Chapter-2-Intro to Electricity and DC Electric Circuits

8/13/2019 Chapter-2-Intro to Electricity and DC Electric Circuits

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Chapter

2

INTRODUCTION TO ELECTRICITY &

DC ELECTRIC C IRCUITS

Objectives

After completing this chapter, you should be able to

1. Demonstrate evidence for the existence of two types of charges.

2. State the Coulombs law of electrostatics and use the Coulombs relation to find electrostatic force

3. Define the electric field, electric potential, emf, and electric current

4. Describe resistance of the conductors and state the Ohms law.

5. Recall and use the relation V = IRto solve its related problems.6. Recall the basic electricity relationships, and equations for Resistances in series and parallel circuits.

7. Construct the voltage and current divider circuits.

8. Compare voltage and current in series and parallel circuits.

9. Investigate and describe qualitatively the relationship among current, voltage, and resistance in a simple

electric circuit

10. State the Kirchhoffs laws and finding currents by applying them.

Outline

Electric charges 25 30

Electric field 31 33

Basics of DC electricity 34 53

Summary of Chapter 2 54 55

Problems and Short Questions 56 59


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2.1 Introduction .......................................................................................................................................... 25

2.2 Electric Charges .................................................................................................................................... 26

2.2.1 Behaviour of electric charges ........................................................................................................ 27

2.3 Coulombs law ...................................................................................................................................... 27

2.3.1 Sample Problems .......................................................................................................................... 29

2.4 Electric Field ......................................................................................................................................... 31

2.4.1 Electric lines of forces ................................................................................................................... 31

2.4.2 Properties of lines of forces .......................................................................................................... 32

2.4.3 Electric field intensity or strength ................................................................................................. 32

2.4.4 Sample problem ............................................................................................................................ 33

2.5 Electric Potential .................................................................................................................................. 34

2.6 Electric current ..................................................................................................................................... 35

2.6.1 Mechanism of flow of currents ..................................................................................................... 36

2.7 Potential Difference ............................................................................................................................. 362.8 Electromotive force (EMF) ................................................................................................................... 37

2.9 Voltage and Current Sources ................................................................................................................ 37

2.9.1 Independent Sources .................................................................................................................... 38

2.9.2 Dependent Sources ....................................................................................................................... 38

2.10 Resistance ........................................................................................................................................... 38

2.10.1 Ohms Law ............................................................................................................................... 39

2.10.2 Resistivity (or) Specific Resistance .............................................................................................. 40

2.10.3 Temperature Coefficient of Resistance ...................................................................................... 41

2.11 Power and Energy .............................................................................................................................. 41

2.12 Resistors ............................................................................................................................................. 43

2.12.1 Fixed type resistors .................................................................................................................... 43

2.12.2 Variable type resistors ............................................................................................................... 43

2.12.3 Special resistors .......................................................................................................................... 44

2.13 Ressitors in DC Ciricuits ...................................................................................................................... 45

2.13.1 Resistors in Series ............................................................................................................................ 45

2.13.1.1 Characteristics of series circuit ................................................................................................ 46

2.13.1.2 Voltage (potential) divider ....................................................................................................... 47

2.13.2 Resistors in Parallel ......................................................................................................................... 48

2.13.2.1 Characteristics of parallel circuit .............................................................................................. 49

2.13.2.2 Current Divider ......................................................................................................................... 49

2.13.2.3 Advantages of parallel circuits ................................................................................................. 50

2.13.3 Resistors in Series and Parallel ........................................................................................................ 50

2.14 Kirchhoffs law .................................................................................................................................... 512.14.1 Kirchhoffs Current Law (KCL) .......................................................................................................... 52

2.14.2 Kirchhoffs Voltage Law (KVL) ......................................................................................................... 53

Summary .................................................................................................................................................... 54

Problems for Chapter 2 .............................................................................................................................. 56

Short Questions for Chapter 2 ................................................................................................................... 59

References .................................................................................................................................................. 60

CONTENTS
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CHAPTER 2 | 2.1 Introduction 25

Engineering Physics (MASC 10002)

2.1 INTRODUCTION

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 ofrubbing, 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 aspositivelycharged.

It was discovered much later that this fluid was actually

composed of extremely small bits of matter called electrons, sonamed in honor of the ancient Greek word for amber.

In the 1780s, Precise measurements of electrical charge

were carried out by the French Physicist Charles Augustin de

Coulomb, using a device called a torsional balancemeasuring the

force generated between two electrically charged objects. This

Fig. 2.1 Static cling, shows the charcomb attracts neutral bits of paper


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26 CHAPTER 2 | 2.2 Electric Charges

Supplementary study material prepared by Dr. R.D. Senthilkumar| Fall 2013

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 rangingfrom 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

surfacesfor 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 ofthe 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.

