Chapter 2 Electrostatics Spring 2010
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Dr. R. D. Senthilkumar Assistant Professor
Middle East College of Information Technology, Oman
2 ELECTROSTATICS
Objectives
After completing this chapter, you should be able to
1. Demonstrate evidence for the existence of two types of charges.
2. State the Coulomb‟s law of electrostatics.
3. Define the electric field and draw the electric force of lines from the charges.
4. Recall and use the Coulomb‟s relation to find electrostatic force between and field around the charges.
5. Describe electric potential.
6. Explain what dielectrics are and list its classifications and applications.
7. Describe capacitors
8. Recall the working principle of capacitors.
9. Determine the total capacitances of the capacitors in series and parallel circuits along with voltage through the
circuits.
Outline
Electrostatics 24 – 33
Dielectrics 34 – 35
Capacitors 35 – 41
Summary of Chapter 2 41 – 42
Problems and Short Questions 43 – 45
2.1 Introduction to Electrostatics .................................................................................................. 24
2.2 Electric Charges ...................................................................................................................... 25 2.2.1 Behaviour of electric charges ........................................................................................... 26
2.3 Coulomb’s law ........................................................................................................................ 26 2.3.1 Sample Problems .......................................................................................................... 28
2.4 Electric Field ........................................................................................................................... 30
2.4.1 Electric lines of forces ...................................................................................................... 30 2.4.2 Properties of lines of forces .............................................................................................. 31
2.4.3 Electric field intensity or strength .................................................................................... 31 2.4.4 Sample problem ........................................................................................................... 33
2.5 Electric Potential ..................................................................................................................... 33 2.6 Dielectrics ............................................................................................................................... 34
2.6.1 Types of dielectrics .......................................................................................................... 34 2.6.2 Dielectric loss ................................................................................................................... 34
2.6.3 Dielectric breakdown ....................................................................................................... 34 2.6.4 Applications of dielectric materials .................................................................................. 35
2.7 Capacitors ................................................................................................................................ 35 2.7.1 Working of the capacitor .................................................................................................. 36 2.7.2 Capacitance ..................................................................................................................... 37 2.7.3 Unit of Capacitance .......................................................................................................... 37
2.7.4 Factors affecting capacitance ........................................................................................... 38
2.7.5 Capacitors in parallel and series ....................................................................................... 39 2.7.5.1 Capacitors in Series ....................................................................................................... 39 2.7.5.2 Capacitors in parallel ..................................................................................................... 40
Summary ....................................................................................................................................... 41 Problems for Chapter 2 ................................................................................................................. 43
Short Questions for Chapter 2 ....................................................................................................... 45 References ..................................................................................................................................... 46
CONTENTS
Engineering Physics (MASC 0003)
2.1 INTRODUCTION TO
ELECTROSTATICS
During the period of 624 BC, Thales of Miletus who was a
Greek philosopher and mathematician discovered that when an
amber rod is rubbed with fur, the rod has the amazing
characteristic of attracting some very light objects such as bits of
paper and shavings of wood. This phenomenon became even
more remarkable when it was found that identical materials, after
having been rubbed with their respected cloths, always repelled
each other. After all, none of these objects was visibly altered by
the rubbing, yet they definitely behaved differently than before
they were rubbed. Whatever change took place to make these
materials attract or repel one another was invisible.
Some experimenters speculated that invisible “fluids” were
being transferred from one object to another during the process of
rubbing, and that these “fluids” were able to produce a physical
force over a distance. Charles Dufay was one the early
experimenters who demonstrated that there were definitely two
different types of changes produced by rubbing certain pairs of
objects together. The fact that there was more than one type of
change manifested in these materials was evident by the fact that
there were two types of forces produced: attraction and repulsion.
The hypothetical fluid transfer became known as a charge.
Benjamin Franklin, who was an American Statesman,
inventor, and philosopher, came to the conclusion that there was
only one fluid exchanged between rubbed objects, and that the
two different “charges” were nothing more than either an excess
or a deficiency of that one fluid. If there is a deficiency of fluid in
the objects after being rubbed, the objects are said to be negatively
charged; if there is an excess of fluid in the objects then the
objects are termed as positively charged.
It was discovered much later that this “fluid” was actually
composed of extremely small bits of matter called electrons, so
named in honor of the ancient Greek word for amber.
In the 1780‟s, Precise measurements of electrical charge
were carried out by the French Physicist Charles Augustin de
Coulomb, using a device called a torsional balance measuring the
Fig. 2.1 Static cling, shows the charged
comb attracts neutral bits of paper
CHAPTER2 | 2.2 Electric Charges 25
Engineering Physics (MASC 0003)
force generated between two electrically charged objects. This
work led to the development of a unit of electrical charge named
in his honor, the coulomb. One Coulomb is defined as the amount of
charge flowing through a conductor in one second when one ampere of
current is flowing through that conductor.
