Chapter1: Coulomb's Law

1. CHAPTER 1:

ELECTRIC CHARGE AND ELECTRIC FIELDS

i)
Electrostatic
ii)
conductors and insulators
iii) Coulomb ’s Law
iv)
Electric Fields
v)
Electric Fields Calculation
vi) Electric Field Lines
vii)
Electric Dipole


1.1 Electrostatic

Electrostatic is a study on the electric charges in the
static or steady state condition. In this chapter, we will
discuss the basic and the fundamental concept of electric
charges, electric fields and their characteristics.

Plastic rods and fur are good for demonstrating
electrostatics.

Benjamin Franklin (1706-1790) suggested charges
negative and positive .

2. “Two positive charges or two negative charges repel
each other. A positive charge and a negative charge
attract each other ”.

Caution: Electric attraction and repulsion
The attraction and repulsion of two charged object are
sometimes summarized as “like charges repel, and
opposite charges attract ”

But keep in mind that the phase
“like charge ” does not
mean that the two charges are exactly identical, only that
both charges have the same algebraic sign (both positive
or both negative). “Opposite charges ” mean that both
objects have an electric charge, and those charges have
different sign (one positive and the other negative).

3. 1.1.1 Electric charge and the structure of matter


The structure of atoms can be described in terms of
three particles: the negatively charged electron , the
positively charged proton , and uncharged neutron (Fig
above).

Proton and neutron in an atom make up a small, very
dense core called the nucleus . (10
-15
m)

Surrounding the nucleus are the electrons
(10
-10
m a far from nucleus).

4. The negative charge of the electron has exactly the
magnitude as the positive charge of proton.

1.1.2 Electric Charge is Conserved

Principle of conservation charge:
i) The algebraic sum of all electric charges in any
closed system is constant
ii) The magnitude of charge of the electron or
proton is a natural unit of charge.

In any charging process, charge is not created or
destroyed: it is merely transferred from one body to
another.

“The electric charge is quantized . (1, 2, 3, 4 …)”

5. 1.2 Conductors and Insulators



Materials that allow easy
passage of charge are called
conductors . Materials that
resist electronic flow are
called
insulators .

The motion of electrons
through conducts and
about insulators allows us
to observe
“opposite
charges attract
” and “like
charges repel.
”


Charging by induction

6. 1.3 Coulomb ’s Law
Charles Augustin de Coulomb (1736-1806) studied the
interaction forces of charged particles in detail in 1784.

7. Point charges
Coulomb found that
i)
The electric force is proportional to
ଶ
ଶ

ii) The electric force between two point charges
depends on the quantity of charge on each body,
which we will denote by q or Q. ( positive or
negative)

iii) The forces that two point charges
ଵand
ଶexert
on each other are proportional to each charge and
therefore are proportional to the product
ଵଶ of
the two charges
ଵଶ


+
-
+
-
+
-
-
+
r

8. Coulomb ’s law state that;

The magnitude of the electric force between two point
charges is directly proportional to the product of the
charges and inversely proportional to the square of the
distance between them.

In mathematical term, the magnitude
F of the force that
each of two point charge
ଵand
ଶa distance r apart
exerts on the other can be expressed as;

૚૛
૛


where
k is a constant.

1.3.1 Electric Constants, k

In SI units the constant, k is
଴
where (
଴- epsilon
nought or opsilon zero)


ଽ ଶଶ


By approximation
ଽ ଶଶ

9. Magnitude of the charge of an electron or proton, e

ିଵଽ


One Coulomb represents the negative of the total charge
of about
ଵ଼
electron.

So that, the electric force is given as
૙
૚૛
૛


Superposition of Forces : holds for any number of
charges. We can apply Coulomb ’s law to any collection of
charges.

10. Example

Two point charges
ଵ and
ଶ , are
separated by a distance of 3.0 cm. Find the magnitude
and direction of
(i)
the electric force that
ଵexerts on
ଶ, and
(ii) the electric force that
ଶ exerts on
ଵ.



Solution
a) This problem asks for the electric forces that two
charges exert on each other, so we will need to use
Coulomb ’s law. After we convert charge to coulombs
and distance to meters, the magnitude of force that
ଵexerts on
ଶ is

ଵ*௢௡*ଶ
଴
ଵଶ
ଶ

ଽ ଶିଶ
ିଽ ିଽ
ଶ

11. Since the two charges have opposite signs, the force is
attractive; that is the force that acts on
ଶis directed
toward
ଵalong the line joining the two charges.

b)
Newton ’s third law applies to the electric force. Even
though the charges have different magnitude, the
magnitude of the force that
ଶ exerts on
ଵis the
same the magnitude of the force that
ଵexerts
on
ଶ. So that
ଶ*௢௡*ଵ

Example

Two point charges are located on the positive x-axis of a
coordinate system. Charge q 1 = 1.0 nC is 2.0 cm from the
origin, and charge q 2 = -3.0 nC is 4.0 cm from the origin.
What is the total force exerted by these two charges on a
charge q 3 = 5.0 nC located at the origin?