Usually, 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 negativecharge. 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 is always very small compared to

the total amounts of positive and negative charges existed in the

Charles Augustin de Coulomb(1736-1806)

Coulomb, French Physicist, pionee

electrical theory, born in Angoulm

W France. After serving as a militaengineer for France, he retired to a

small estate and devoted himself to

research in magnetism, friction, an

electricity. In 1777 he invented the

torsion balance for measuring the f

of magnetic and electrical attraction

With this invention, Coulomb was

to formulate the principle, now kno

as Coulomb's law, governing the

interaction between electric charge

Thecoulomb

, the unit of electricalcharge, is named after him.


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CHAPTER 2 | 2.3 Coulombs law 27


Fig. 2.2 shows the (a) repulsive and attractive forces between two same aopposite charges respectively.

Fig. 2.3 shows the conservation ofcharges. Here one neutron, one protand one pion are produced when twprotons are combined together

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 glassbecomes 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 attracteach

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 aspositive 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 1.60210-19C and the protoncharge is +1.60210-19C.

2. Like charges repel each other3. Unlike charges attract each other4. Electric charge is quantised- any charge q can be written

as a integer multiple of the fundamental charge e =

1.60210-19 C. (i.e., charge of particles are either 0, 1e,

2e, 3e, 4e, etc.).

5. Charge is conserved- That is, during any process, the netelectric charge of an isolated system remains constant.

2.3 COULOMBS 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 (0Proton (+e

Pion (+e)Proton (+e) + Proton (+e)


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28 CHAPTER 2 | 2.3 Coulombs law


between any two charged particles depends on the following

three points:

1. The distance between the particles2. The magnitude of the charges3. 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 Coulombs 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 andinversely 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

orrqqF

rFandqqF

2

21

221

1

Newtonr

qqkF

r

2

21

(2.1)

where k is called proportionality constant or electrostatic constant.

The value of kis 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, separaby distance r, repel each other if the

charges are (a) both positive and (b)both negative. (c) They attract eachother if their charges are of oppositesigns.

Fig. 2.5

Object A

Force ofA on B

Force ofB on A

Object B

(a) Repulsion

(b) Repulsion

(c) Attraction

r

Object A

Force ofA on B

Object B

(a) Repulsion

(b) Repulsion

(c) Attraction

r

q1q2

r


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CHAPTER 2 | 2.3 Coulombs law 29


The electric force F in air medium (free space) is given by

Newton

r

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 = - 1.60210-19C.

Hence, 1819

1024.610602.1

1

C

Cn

That is 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 polyethylenewhose 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,

N

r

qqkF

r

68.0)02.0(3.2

102.01035.0109

)(Force

2

669

2

21

2 Relative permittivity (r) ratio between absolute permittivity () of insulating

materials and the absolute permittivity of free space or vacuum (0= 8.85410-12

C2/Nm2). i.e. r= /0.


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30 CHAPTER 2 | 2.3 Coulombs law


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 distanceof 5.1210-15m.

Solution:

Charge of proton is Cq 191 10602.1

An -particle is made up of two protons and two neutrons and hence its charge is

CCq 19192 10204.310602.12


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CHAPTER 2 | 2.4 Electric Field 31


(a)

(b)

Fig.2.6 shows the direction of electrlines of forces in (a) a positive chargand (b) a negative charge.

The force of attraction between the proton and -

particle is

Nr

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

Newtons law of gravitation3.

We all know that force acting on a particle changes itsmotion. 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 asthe 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 orelectric 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.

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

3Newtons Law of Gravitationstates 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. 221

d

mmGF

-


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32 CHAPTER 2 | 2.4 Electric Field


(a)

(b)

Fig. 2.7 Imaginary lines with arrowheads show direction along whichhypothetical positive charges would

move (a) Two positively chargedparticles, (b) A negatively and apositively charged particles.

Fig. 2.8

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 inthe 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 andterminate 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 movetowards the negatively charged particle along the electric line

of force.

5. Two lines of forces moving in the same direction repel eachother while moving in the opposite direction attracts eachother.

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

F = qE (2.4)

According to the Coulombs law, the force between the

charges Qand qat a distance r is given by

q Q

r


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CHAPTER 2 | 2.4 Electric Field 33


204 r

qQF

r

(2.5)

where, F electric field,

Q, q charges of the particles,

0 permittivity of free space = 8.8510-12C2/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

(2.6)

If the medium is air (r= 1 for air),

CNr

Qk

r

QE /

4 220

(2.7)

where, k = 229

0

/1099.84

1CNm

.