Nowadays, electrostatics has many applications ranging
from the analysis of phenomena such as thunderstorms to the
study of the behaviour of electron tubes. That is, it plays an
important role in modern design of electromagnetic devices
whenever a strong electric field appears. For example, an electric
field is of paramount importance for the design of X-ray devices,
lightning protection equipment and high-voltage components of
electric power transmission systems. In the area of solid-state
electronics, dealing with electrostatics is inevitable. Electrostatics
can also be used in relation to transport and holding of particles to
surfaces – for example: electrostatic precipitation, paint spraying,
electrostatic clamping, fly-ash collection in chimneys, laser
printing, photocopying, and particle alignment (ex. flocking).
2.2 ELECTRIC CHARGES
A spark will be produced if your finger were kept closer to
the metal doorknob while walking across a carpet during dry
weather. Television advertising has alerted us to the problem of
“Static cling” in clothing. Besides that, lightning is familiar to
everyone. Each of these phenomena indicates a tiny glimpse of
the vast amount of electric charge that is stored in the familiar
objects that surround us and in our own bodies. Electric charge is
an intrinsic characteristic of the fundamental particles like electrons and
protons in the atoms which making up those objects; that is, it is a
characteristic that automatically accompanies those particles
wherever they exist.
Normally, huge amount of charge in an everyday object is
hidden because the object contains equal amount of the two kinds
of charge: positive charge and negative charge. When such charges
are balanced, it contains no net charge and the object is said to be
electrically neutral. On the other hand, if the two types of charge
are not in balance, then there is a net charge and the object is said
to be charged. The imbalance, generally, is always very small
compared to the total amounts of positive and negative charges
Charles Augustin de Coulomb
(1736-1806)
Coulomb, French Physicist, pioneer in
electrical theory, born in Angoulême,
W France. After serving as a military
engineer for France, he retired to a
small estate and devoted himself to
research in magnetism, friction, and
electricity. In 1777 he invented the
torsion balance for measuring the force
of magnetic and electrical attraction. With this invention, Coulomb was able
to formulate the principle, now known
as Coulomb's law, governing the
interaction between electric charges. The coulomb, the unit of electrical
charge, is named after him.
26 CHAPTER2 | 2.3 Coulomb‟s law
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
Fig. 2.2 shows the (a) repulsive and (b)
attractive forces between two same and
opposite charges respectively.
Fig. 2.3 shows the conservation of
charges. Here one neutron, one proton,
and one pion are produced when two
protons are combined together
existed in the object.
Charged objects, which are in nearer to each other,
interact by exerting forces on one another. To demonstrate this,
we first charge a glass rod by rubbing one end with silk. During
this process, electrons will be transferred to silk and glass
becomes positively charged. We now suspend the charged glass
rod from a thread to isolate electrically it from its surrounding so
that its charge cannot be changed. If we bring a second, similarly
charged, glass rod nearby (Fig. 2.2a), the two rods repel each
other; that is, each rod experiences a force directed away from the
other rod. However, if we rub a plastic rod with fur and bring it
near the suspended glass rod (Fig. 2.2b), the two rods attract each
other; that is, each rod experiences a force directed toward the
other rod. The reason for attracting these two rods is that
plastic rod is negatively charged while rubbing with fur as
positive charges are transferred into fur.
The above demonstrations reveal that charges with same
electrical sign repel each other, and charges with opposite electrical signs
attract each other.
2.2.1 Behaviour of electric charges
1. Charge of electron is –1.602×10-19C and the proton
charge is +1.602×10-19C.
2. Like charges repel each other
3. Unlike charges attract each other
4. Electric charge is quantised - any charge q can be written
as a integer multiple of the fundamental charge e =
1.602×10-19 C. (i.e., charge of particles are either 0, 1e,
2e, 3e, 4e, etc.).
5. Charge is conserved - That is, during any process, the net
electric charge of an isolated system remains constant.
2.3 COULOMB’S LAW
In 1785, Coulomb studied the electric attraction and
repulsion quantitatively and prepared the law that governs them.
This law describes the electrostatic force between two point
charges at certain distance at rest (or nearly at rest). According
to Coulomb, the magnitude of the force of attraction or repulsion
Neutron (0)
Proton (+e)
Pion (+e)
Proton (+e) + Proton (+e)
CHAPTER2 | 2.3 Coulomb‟s law 27
Engineering Physics (MASC 0003)
between any two charged particles depends on the following
three points:
1. The distance between the particles
2. The magnitude of the charges
3. Nature of the medium between two charges
Based on the experimental measurements of the force
between two charges, Coulomb derived the following laws,
known as the Coulomb‟s law of electrostatics.
