with a third charge Q =+4x10-6C.
Find the magnitude and direction of the resultant force on Q.
Solution:
q1
0.3m
0.4m
0.3m
Q
q2
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Example:
Three charges are placed on the corners of an equilateral triangle (0.2m) as shown in Fig.
What is the magnitude and direction of the net force on the 2 micro-Coulomb charge?
Solution:
-1mc
+2mc
+3mc
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Dr. Loai Afana
Example:
Three point charges are located at the corners of an equilateral triangle as shown in Fig.
Calculate the resultant electric force on the 7.0 µC
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Example
Three point charges are arranged as shown in Figure.
1. Determine the magnitude of the electric force at 5nC charge.
2. What is the direction of the electric force at this charge?
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Dr. Loai Afana
Example:
Two identical small charged spheres, each having a mass of 3.0 *10-2 kg,
hang in equilibrium as shown in Figure.
The length of each string is 0.15 m, and the angle θ = 5.0°.
Find the magnitude of the charge on each sphere.
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Example:
In Figure, two conducting balls of identical mass m and identical charge q hang
from non-conducting strings of length L.
Assume that Ө is so small that (tan Ө) can be replaced by its approximate equal (sin Ө).
(a) Show that the equilibrium separation (x) of the balls gives by:
 q2L 
mgx
1 q

x

2
2L
4e o x
2
e
mg
o


2
1/ 3
(b) If L = 120 cm, m = 10 g, and x = 5.0 cm, what is |q|?
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Dr. Loai Afana