Solution
Find the magnitude of
ଵ*௢௡*ଷ

12. ଵ*௢௡*ଷ
଴
ଵଶ
ଶ

ଽ ଶିଶ
ିଽ ିଽ
ଶ

ିସ

*(this force has a negative x-component because q 3 is
repelled by q 1)


Then Find the magnitude of
ଶ*௢௡*ଷ
ଶ*௢௡*ଷ
଴
ଵଶ
ଶ

ଽ ଶିଶ
ିଽ ିଽ
ଶ

ିହ

*(this force has a positive x-component because q 3 is
attracted by q 2)

So the sum of x-component is
௫
There are no y or z- components. Thus the total force on q 3 is
directed to the left, with magnitude.

13. 1.4 Electric Fields

We defined the electric field
at point as the electric
force experienced by a test charge q0 at the point,
divided by the charge q0.

଴


The direction of and is the same.

14. Electric field of a point charge

Consider we have a charge q as a point source. If we
place a small test charge q0 at the field point, P at a
distance r from the point source, the magnitude F0 of
the force is given by Coulomb ’s law


଴
ଵ
ସగఢ
బ
ȁ௤௤
బ
ȁ
௥
మ


so that, the
magnitude of electric field , E is
଴
ଶ


But the direction of and is the same. Then the
electric field vector is given as ,
଴
ଶ

is a vector unit in r direction.

15. Example 1: Electric-field magnitude for a point charge
What is the magnitude of electric field at a field point 2.0
m from a point charge q = 4.0 nC ?

Solution
We are given the magnitude of charge and the distance
from the object to the field point, so by using
ͳ
Ͷߨ߳
Ͳ
ȁݍȁ
ݎ
ʹ

we could calculate the magnitude of
଴
ଶ

16. ଽ ଶିଶ
ିଽ
ଶ



Example 2: Electric Field Vector for a point charge.

A point charge
q = -8.0 nC is located at the origin. Find
the electric-field vector at the field point x = 1.2 m,
y = -1.6 m?

Solution


The vector of field point
P is

The distance from the charge at point source, S to the
field point, P is
ଶ ଶ ଶ ଶ

The vector unit,

17. ௥Ԧ
ȁ௥ȁ
ଵǤଶపƸିଵǤ଺ఫƸ
ଶǤ଴

Hence the electric-field vector is
ܧሬԦൌ
ͳ
Ͷߨ߳
଴
ݍ
ݎ
ଶ
ݎƸ
****ൌ*ሺͻǤͲ*ൈͳͲ
ଽ
ܰ*݉
ଶ
ܥ
ିଶ
ሻ
െͺǤͲൈͳͲ
ିଽ
*ܥ
ሺʹǤͲሻ
ଶ
*ሺͲǤ͸ଓƸെͲǤͺଔƸሻ
****ൌെͳͳ
ܰ
ܥ
ଓƸ൅ͳͶ
ܰ
ܥ
ଔƸ

Example 3: Electron in a uniform field

When the terminals of a battery are connected to two
large parallel conducting plates, the resulting charges on
the plate cause an electric field ܧሬ
Ԧ in the region between
the plates that is very uniform.


If the plate are horizontal and separated by 1.0 cm and
the plate are connected to 100 V battery, the magnitude
of the field is E = 1.00 x 10
4
N/C. Suppose the is
vertically upward,

18. a) If an electron released from rest at the upper plate,
what is its acceleration?
b)
What speed does the electron acquire while
traveling 1.0 cm to lower plate?
Given electron charge is
ିଵଽ
and
mass
௘
ିଷଵ


Solution:
a) Noted that
is upward but is downward because the
charge of electron is negative. Thus Fy is negative. Because
Fy is constant, the electron moves with constant
acceleration ay given by,
௬
௬
௘ ௘
ିଵଽ
ସ
ିଷଵ
ଵହ ଶ

b) The electron starts from rest, so its motion is in the y
–direction only. We can find the electron ’s speed at
any position using constant-acceleration
formula
௬
ଶ
଴௬
ଶ
௬ ଴. We have
଴௬
ଶ

and y0 = 0 so speed
௬when y = -1.0 cm.