First law: Like charges repel each other and unlike charges
attract each other.
Second law: The force exerted between two charges directly
proportional to the product of their strength and
inversely proportional to the square of the distance
between them.
Let q1 and q2 be two charged particles and „r‟ is the
distance between them as shown in Fig. 2.5, then electrostatic
force between two charged particles can be written as
orr
qqF
rFandqqF
2
21
221
1
Newtonr
qqkF
r
2
21
(2.1)
where “k” is called proportionality constant or electrostatic constant.
The value of k is given by,
229 /1099.84
1CNmk
(2.2)
where, 0 – permittivity1 of free space, and
r – relative permittivity2 of the medium between two
charges
1 Permittivity is the property of a medium and affects the magnitude of force
between two point charges.
Fig. 2.4 Two charges objects, separated by distance r, repel each other if their
charges are (a) both positive and (b)
both negative. (c) They attract each
other if their charges are of opposite
signs.
Fig. 2.5
Object A
Force of A on B
Force of B on A
F
F
Object B
F
F
F
F
(a) Repulsion
(b) Repulsion
(c) Attraction
r
Object A
Force of A on B
F
F
Object B
F
F
F
F
(a) Repulsion
(b) Repulsion
(c) Attraction
r
q1 q2
r
28 CHAPTER2 | 2.3 Coulomb‟s law
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
The electric force F in air medium (free space) is given by
Newtonr
qqkF
2
21 (r = 1 for air) (2.3)
Generally, forces are measured in Newton; hence, the
electrostatic force between two charged particles is also measured in
Newton (N).
2.3 .1 Sample Problems
1. How many electrons are required to have a charge of one
Coulomb?
Solution:
Charge of an electron is e = - 1.602×10-19 C.
Hence, 18
191024.6
10602.1
1
C
Cn
That is 6.24 x 1018 electrons are required to have a charge of one
Coulomb.
2. Two charges, +0.35C and +0.2C, are embedded 2cm apart in a
block of polyethylene whose relative permittivity (r) is 2.3.
a) What is the magnitude and direction of the force acting on
each charge?
b) What would be the magnitude if the two charges were in
vacuum?
Solution (a):
As the charges are embedded in the medium of polyethylene,
2 Relative permittivity (εr) ratio between absolute permittivity (ε) of insulating
materials and the absolute permittivity of free space or vacuum (ε0= 8.854×10-12
C2/Nm
2). i.e. εr = ε /ε0.
CHAPTER2 | 2.3 Coulomb‟s law 29
Engineering Physics (MASC 0003)
N
r
qqkF
r
68.0)02.0(3.2
102.01035.0109
)( Force
2
669
2
21
Hence, the force acting on each charge is 0.68 N. Since both the
charges are positive, force is acting away from the other charge.
Solution (b):
As both the charges were in the vacuum, its relative permittivity
(r) is 1. [r is 1 for air and vacuum].
Therefore, Force (F) = 2
21
r
qqk
N6.1)02.0(
102.01035.01092
669
3. What would be the force of attraction between two 1 C
charges separated by distance of (a) 1 m and (b) 1 km?
Solution (a):
N
r
qkqFForce
9
2
9
2
21
1091
1109
)(
Solution (b):
N
r
qkqFForce
3
23
9
2
21
109)101(
1109
)(
4. Calculate the electrostatic force between an -particle and a
proton separated by a distance of 5.12×10-15m.
Solution:
Charge of proton is Cq 19
1 10602.1
An -particle is made up of two protons and two neutrons and
hence its charge is
30 CHAPTER2 | 2.4 Electric Field
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
CCq 1919
2 10204.310602.12
The force of attraction between the proton and -
particle is
N
r
qqkF 5.17
1012.5
10204.310602.11099.8
215
19199
2
21
2.4 ELECTRIC FIELD
The concept of electric field was introduced by the British
Physicist and Chemist, Michael Faraday. The electric field force
acts between two charges, in the same way that the gravitational
field force acts between two masses, which could be explained by
Newton‟s law of gravitation3.
We all know that force acting on a particle changes its
motion. In some cases, a particle experiences a force when
another body comes in contact, while in other cases; the particle
experiences a force due to a field such as electric, magnetic and
gravitational fields. Hence, electric field is defined as the space
in which an electric charge experiences a force. That is the
space between and around the charged bodies in which their
influence is felt is called an electric field or electric field of force.
2.4.1 Electric lines of forces
When a small positively charged body is placed in an
electric field, it experiences a force in a field direction. If the
charged body is less in weight and free to move, it will start
moving in the direction of force and the path in which this
charged body moves is called line of force.