To determine the electric field intensity due to a group ofpoint 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 (2.8)

2.4.4 Sample problem

5. Find the electric field from a point charge of 30 C at adistance of 5 m.

To solve this question we shall consider the Fig.2-9.

p

q =3010-6

C

5 m

Fig. 2.9


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34 CHAPTER 2 | 2.5 Electric Potential


Fig.2.10 Electric potential due to anelectric field

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

doneworkV

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

voltif 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 doneto 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.

A B

r

E


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CHAPTER 2 | 2.6 Electric current 35


2.6 ELECTRIC CURRENTIt is well known that water always flows from higher pressure level to the lower

pressure level and this is called water flow. Similarly, when a conductor has different levels of

electricity at their two ends, charge also flows from one to other end of the conductor. This

charge flow is called flow of electric current.

The American physicists Tolman and Stewart experimentally confirmed that the

electric current in a conductor is carried by electrons present in that conductor. A metallic

wire contains a large number of electrons in random directions. The electrons in the outer

most orbits are loosely bound with the nucleus compared to the electrons at inner most orbits.

The outer electrons are called free electrons since they can be easily removed from the orbit.

The flow of free electrons in a definite direction in a conductor forms electric current.

Normally, an external force is required to move these free electrons in a definite

direction and this force is called electromotive force(EMF). The EMF is not a force, but it is thework done in moving an unit charge from one end to the other. Therefore, because of EMF,

different levels of electricity are found at the ends of a conductor, i.e., at different electric

potentials. The electric current will flow in a conductor as long as its two ends are at different

electric potentials.

Generally, the electric current (I) is definedasthe rate of flow of charge through any section of a

conductor. Thus,

I =

If charge dqpasses through a wire in time dt, then the current I defined as

= (2.9)In the above relation, the charge is measured in coulomb and time in second; therefore current

is measured in ampere. Hence,

= =

The unit of current ampere is defined as the current which flows through a conductor when one

coulomb of charge flows through that conductor in one second.


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36 CHAPTER 2 | 2.7 Potential Difference


2.6.1 Mechanism of flow of currents

The flow of electric current can be explained by referring

to Fig. 2.11. The copper wire has a large number of free

electrons. When the voltage or electric pressure is applied to the

copper wire, free electrons will start moving towards thepositive terminal round the circuit as shown in Fig. 2.11. This

directed flow of electrons is called electric current. The

direction in which the positive charge will flow is called

conventional current. Since the flow of electric current is

attributed to flow of electrons, the direction of electric current is

opposite to the direction of flow of electrons:

2.7 POTENTIAL DIFFERENCE

The potential difference indicates the electric state of a body.The difference in the potentials of two charged bodies is

called potential difference. If two bodies of different potentials

are connected together a redistribution of charge will takes place

and some charge will move from higher potential to lower

potential, i.e., current flows from higher potential to lower

potential.

Consider two bodies A and B having potentials of 5 volts and 3

volts respectively as shown in Fig. 3.2a. Each coulomb of charge

on body A has energy of 5 joules while each coulomb of chargeon body B has energy of 3 joules. Hence, body A is at higher

potential than the body B. If these two bodies are joined

through a conductor as shown in Fig. 3.2b, then electrons will

flow4from body B to body A. The flow of current stopped when

the two bodies attain the same potential. Hence, the current will

flow in a circuit if potential difference exists. Generally, the

potential difference is called voltage. The unit of potential

difference is volt.

The voltage, vab, between two points a and b in an electriccircuit can be mathematically described as,

= 4The conventional current flow will be in the opposite direction, i.e. from body

A to body B.


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CHAPTER 2 | 37


where w is energy in joules (J) and is qcharge in coulombs (C).

The voltage vabor simply vis measured in volts (V),

Hence, 1 volt = 1 joule/coulomb = 1 newton meter/coulomb

Thus, the potential difference or voltage is the energy (w)required to move a unit charge (q) through an element,

measured in volts (V).

Figure 2.13 shows the voltage across an element connected to

points a and b. The plus (+) and minus ( ) signs are used to

define reference direction or voltage polarity. The vab can be

interpreted in two ways: (1) point a is at a potential of vab volts

higher than point b, or (2) the potential at point a with respect to

point b is vab. It follows logically that in general

= For example, in Fig. 2.14, we have two representations of the

same voltage. In Fig. 2.14(a), point a is+9Vabove point b; in Fig.

2.14(b), point b is 9 V above point a. We may say that in Fig.

