chapter 2 electrostatics spring 2010

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


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


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


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


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


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


(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


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


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


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


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


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

+ +

+



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.



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



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



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


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


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


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


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

senthil

Cross-Out

senthil

Replacement Text

change to cm

senthil

Cross-Out

senthil

Replacement Text

change the unit to mm

CHAPTER2 | Short Questions for Chapter 2 45


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


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

http://www.glenbrook.k12.il.us/gbssci/Phys/Class/estatics/estaticstoc.html

http://phet.colorado.edu/simulations/index.php?cat=Electricity_Magnets_and_Circuits

http://www.internet4classrooms.com/physics.htm

chapter 2 electrostatics spring 2010

Documents

electric potential

behaviour of electric

electric force of lines

electric field intensity

electric lines of forces

types of charges

coulombs law of electrostatics

types of dielectrics