3 Newton’s Law of Gravitation states that every matter that has a mass attracts
other matters with a force that is directly proportional to the product of their
masses and inversely proportional to the square of the distance between the
centers of gravity of the two matters. i.e. 2
21
d
mmGF
(a)
(b)
Fig.2.6 shows the direction of electric
lines of forces in (a) a positive charge
and (b) a negative charge.
-
CHAPTER2 | 2.4 Electric Field 31
Engineering Physics (MASC 0003)
(a)
(b)
Fig. 2.7 Imaginary lines with arrow
heads show direction along which
hypothetical positive charges would
move (a) Two positively charged
particles, (b) A negatively and a
positively charged particles.
Fig. 2.8
Therefore, electric line of force can be defined as the path
along which a unit positive charge would tend to move when free
in an electric field.
A charged body is generally represented by lines which
are referred to as electrostatic lines of force. These lines are
imaginary and are used just to represent the direction and
strength of the field. Electric force lines originate from a positive
charge and ends at a negative charge. The number of lines of
force from a unit charge of „q‟ Coulomb will be equal to „q‟ (i.e.
equal to magnitude of the charge of the particle). The lines of
forces for a positive and negative charge are separately shown in
the Fig.2-6.
The lines of forces for two equal and similar charges and
for two equal and dissimilar charges separated by a distance are
shown in Fig.2-7.
2.4.2 Properties of lines of forces
The properties electrostatic lines of force are given below:
1. Electric force lines originate from a positive charge and
terminate on a negative charge.
2. They do not cross each other.
3. Lines of forces are always perpendicular to the surface of the
charged body at the point where they originate or terminate.
4. A unit positive charge, which is free to move, will move
towards the negatively charged particle along the electric line
of force.
5. Two lines of forces moving in the same direction repel each
other while moving in the opposite direction attracts each
other.
2.4.3 Electric field intensity or strength
Electric field intensity at a given point is defined as equal
to the force experienced by a positive unit charge place at that
point. It is denoted by the letter E.
Let the electric field intensity due to a charge „q‟ at a
distance „r‟ be E. If a charge „Q‟ Coulomb is placed at this point
(Fig.2-8), it will experience a force
q Q
r
32 CHAPTER2 | 2.4 Electric Field
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
F = qE (2.6)
According to the Coulomb‟s law, the force between the
charges Q and q at a distance „r‟ is given by
2
04 r
qQF
r (2.6)
where, F – electric field,
Q, q – charges of the particles,
0 – permittivity of free space = 8.85×10-12 C2/Nm2,
r – relative permittivity which depends on nature of
the medium and
r – distance between the charges.
From the equations (2.3) and (2.4), we have
CNr
QE
orr
qQqE
r
r
/4
4
20
20
(2.7)
If the medium is air (r = 1 for air),
CNr
Qk
r
QE /
4 22
0
(2.8)
where, k = 229
0
/1099.84
1CNm
.
To determine the electric field intensity due to a group of
point charges, we first calculate the electric field intensity of each
charge at the given point assuming only that charge present and
add up all these intensities vectorially, i.e.,
20
220
2
210
1
4.......
44 nr
n
rr r
Q
r
Q
r
QE
E = E1 + E2 + E3 + …+En (2.7)
CHAPTER2 | 2.5 Electric Potential 33
Engineering Physics (MASC 0003)
Fig.2.10 Electric potential due to an
electric field
2.4 .4 Sample problem
5. Find the electric field from a point charge of 30 C at a
distance of 5 m.
To solve this question we shall consider the Fig.2-9.
Electric field from a point charge at a distance 5 m is
E = CNr
qk /
2
CN
CN
/1008.1
/)5(
1030109
4
2
69
2.5 ELECTRIC POTENTIAL
Definition: The potential at any point is defined as the amount of
work done, against the field, in moving an unit positive charge
from infinity to that point. The symbol for potential is “V” and
the unit is joule per coulomb (J/C) or volt (V).
When a body is charged, work is done in charging it.
This work done is stored in the body in the form of potential
energy. The charged body has the capacity to work by moving
other charges either by attraction or repulsion. The ability of the
charged body to do work is called electric potential. Generally,
electric potential is a measured as a ratio between work done by
the body and its charge. i.e.,
Electric Potential, C
J
Q
W
charge
donework V
The work done is measured in Joules and charge in
Coulombs. Hence, the unit of electric potential is Joules/Coulomb
or volt. If W = 1 joule, and Q = 1 Coulomb, then V = 1/1 = 1
volt. Therefore, a body is said to have an electric potential of 1
volt if 1 joule of work is done to give it a charge of 1 Coulomb.