2.14(a), there is a 9-V voltage drop from a to b or equivalently a

9-V voltage rise from b to a. In other words, a voltage drop from

a to b is equivalent to a voltage rise from b to a.

2.8 ELECTROMOTIVE FORCE (EMF)

The force which creates the pressure that causes the current to

flow through a conductor is called electromotive force (EMF).

The EMF of a cell is defined as equal to the potential difference

between the terminals of the cell in an open circuit, i.e., when no

current is drawn from the cell. In case, some current is drawn i.e.,

the cell is connected in a closed circuit the potential difference of

the terminals will not be equal to the EMF. The unit of EMF is

also volt.

2.9 VOLTAGE AND CURRENT SOURCES

The voltage or current sources that generally deliver power tothe circuit connected to them. There are two kinds of sources:independent and dependent sources.


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38 CHAPTER 2 | 2.10 Resistance


2.9.1 Independent Sources

An ideal independent source, which is an active element,

provides a specified voltage or current that is completely

independent of other circuit variables.

That is, an ideal independent voltage source delivers to the

circuit whatever current is necessary to maintain its terminal

voltage.

Physical sources such as batteries and generators are examples

for the ideal voltage sources. Figure 2.15 shows the symbols for

independent voltage sources. Notice that both symbols in Fig.

2.15(a) and (b) can be used to represent a dc voltage source, but

only the symbol in Fig. 2.15(a) can be used for a time-varying

voltage source.

An ideal independent current source is an active element that

provides a specified current completely independent of the

voltage across the source. The symbol for an independent current

source is displayed in Fig. 1.12, where the arrow indicates the

direction of current i.

2.9.2 Dependent Sources

An ideal dependent (or controlled) source is an active element inwhich the source quantity is controlled by another voltage or

current.

Dependent sources are usually designated by diamond-shapedsymbols, as shown in Fig. 2.17. Since the control of thedependent source is achieved by a voltage or current of someother element in the circuit, and the source can be voltage orcurrent. Dependent sources are useful in modeling elements suchas transistors, operational amplifiers and integrated circuits.

2.10 RESISTANCE

If we apply the same potential difference between the ends of

geometrically similar rods of copper and glass, the currents

passing through these rods are varied. These variations of

currents are due to the internal characteristic of conductor called

electrical resistance.


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CHAPTER 2 | 2.10 Resistance 39


The opposition to the flow of electric current offered by a

material is called resistance.

2.10.1 Ohms Law

A German Physicist, George Ohm derived an importantrelation which is known as Ohms law.

Ohms law states that at uniform temperature, current

flowing through a conductor is always proportional to the

potential difference between its two ends.

If I is the current passing through a conductor and V is

the potential difference between its ends, then

V (or)V = (2.12)where R is the proportional constant known as electrical

resistance of the conductor.

Therefore, the resistance (R) between any two points of a

conductor is,

R =

V

I (2.13)

As the current is due to the flow of free electrons, we could also

say that resistance is the opposition offered by the substance to

the flow of free electrons. This opposition occurs since atoms and

molecules of the substance block the flow of free electrons. Some

metals such as silver, copper, aluminium etc. offer very low

resistance while the substances such as glass, rubber, mica, dry

wood etc. offer high resistance to the electric current.

It may be noted that resistance is the electric friction offered by

the substance and causes heat with the flow of electric current.

The moving electrons collide with atoms or molecules of the

substance; each collision results the smaller amount of heat.

The unit of resistance is ohm and it is symbolically

represented as . A wire is said to have a resistance of 1 ohm if a

Fig. 2.18

R = 1

1 A

1 V

V

I R

Relation between voltage, curren

and resistance.


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40 CHAPTER 2 | 2.10 Resistance


p.d. of 1 volt across its ends causes 1 ampere of current to flow through it (Fig. 2.14).

2.10.2 Resistivity (or) Specific Resistance

The resistance R of any conductor is

i) directly proportional to its length (L),ii) inversely proportional to its area of cross-section (A),iii) depends on the nature of material, andiv) depends on the temperature.

Therefore,

R (2.14)

where (Rho) is the constant known as specific resistanceor resistivityof the material. Its value

depends on the nature of material. If l= 1 m and a = 1m2, then R = . Hence specific resistance

or resistivity of a material is the resistance offered by 1 meter length of wire of material having 1m2 area

of cross-section. The unit of resistivity is ohm-m.

. = =

The following table provides the resistivity of some materials.