หݒ
௬หൌටʹܽ
௬ݕൌටʹቀെͳǤ͹͸*ൈͳͲ
ଵହ
݉
ݏ
ଶ
ቁሺെͳǤͲ*ൈͳͲ
ିଶ
݉ሻ
ൌͷǤͻ*ൈͳͲ
଺
݉Ȁݏ

19. Example 4: An electron trajectory

If we launch an electron into the electric field of Example
3 with an initial horizontal velocity v0, what is the
equation of its trajectory?

Solution
The acceleration is constant and in the y-direction. Hence
we can use the kinematic equation for 2-dimensional
motion with constant acceleration.

଴ ଴௫
ଵ
ଶ
௫
ଶ
and
଴ ଴௬
ଵ
ଶ
௬
ଶ


We have a
x=0 and ay = (-e)E /m . at t =0 , x0 =y0=0, v0x = v0
and v0y=0, hence at time t,

଴ and
ଵ
ଶ
௬
ଶ
ଵ
ଶ
௘ா
௠
ଶ


Eliminating t between these equations, we get
଴
ଶ
ଶ

20. 1.5 Electric Fields Calculation

In real situations, we encounter charge that is distributed
over space. To find the field caused by a distribution, we
imagine the distribution to be made up of many point
charges, q1,q2,q3…….qn. At any given point P, each point
charge produces its own electric field
ଵଶଷ ௡,
so a test charge q 0 placed at P experiences a force
ଵ ଴ଵ
from charge q1
and a force
ଶ ଴ଶ from charge q2 and so
on.
From the principle of superposition of forces, the total
forces
଴that the charge distribution exerts on the q0 is the
vector sum of these individual forces,
଴ ଵ ଶ ଷ ଴ଵ ଴ଶ ଴ଷ
Then the total electric field at point P,
଴
଴
ଵ ଶ ଷ

21. Example 1:
Point charge q1 and q2 of +12nC and -12nC respectively,
are placed 0.10 m apart. This combination of two charges
with equal magnitude and opposite sign is called an
electric dipole . Compute the electric field caused by q1,
the field caused by q2, and total field (a) point a, (b) at
point b, and (c) at point c.


Solution
a) At point a: the electric field,
ଵ caused by the
positive charge q1 and the field
ଶ caused by the
negative charge q2 are both directed toward the
right. The magnitude of
ଵ and
ଶ are;
หܧ
ଵ
ሬሬሬሬԦหൌ
ͳ
Ͷߨ߳
଴
ȁݍ
ଵȁ
ݎ
ଶ
ൌ*ൌሺͻǤͲ*ൈͳͲ
ଽ
ܰ*݉
ଶ
ܥ
ିଶ
ሻ
ȁͳʹൈͳͲ
ିଽ
*ܥȁ
ሺͲǤͲ͸݉ሻ
ଶ

22. ൌ͵ǤͲ*ൈͳͲ
ସ
*ܰȀܥ

หܧ
ଶ
ሬሬሬሬԦหൌ
ͳ
Ͷߨ߳
଴
ȁݍ
ଶȁ
ݎ
ଶ
ൌ*ൌሺͻǤͲ*ൈͳͲ
ଽ
ܰ*݉
ଶ
ܥ
ିଶ
ሻ
ȁͳʹൈͳͲ
ିଽ
*ܥȁ
ሺͲǤͲͶ݉ሻ
ଶ

ൌ͸Ǥͺ*ൈͳͲ
ସ
*ܰȀܥ

The component of
ଵ and
ଶ are;
ܧ
ଵ௫ൌ*͵ǤͲ*ൈͳͲ
ସ
ே
஼
and ܧ
ଵ௬ൌ*Ͳ
ܧ
ଶ௫ൌ*͸Ǥͺ*ൈͳͲ
ସ
ே
஼
and ܧ
ଶ௬ൌ*Ͳ
Hence at point a the total electric field

௔ ଵ ଶ has components.