Therefore, when we say that a body has an electric
potential of 4 volts, it means that 4 joules of work has been done
p
q =30×10-6
C
5 m
Fig. 2.9
A B
r
E
34 CHAPTER2 | 2.6 Dielectrics
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
to charge the body to 1 coulomb. In other words, every coulomb
of charge possesses energy of 4 joules. The greater the
joules/coulomb on a charged body, the greater is its electric
potential.
2.6 DIELECTRICS
A dielectric, or electric insulator, is a substance that is a
poor conductor of electricity, but an efficient supporter of
electrostatic fields. If the flow of current between opposite
electric charge poles kept to a minimum while the electrostatic
lines of flux are impeded or interrupted, an electrostatic field can
store energy. This property is useful in capacitors.
2.6.1 Types of dielectrics
Dielectric materials can be solids, liquids, or gases. In
practice, most dielectric materials are solid. They are porcelain
(ceramic), mica, glass, plastics, and the oxides of various metals.
Dry air is an excellent dielectric, and is used in variable
capacitors and some types of transmission lines. Distilled water
is fair dielectric. In addition, a high vacuum can also be a useful,
lossless dielectric even though its relative dielectric constant is
only unity.
2.6.2 Dielectric loss
When the a.c. voltage is applied to a dielectric material,
the electrical energy is absorbed by the material and is dissipated
in the form of heat. This dissipation of energy is called dielectric
loss. Since this involves heat generation and heat dissipation,
this assumes a dominating role in high voltage applications.
2.6.3 Dielectric breakdown
Every insulator (dielectric) can be forced to conduct
electricity. This phenomenon is known as dielectric breakdown.
The most important mechanism for the breakdown is that
some free carriers (for example caused by impurities) are
accelerated in the field so much that they can ionize other atoms
CHAPTER2 | 2.7 Capacitors 35
Engineering Physics (MASC 0003)
and generate more free carriers. Then the breakdown proceeds
like an avalanche.
2.6.4 Applications of dielectric materials
Dielectrics are very widely used as insulating materials to
provide electrical insulation to electrical and electronic
equipments.
1. Plastic or rubber is used as insulator for the electrical
conductors made of aluminium or copper which are used for
electric wiring.
2. In heater coils ceramic beads are used to avoid short
circuiting as well as to insulate the outer body from electric
current.
3. In electric iron, mica or asbestos insulation is provided to
prevent the flow of electric current to the outer body of the
iron.
4. In transformers as well as in motor and generator windings
varnished cotton is used as insulator.
5. A very important application of dielectric materials is their
use as energy storage capacitors.
2.7 CAPACITORS
Capacitors4 (also known as Condenser) are components
designed for storing the electric charge. A capacitor is made by
using two conductors that are electrically separated by a
dielectric material (i.e. isolated electrically) from each other and
from their surroundings.
Capacitors have a variety of uses because there are many
applications that involve storing charge. A good example is
computer memory, but capacitors are found in all sorts of
electrical circuits, and are often used to minimize voltage
fluctuations. Another application is a flash bulb for a camera,
which requires a lot of charge to be transferred in a short time.
4 The name is derived from the fact that this arrangement has the capacity to
store charge. The name condenser is given to the device due to the fact that
when potential difference is applied across it, the electric lines of force are
condensed in the small space (dielectric) between the plates.
36 CHAPTER2 | 2.7 Capacitors
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
Fig. 2.11 (a) shows the capacitor
without charges
Fig. 2.11 (b) shows the movement of
electrons from Plate A to B.
Fig. 2.11 (c) shows the voltage V across
the plates A and B equals with battery
voltage V.
Fig. 2.11(d) shows the capacitor with
charges
Batteries are good at providing a small amount of charge for a
long time, so charge is transferred slowly from a battery to a
capacitor. The capacitor is discharged quickly through a flash
bulb, lighting the bulb brightly for a short time.
2.7.1 Working of the capacitor
Let us consider a capacitor which has two parallel plates
A and B is connected across a battery of V volts as shown in
Fig.2.11. If the switch „S‟ is in open as shown in Fig. 2.11(a), the
capacitor plates are neutral i.e., there is no charge on the plates.
When the switch is closed, the electrons from plate A will be
attracted by the positive terminal of the battery and these
electrons are storing on the plate B as shown in Fig. 2.11(b).
Hence, plate A attains more and more positive charge and plate B
gets more and more negative charge. This action is referred to
as charging a capacitor because the capacitor plates are becoming
charged. This process of electron flow or charging continues till
potential difference across capacitor plates becomes equal to
battery voltage V. When the capacitor is charged to the battery
voltage V, the current flow ceases as shown in Fig. 2.11(c). If
now the switch is opened as shown in Fig. 2.11(d), the capacitor
plates will retain the charges. Thus the capacitor plates which
were initially neutral now have charges on them. This shows
that a capacitor stores charges. The following points may be
noted about the action of a capacitor:
1) When a d.c. potential difference is applied across a capacitor,
a charging current will flow until the capacitor is fully
charged when the current will cease. This whole charging
process takes place in a very short time, a fraction of a second.