Sl. No. Material Nature

Resistivity

(-m) at room

temperature

1 Copper Metal 1.7 10-8

2 Iron Metal 9.68 10-8

3 Maganin Alloy 48 10-8

4 Nichrome Alloy 100 10-8

5 Pure silicon Semiconductor 2.5 103

6 Pure germanium Semiconductor 0.6

7 Glass Insulator 1010to 1014

8 Mica Insulator 1011to 1015


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CHAPTER 2 | 41


2.10.3 Temperature Coefficient of Resistance

Normally, temperature affects the resistance of all the substances. The value of

resistance usually decreases in insulators while increasing the temperature, but in the case of

conductors the resistance increases when its temperature increases.

Consider a conductor having resistance R0at 0C. After heating the conductor to tC,

its resistance becomes Rt. Then the change in resistance (RtR0) is

i) directly proportional to the initial resistanceii) directly proportional to the rise in temperature andiii) depends on the nature of material

Therefore,

(RtR0) (R0 t) (or)(RtR0)= (R0 t) (or)

= (2.15)

where (alpha) is a constant called temperature co-efficient of resistanceat 0C. Its value depends

on the nature of material and temperature. From the above relation the temperature coefficient

of resistance can be defined as the ratio of increase in resistance per unit rise in temperature to

the original resistance. It can also be defined as the change in resistance per unit resistance, per

unit rise in temperature. The unit of is per C.If we rearrange the above equation, we get

= + (2.16)From this equation, the resistance of a conductor at any temperature can be found.

2.11 POWER AND ENERGY

Although current and voltage are the two basic variables in an electric circuit, they are not

sufficient by themselves. Thus we need to know how much power an electric device can handle,for practical purposes. To relate power and energy to voltage and current, we recall from

physics that:

Power is the time rate of expending or absorbing energy, measured in watts (W).


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42 CHAPTER 2 | 2.11 Power and Energy


.

Hence, mathematically,

=

where, p is the power in watts (W), is energy in Joules (J), t isthe time in seconds (S). From eq. 2.9, 2.10and 2.17, we get,

= =

= (2.18)

or

= (2.19)

Thus, the power absorbed or supplied by an element is the

product of the voltage across the element and the current

through it. The power p in Eq. (2.18) is a time-varying quantity

and is called the instantaneous power. If the power has a

+sign, power is being delivered to or absorbed by the element.If, on the other hand, the power has a sign, power is beingsupplied by the element.

Let us look into the relationship between current i and voltage v

in Fig. 2.19(a). In Fig. 2.19 (a), current enters through thepositive polarity of the voltage. In this case, = + or implies that the element is absorbing power. As thecurrent leaves through the positive polarity of the voltage, in

Fig. 2.19 (b), = or . In other words, in Fig. 2.19(b), the element is releasing or supplying power.

According to the conservation of energy, the algebraic sum of

power in a circuit, at any instant of time, must be zero:

= (2.20)

This says that the total power supplied to the circuit is equal to

the total power absorbed.


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CHAPTER 2 | 43


(a)

(b)

Fig. 2.20 Fixed resistors: (a) Wwound resistor, (b) Carbon film resi

Fig. 2.21 Symbol of fixed resistor

Fig. 2.22 Circuit symbol for: (a) a varresistor in general, (b) Potentiometer

From Eq. (2.18), the energy absorbed or supplied by an element

from time to time is

= =

(2.21)

The electric power utility companies measure energy in watt-

hours (Wh), where

1 Wh = 3,600 J

2.12 RESISTORS

An electric resistoris a two-terminal passive component used to

oppose and limit current. A resistor works on the principle of

Ohms Law. There are two types of resistors, namely, fixed and

variable.

2.12.1 Fixed type resistors

Most resistors are of the fixed type, meaning their resistance

remains constant. The wire-wound and composition are two

common types of fixed resistors which are shown in Fig. 2.20.

The composition (carbon) resistors are used when large

resistance is needed. The circuit symbol in Fig. 2.21 is for a fixed

resistor.

2.12.2 Variable type resistors

Variable resistors have adjustable resistance. The symbol for avariable resistor is shown in Fig. 2.22(a). A common variable

resistor is known as a potentiometer or pot for short, with the

symbol shown in Fig. 2.22(b). The pot is a three-terminal

element with a sliding contact or wiper. By sliding the wiper, the

resistances between the wiper terminal and the fixed terminals

vary. Like fixed resistors, variable resistors can either be of wire-

wound or composition type, as shown in Fig. 2.23. Although

Energy is the capacity to do work, measured in joules (J).


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44 CHAPTER 2 | 2.12 Resistors


resistors like those in Figs. 2.20 and 2.23 are used in circuit designs, today most circuit

components including resistors are either surface mounted or integrated, as typically shown in

Fig. 2.24.