௔௫ ଵ௫ ଶ௫
Ͷ
ܰ
ܥ

௔௬ ଵ௬ ଶ௬
At point
a the total field has magnitude 9.8
ସ
ே
஼

and is directed toward the right.
௔
Ͷ
ܰ
ܥ


b)
At point a: the electric field,
ଵ caused by the
positive charge q1 is directed toward left and the
field
ଶ caused by q2 is directed toward the right.
The magnitude of
ଵ and
ଶ are;
หܧ
ଵ
ሬሬሬሬԦหൌ
ͳ
Ͷߨ߳
଴
ȁݍ
ଵȁ
ݎ
ଶ
ൌ*ൌሺͻǤͲ*ൈͳͲ
ଽ
ܰ*݉
ଶ
ܥ
ିଶ
ሻ
ȁͳʹൈͳͲ
ିଽ
*ܥȁ
ሺͲǤͲͶ݉ሻ
ଶ

ൌ͸Ǥͺ*ൈͳͲ
ସ
*ܰȀܥ

23. หܧ
ଶ
ሬሬሬሬԦหൌ
ͳ
Ͷߨ߳
଴
ȁݍ
ଶȁ
ݎ
ଶ
ൌ*ൌሺͻǤͲ*ൈͳͲ
ଽ
ܰ*݉
ଶ
ܥ
ିଶ
ሻ
ȁͳʹൈͳͲ
ିଽ
*ܥȁ
ሺͲǤͳͶ݉ሻ
ଶ

ൌͲǤͷͷ*ൈͳͲ
ସ
*ܰȀܥ

The component of
ଵ and
ଶ are
ܧ
ଵ௫ൌ*െ͸Ǥͺ*ൈͳͲ
ସ
ே
஼
and ܧ
ଵ௬ൌ*Ͳ
ܧ
ଶ௫ൌ*ͲǤͷͷൈͳͲ
ସ
ே
஼
and ܧ
ଶ௬ൌ*Ͳ
Hence at point a the total electric field
ܧ
௕
ሬሬሬሬԦൌܧ
ଵ
ሬሬሬሬԦ*൅ܧ
ଶ
ሬሬሬሬԦ has components

ܧሬ
Ԧ
௕௫ൌܧ
ଵ௫൅ܧ
ଶ௫ൌሺെ͸Ǥͺ൅*ͲǤͷͷ*ሻ*ൈͳͲ
ସ
ே
஼

ܧሬԦ
௕௬ൌܧ
ଵ௬൅ܧ
ଶ௬ൌͲ൅Ͳ

At point
b the total field has magnitude 6.2
ସ
ே
஼

and is directed toward the right.
௕
Ͷ
ܰ
ܥ


c)
At point c, both
ଵ and
ଶ have same magnitude,
since this point is equidistant from both charges
and charge magnitude are the same;
หܧ
ଵ
ሬሬሬሬԦหൌหܧ
ଶ
ሬሬሬሬԦหൌ
ͳ
Ͷߨ߳
଴
ȁݍ
ଵȁ
ݎ
ଶ
ൌ*
ൌሺͻǤͲ*ൈͳͲ
ଽ
ܰ*݉
ଶ
ܥ
ିଶ
ሻ
ȁͳʹൈͳͲ
ିଽ
*ܥȁ
ሺͲǤͳ͵݉ሻ
ଶ

ൌ͸Ǥ͵ͻ*ൈͳͲ
ସ
*ܰȀܥ

24. The direction of
ଵ and
ଶ are shown in Figure.
The x-component of the both vector as the same

ܧ
ଵ௫ൌܧ
ଶ௫ൌܧ
ଵ…‘•ߙൌ͸Ǥ͵Ͳ*ൈͳͲ
ଷ
ே
஼
…‘•ߙ*ൌ͸Ǥ͵Ͳ*ൈͳͲ
ଷ
ே
஼
*ቀ
ହ
ଵଷ
ቁൌ
*ʹǤͶ͸ൈͳͲ
ସ
ே
஼

From symmetry the y-component are equal and
opposite direction so add to zero.
ܧሬԦ
௖௫ൌܧ
ଵ௫൅ܧ
ଶ௫ൌʹሺʹǤͶ͸*ሻൈͳͲ
ଷ
ே
஼
ൌͶǤͻൈͳͲ
ଷ
ே
஼

ܧሬԦ
௖௬ൌܧ
ଵ௬൅ܧ
ଶ௬ൌͲ

So at point c the total electric field has magnitude
ͶǤͻൈͳͲ
ଷ
ே
஼
and its direction toward the right.
௖
͵
ܰ
ܥ


1.6 Electric Field Lines

Electric field lines can be a big help for visualizing electric
fields and making them seem more real. An electric field
line is an imaginary line or curve drawn through a region
of space so that its tangent at any point is in the
direction of the electric field vector at that point.

25. Figure below shows some of the electric field lines in a
plane (a) a single positive charge, (b) two equal-
magnitude charges, one positive and one negative
(dipole), (c) two equal positive charges.

Diagram called
Field Map .