Thus a capacitor once charged, prevents the flow of direct
current.
2) The current does not flow through the capacitor ie., between
the plates. There is only transference of electrons from one
plate to the other.
3) When a capacitor is charged, the two plates carry equal and
opposite charges (say +Q and –Q). This is expected because
one plate loses as many electrons as the other plate gains.
Thus charge on a capacitor means charge on either plate.
A B
S V
A B
S V
+ +
+
A B
S V
+ +
+
V
A B
S V
+ +
+
CHAPTER2 | 2.7 Capacitors 37
Engineering Physics (MASC 0003)
2.7.2 Capacitance
When a voltage is applied to the capacitor, it is charged
and the conductors or plates have equal but opposite charges5 of the
magnitude q. The charge (q) stored in a capacitor is proportional
to the potential difference (V) between the two plates. That is
q V (or) q = C V C = q/V (2.8)
where „C‟ is the proportionality constant known as capacitance of
the capacitor. Hence “the ratio between the charge on
capacitor plates and the potential difference across the plates
is called capacitance of the capacitor”. Its value depends only
on the size of the plates and not on their charge or potential
difference. The greater the capacitance, the more charge is
required.
2.7.3 Unit of Capacitance
The unit of the capacitance is farad (F) which equal to one
coulomb per volt. That is,
1 farad = 1 F = 1 coulomb per volt = 1 C/V (2.9)
As the farad is a very large unit, submultiples of the farad,
such as the millifarad (1mF = 10-3 F), microfarad (1F = 10-6 F)
and the picofarad (1pF = 10-12 F) are practically used.
A capacitance for any pair of separated conductors can be
found with this formula:
d
A C 0 r (in a medium) (2.10)
d
A C 0 (in air) (2.11)
5 That is one plate of the capacitor is positively charged, while the other is
negatively charged. Since the plates in the capacitor are conductors, all points
on a plate are at the same electric potential. Moreover, there is a potential
difference between the two plates.
38 CHAPTER2 | 2.7 Capacitors
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
Fig. 2.12 (a)
Fig. 2.12 (b)
Fig. 2.12 (c)
where,
C = capacitance in Farads
0 = absolute permittivity
r = relative permittivity of the medium (r = Cmed / Cair)
A = area of plate overlap in square meters
d = distance between plates in meters
A capacitor can be made variable rather than fixed in value by
varying any of the physical factors determining capacitance. One
relatively easy factor to vary in capacitor construction is that of plate
area, or more properly, the amount of plate overlap.
2.7.4 Factors affecting capacitance
The capacitance of a capacitor depends upon the
following:
1. Plate Area: All other factors being equal, greater plate area
gives greater capacitance; less plate area gives less capacitance.
Explanation: Larger plate area can hold greater charge for a
given p.d. and hence capacitance will be high (Fig. 2.12(a)).
2. Plate Spacing: All other factors being equal, larger plate
spacing gives less capacitance; closer plate spacing gives greater
capacitance.
Explanation: When the plates are brought closer, the
electrostatic filed between the plates is intensified and hence
capacitance increases (Fig. 2.12(b)).
3. Dielectric Material: All other factors being equal, greater
permittivity of the dielectric gives greater capacitance; less
permittivity of the dielectric gives less capacitance.
Explanation: Although it's complicated to explain, some
materials offer less opposition to field flux for a given amount of
field force. Materials with a greater permittivity allow for more
field flux (offer less opposition), and thus a greater collected
charge (Fig. 2.12(c)).
Less capacitance
More capacitance
Less capacitance
More capacitance
More capacitance
glass
Less capacitance
air
CHAPTER2 | 2.7 Capacitors 39
Engineering Physics (MASC 0003)
Fig. 2.13 shows the capacitor in series
connection.
2.7.5 Capacitors in parallel and series
When there is a combination of capacitors in a circuit, we
can sometimes replace that combination with an equivalent
capacitor – that is, a single capacitor that has the same
capacitance as the actual combination of capacitors. With such a
replacement, we can simplify the circuit, affording easier
solutions for unknown quantities of the circuit. Here we discuss
some basic combinations of capacitors.
2.7.5.1 Capacitors in Series
When the elements are connected end-to-end in circuits called
series circuit.