Fig. 2. 23 Variable Resistors Fig. 2. 24 Surface Mount Resistors

(a) Composition type, (b) slider pot Surface mount resistors are soldered ontop of the circuit board and are

identified by number rather than color

bands.

2.12.3 Special resistors

Thermistorsare special resistors whose resistance changes with the temperature. There are

two types of thermistors, namely, positive temperature coefficient (PTC) and negative

temperature coefficient (NTC). In PTC, resistance increases with increase in temperature,

while in NTC, the resistance decreases with the increase in temperature.

PTCs are mostly used as current limiter for circuit protection. As the heat dissipation of

resistor increases, the resistance is increased thereby limiting the current. The NTCs are

mostly used for temperature sensing, replacement of fuses in power supply protection and for

low temperature measurements of up to 10K. An NTC can be replaced by a transistor with a

trimmer potentiometer. These are constructed using sintered metal oxides in ceramic matrix.


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CHAPTER 2 | 2.13 Ressitors in DC Ciricuits 45


Light Dependent Resistor (LDR) is another type of special

resistor. LDRs have cadmium sulfide zigzag tack whose

resistance decreases as the light intensity incident on it increases.

In the absence of light, its resistance is in mega ohms but on the

application of light, the resistance falls drastically. These resistors

are used in many consumer items such as camera light meters,

street lights, clock radios, alarms, and outdoor clocks.

2.13 RESSITORS IN DC CIRICUITS

Generally, the closed path in which Direct Current (DC.) flows

is called DC circuit.

A DC circuit basically have a DC power source (e.g.battery etc.), conductors which are used to carry current and the

load. Fig. 2.26 shows a bulb connected to a battery through the

conducting wires. The direct current starts from the positive

terminal of the battery and comes back to the same terminal

through the bulb as shown in Fig. 3.4. In this circuit, bulb is the

load. The path ABCDA is a DC circuit. Generally, resistances

act as load in a DC circuits and these may be connected in series

or parallel or series parallel. According to the connection of

resistances in the circuits, it can be classified as:

1) Series circuits2) Parallel circuits3) Seriesparallel circuits

2.13.1 RESISTORS IN SERIES

The circuit in which elements are connected end-to-end is called

series circuit.

Consider two resistors R1and R2are connected in series across abattery of V volts as shown in Fig. 2.27 a. The equivalent

resistance circuit is shown in Fig. 2.27b. By current

conservation, the same current (I) is flowing through each

resistor and the voltage (V) across each resistor is different.

Therefore, the total voltage drop from a to c across both

Bulb

V

A

B

I

Fig. 2.26

V2V1

V

R1 R2a b c

I

Fig. 2.27a shows the resistors R1an

R2are connected in series circuit


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46 CHAPTER 2 | 2.13.1 Resistors in Series


resistors is the sum of the voltage drops across the individual

resistors.

According to Ohms law, the voltage drop across the each resistor

is

V1= IR1 and V2= IR2. Therefore,

V = V1+ V2 or

V = IR1 + IR2= I (R1 + R2) or

V

I12

The two resistors in series can be replaced by one

equivalent resistor RTwith the identical voltage drop V = I RTwhich implies that

RT = R1 + R2

Hence, when a number of resistances are connected in

series, the total resistance is equal to the sum of individual

resistances.

The above argument can be extended to N resistors

connected in series. The equivalent resistance is just the sum ofthe individual resistances,

R=R+R+ +R= R=

2.13.1.1 Characteristics of series circuit

A series resistive network has the following

characteristics:

1.

Total resistance is equal to the sum of individual resistanceacross the each resistor.

2. Current flows through all the resistors are same.3. Voltage drop across each resistor is different.4. Sum of voltage drops is equal to the voltage supplied.5. Sum of power supplied by the source is equal to the power

dissipated in the components.

V

RT

a cI

Fig. 2.27b shows the circuit with

total resistance


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CHAPTER 2 | 2.13.1 Resistors in Series 47


2.13.1.2 Voltage (potential) divider

Two resistors that are connected in series can be used as voltage

divider. A Voltage divider circuit is shown in Fig. 2.28. It

consists of a battery which is connected with two resistors R1andR2 in series. The output voltage (divided voltage) is taken

between the two points B and C. The total resistance (RT) in the

circuit is

RT= R1+ R2

Then, I = VR+R =VR (since, V I, Ohms law)

Therefore, Vout= V2 = I R2(i.e. voltage across the resistor R2)

V = VR R ( I =VR)

The above relation is called voltage divider formula. Therefore in

a voltage divider circuits, the input voltage (Vin) is divided into

two parts one part across the resistor R1 and the other part

across the resistor R2. The output taken across the resister R2is

just a fraction of the input voltage and that fraction being

completely determined by the values of the resistances R1and R2.