Let a p.d6. of V volts be applied across the combination of
capacitors C1, C2, C3 and let V1, V2, and V3 be the p.d. across the
capacitors in Fig.2.13. Let C be the equivalent capacitance of this
combination.
In series, voltage across each capacitor is difference and
the charge on each capacitor is same.
Now, V = V1 + V2 + V3 (2.12)
We know, V = charge / capacity, therefore
V1 = Q/C1, V2 = Q/C2, V3 = Q/C3 and V = Q/C
Substitute the values of V1, V2, V3 and V in eq. (2), we have
𝑄
𝐶=
𝑄
𝐶1+
𝑄
𝐶2+
𝑄
𝐶3 (2.13)
or
1
𝐶=
1
𝐶1+
1
𝐶2+
1
𝐶3 (2.14)
or, 𝑐 =𝐶1𝐶2𝐶3
𝐶2𝐶3+𝐶3𝐶1+𝐶1𝐶2 (2.15)
6 When the potential difference (p.d.) V is applied across several capacitors
connected in series, the capacitors have identical charges q. The sum of the
potential differences across all the capacitors is equal to the applied p.d. V.
V
+Q - Q
V1 V2 2
V3 3
C3 C1
+Q +Q - Q - Q
C2
40 CHAPTER2 | 2.7 Capacitors
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
Fig. 2.14 shows the capacitor in parallel
connection. In parallel, voltage across
each capacitor is same and the charge
on each capacitor is different.
2.7.5.2 Capacitors in parallel
When the elements are connected side-to-side in circuits called
parallel circuit.
Consider three capacitors of capacitance C1, C2 and C3 and
a p.d. of V volts be applied across the combination as shown in
Fig. 2.14. If Q is the total charge given, all the three capacitors
will share this charge depending upon the capacity of individual
capacitors. Let Q1, Q2, and Q3 be the charge on these capacitors
respectively, then the total charge
Q = Q1 + Q2 + Q3 (2.16)
Since, charge Q = C V, we have
Q1 = C1V. Q2 = C2V, and Q3 = C3V. (As the potential across each
capacitor is the same). Substituting these values of Q, Q1, Q2, and
Q3 in eq. (5), we have
CV = C1V+C2V+C3V (or) C = C1 + C2 + C3 (2.17)
Thus the total capacitance of the combination of a number of
capacitors in parallel equals to the algebraic sum of the capacities
of individual capacitors.
V
+Q1
C3
- Q1
C2
C1
+Q2
+Q3
- Q2
- Q3
CHAPTER2 |Summary 41
Engineering Physics (MASC 0003)
SUMMARY
1. Electrostatics: A study about electric charges, electric forces and electric fields at rest.
2. Electric charge: An intrinsic characteristic of the fundamental particles in the atoms.
Electric charges may be positive or negative. It is measured in the unit of Coulomb.
3. Coulomb: One Coulomb is the amount of charge flowing through a conductor in one
second when one ampere of current is flowing through that conductor.
4. Coulomb’s law: (1) Like charges repel and unlike charges attract each other. (2) The
electrostatic force experienced between two charged particles is directly proportional to
the product of their strength and inversely proportional to the square of the distance
between them.
5. Electrostatic force (F) between two charged particles:
Newtonr
qqkF
r
2
21
(for medium)
Newtonr
qqkF
2
21 (r = 1 for air)
6. Electric field (E): The space in which an electric charge experiences a force.
Normally the space between and around the charged bodies is called electric field.
7. Electric lines of forces: These are the lines drawn virtually that indicates the
movement of an unit positive charge in the electric field.
8. Electric potential (V): The amount of work done in moving an unit positive charge
from infinity to a point in the opposite direction to the electric field. Its unit is J/C or
Volt (V).
9. Dielectric: A substance which has two different electric charges (dipole). Normally,
these substances are poor conductor of electricity.
10. Dielectric loss: When the electrical energies are applied to the dielectric materials, they
are absorbed by the materials and dissipated in the form of heat, as dielectrics are
insulators. This phenomenon is called dielectric loss.
11. Dielectric breakdown: The phenomenon of converting dielectrics to conduct
electricity is known as dielectric breakdown.
12. Capacitor: A device which is made by two conductors separated by a dielectric
material used to store and discharges the electrical energies.
42 CHAPTER2 | Summary
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
SUMMARY CONT’D
13. Capacitance: The ratio between the charge on capacitor plates and the p.d. across the
plates is called Capacitance of the capacitor (i.e. C = Q/V). Its unit is Farad (F).
Practically it is determined using the relation, 𝐶 =𝜀𝑜𝜀𝑟𝐴
𝑑𝐹.
14. Capacitors in series: Equivalent capacitance of the capacitors in series can be calculated
using the relation,𝐶𝑒𝑞 =1
1
𝐶1 +
1
𝐶2 +
1
𝐶3 +⋯+
1
𝐶𝑛 .
15. Capacitors in parallel: Equivalent capacitance of the capacitors in parallel can be
calculated using the relation,𝐶𝑒𝑞 = 𝐶1 + 𝐶2 + 𝐶3 + ⋯+ 𝐶𝑛 .
CHAPTER2 | /Problems for Chapter 2 43
Engineering Physics (MASC 0003)
PROBLEMS FOR CHAPTER 2
1. Two spheres charged with equal but opposite charges experience a force of 103 Newtons
when they are placed 10 cm apart in a medium of relative permittivity is 5. Determine the
charge on each sphere.
2. A point charge of C6100.3 is 12cm distant from a second point charge of
C6105.1 . Calculate the magnitude of the force between them.
3. What must be the distance between the point charge Cq 261 and point charge
Cq 472 in order that the attractive electrical force between them has a magnitude of
5.7N?
4. The average distance r between the electron and the central proton in the hydrogen atom
is m11103.5 . What is the magnitude of the average electrostatic force that acts between
these particles?
5. Two charges, q1 = +.35 C and q2 = +0.2 C are embedded 2 cm apart in a block of
polyethylene (r = 2.3).
a) Determine the electric field due to q1 on q2.
b) What would be the electric field due to q1 on q2 if the two charges were in vacuum?
6. A small uniformly charged sphere has a total charge of 1.4 ×10-8 C.
a) What is the electric field at a point 5 mm away from the sphere?
b) What force would act on a point charge of -1×10-9C at this point?
7. Two charges, q1 = 5C and q2 = 7C, are located 15 cm away from the point P. Determine
the electric field at the point P by the charge A and B.
8. Three capacitors have capacitances of 5 F, 10 F, and 13 F respectively. Determine the
equivalent capacitance when they are connected (i) in series and (ii) in parallel.
44 CHAPTER2 | /Problems for Chapter 2
Supplementary Lecture notes prepared by Dr. R.D. Senthilkumar | Spring 2009
9. A parallel plate capacitor in which the plates are 2.0 m long and 1.0 cm wide and are separated
by a 0.017 mm air gap.
a) Find the capacitance.
b) What is the charge on each plate if the capacitor is connected to a 12-V battery?
c) If a dielectric which is 0.017 mm thick and of relative permittivity 2.3 is inserted between
the plates of the capacitor, Recalculate the part (a), and (b).
10. Circuits show the capacitors in parallel and series in Fig. 1 and Fig. 2 respectively. For each
capacitor calculate (a) the charge on it, (b) the p.d. across it. What is the total capacitance for
each circuit?
Fig. 1 Fig. 2
CHAPTER2 | Short Questions for Chapter 2 45
Engineering Physics (MASC 0003)
SHORT QUESTIONS FOR CHAPTER 2
1. Write a short note on Electrostatics.
2. Define Electric charge. List the properties of electric charge.
3. State the Coulomb‟s law of electrostatics and derive the relation to find force between two
charged particles.
4. What is electric field? Write the relation of electric field intensity.
5. What are the properties of electric lines of forces?
6. Define the following terms:
a. Coulomb b. Electric Potential c. Dielectric loss d. Dielectric breakdown e. Capacitance
7. What are the dielectric materials? Mention its types with examples.
8. Write the applications of dielectric materials.
9. Write a short note on Capacitors.
10. What are the three factors affecting capacitance of the capacitors. Briefly explain the each.
11. What are series and parallel circuits? Explain with diagrams.
12. Write the formula for the equivalent capacitances of two capacitors in series and parallel.
46 CHAPTER2 | References
Engineering Physics (MASC 0003)
REFERENCES
1. Halliday, Resnick and Walker, Fundamentals of Physics (6th
ed). John Wiley & Sons, Inc.,
New York (2001). ISBN: 9971-51-330-7.
2. V.K. Metha and Rohit Metha, Basic Electrical Engineering. S. Chand & Co Ltd., New Delhi,
India (2002). ISBN: 81-219-0871-X.
3. R.K. Gaur and S.L. Gupta, Engineering Physics (8th
ed). Dhanpat Rai Publications (P) Ltd.,
New Delhi, India (2003).
For Additional reading:
1. The Tutorials on Electrostatic force and field at online:
http://www.glenbrook.k12.il.us/gbssci/Phys/Class/estatics/estaticstoc.html
2. Some simulations on electrostatics at online:
http://phet.colorado.edu/simulations/index.php?cat=Electricity_Magnets_and_Circuits
3. Internet for Classroom (This site has plenty of resources about physics including the
animations which could explain the basics of physics):
http://www.internet4classrooms.com/physics.htm
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