For example,

If Vin= 12V, R1= 10kand R2 = 20k, then

Vout= (2/3) (12)= 8V.

If R1= 2kand R2= 680for the same input voltage,

then

Vout= (0.25) (12) = 3V.

Voltage dividers are widely used in electric meter circuits, where

specific combinations of series resisters are used to divide a

voltage into accurate values.

Fig. 2.28 Voltage divider


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48 CHAPTER 2 | 2.13.2 Resistors in Parallel


2.13.2 RESISTORS IN PARALLEL

A parallel circuit is a branched arrangement in which two or

more resistors are connected side by side across as shown in

figure 2.29. Parallel connections are also called multipleor shunt

connections.

Consider two resistors R1and R2are connected in parallel

across a battery of V volts as shown in Fig. 2.29 (a). By current

conservation, the current (I) across the resistors is not same and

must be divided into I1 (current through the resistor R1) and I2

(current through the resistor R2). But, the voltage across each

resistor is the same.

According to the Ohms law, current through each

resistance is

I1V

1 and I2

V

2.

Therefore, the total current passes in the circuit is

I I1I2V

1

V

2= V ( 1

1

1

2)

The two resistors in parallel can be replaced by one

equivalent resistor RT with V = IRT (Fig. 2.29(b)). Comparing

these results, the equivalent resistance for two resistors that areconnected in parallel is given by

1

T= ( 1

1

1

2)

This result easily generalizes to N resistors connected in

parallel

R =

R +

R + +

R =

R

=

When one resistance R1 is much smaller than the other

resistance Ri, then the equivalent resistance RTis approximately

equal to the smallest resistor R1. In the case of two resistors,

R= RRR+R RR

R = R

Branches

V

Fig. 2.29(a) shows the resistances

R1and R2in parallel circuit

I1

I

R1

I2

Fig. 2.29 (b) shows the total

resistance

V

I

I


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CHAPTER 2 | 2.13.2 Resistors in Parallel 49


2.13.2.1 Characteristics of parallel circuit

The characteristics of a parallel resistive network are following:

1. Voltage across each branch is the same.2. Current through each branch is different.3. The sum of branch currents is equal to the total current in the

parallel circuit.

4. The reciprocal of the total resistance equals the sum of thereciprocals of the individual branch resistances.

5. The total power consumed in a parallel circuit is equal to thesum of the power consumed by the individual resistors.

2.13.2.2 Current Divider

The resistors connected in parallel provide the mechanismof current dividers.

Current is inversely proportional to resistance.

Therefore, current through each branch of a parallel circuit

shown in Fig. 3.8 can be determined by setting up an inverse

formula like the one given below:

From the Fig. 2.30, the total resistance in the parallel

circuit is given by

T12

12= = V

Hence, the voltage in the parallel circuit is given by

V = ITRT

The current (I1) passing through the resistor R1 is,

(since, V = ITRT )

Similarly,

In general, n

n

The above relation is called as current divider formula.

Fig. 2.30

I = 6 mA

V

I1

R1 R22K


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CHAPTER 2 | 2.14 Kirchhoffs law 51


Now the circuit becomes as serial circuit (Fig. 2.31 (b)).

In the second step, the total resistance (RT) can be found by

adding the individual resistances as we seen in the serial circuit

early. Therefore,

RT= R1+ RTP+ R4= 5 + 2 + 1 = 8

Third step is to calculate total current supplied by the

battery by using Ohms law.

IT= Vin/RT= 12 / 8 = 1.5A

Now the branch currents I1 and I2 and various voltage

drops can be calculated as follows:

I1 = = A

= = or

I2= ITI1= 1.5 1 = 0.5 A

Voltage drop across R1(V1) = IR1 = 1.5 5 = 7.5 V

Voltage drop across R2(V2) = I1 R2= 1 3 = 3V

It should be noted that the voltage drops across R2 and

R3 are same as they are connected in parallel.

Voltage drop across R4(V4) = I R4= 1.5 1 = 1.5V.

Therefore, sum of various drops = 7.5 + 3 + 1.5 = 12V

which is equal to the input voltage given to the circuit. (Since

voltage drop across the parallel circuit is same, only the drop

across the R2is considered to calculate the sum of voltage drops

in the circuit).

2.14 KIRCHHOFFS LAW

We have seen the circuits in which various components

are connected in series, parallel and in series-parallel with a

single voltage source. If a circuit has two or more batteries


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52 CHAPTER 2 | 2.14.1 Kirchhoffs Current Law (KCL)


connected in its different branches, rules of series and parallel

circuits are inapplicable. Such large electric circuits can be easily

solved by the Kirchhoffs Current and Voltage laws which were

formulated by the German physicist Gastav Robert Kirchhoff in

1847.

2.14.1 KIRCHHOFFS CURRENT LAW (KCL)

This is Kirchhoffs first law which is based on the law of

conservation of charge, which means the algebraic sum of

charges within a system cannot change.

According to this statement, the total current leavinga node is

equal to the total current enteringthat node.

To explain this rule, consider the network in which four currents

I1, I2, I3and I4meeting at a junction A as shown in Fig. 2.32.

Let us assume that all the currents entering the junction as

positive signwhereas those leaving as negative sign. Hence

the currents I1 and I2 will be taken as positive whereas thecurrents I3and I4will be taken as negative.

According to the KCL,

I1+ I2+ (I3) + (I4) = 0I1+ I2I3 I4= 0

I = - at a nodeAlso, I1+ I2 = I3+ I4

That is, incomingcurrents = outgoingcurrents, or

Iin= Iout at a node

A

Fig. 2.32 Currents at a node

illustrating KCL

I1

I2

I3

I4

Kirchhoffs Current Law (KCL) states that the algebraic sum

of currents meeting at a point (or node) in any electric circuit

is zero.


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CHAPTER 2 | 2.14.2 Kirchhoffs Voltage Law (KVL) 53


2.14.2 KIRCHHOFFS VOLTAGE LAW (KVL)

This is Kirchhoffs second law which is based on the principle of

conservation of energy.

To illustrate KVL, let us consider the circuit in Fig. 2.33. The

sign on each voltage is the polarity of the first terminal as we

travel around the loop. We can start with any branch and go

around the loop either clockwise or counterclockwise. Suppose

we start with the voltage source and go clockwise around theloop as shown; then voltages would be v1,+v2,+v3,v4, and +v5,

in that order. For example, as we reach branch 3, the positive

terminal is met first; hence we have+v3. For branch 4, we reach

the negative terminal first; hence, v4. Thus, KVL yields

+ + + = Rearranging terms gives

+ + = + which may be interpreted as

Sum of voltage drops = Sum of voltage rises,

which is an alternate form of KVL.

Kirchhoffs Voltage Law (KVL) states that the algebraic sum

of all voltages in a closed path (or loop or mesh) is zero.


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54 CHAPTER 2 | Summary


SUMMARY

1. 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.

2. 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.

3. Coulombs 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.

4. Electrostatic force (F) between two charged particles:

(for medium)

( r = 1 for air)

5. 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.

6. Electric lines of forces: These are the lines drawn virtually that indicates the

movement of an unit positive charge in the electric field.

7. 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).

8. Ohms Law:

At the uniform temperature, current flowing through a conductor is always

proportional to the potential difference between its two ends. i.e. V = I R. =

=

9. Resistivity : =

R

10. Temperature coefficient of resistance: = 11. Total Resistance in series: RT= R1+ R2+ R3+ + Rn


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56 CHAPTER 2 | Problems for Chapter 2


PROBLEMS FOR CHAPTER 2

1. Two spheres charged with equal but opposite charges experience a force of 103 Newtonswhen 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 ofC6105.1 . Calculate the magnitude of the force between them.

3. What must be the distance between the point charge Cq 261 and point chargeCq 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 atomis m

11

103.5

. What is the magnitude of the average electrostatic force that acts betweenthese particles?

5. Two charges, q1 = +.35 C and q2 = +0.2 C are embedded 2 cm apart in a block ofpolyethylene (r= 2.3).

a) Determine the electric field due to q1on q2.

b) What would be the electric field due to q1on q2if the two charges were in vacuum?

6. A small uniformly charged sphere has a total charge of 1.4 10-8C.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 -110-9C at this point?

7. Two charges, q1 = 5C and q2= 7C, are located 15 cm away from the point P. Determinethe electric field at the point P by the charge A and B.

10.Determine the resistance of an electric light bulb if there is a current of 0.8A and a p.d. of120V. Ans: 150Ohms

11.The high voltage in a TV receiver is 17kV. The maximum allowable current is 150 A.Find the least permissible value of load resistance. Ans: 113.3 106.

12.What is the p.d. across a toaster of 13.7Ohms resistance when the current through it is8.75 A? Ans: 120V

13.An electric kettle takes a voltage of 12.8V and supplies a current of 3.2A. Determine theresistance of the circuit. Ans: 4.


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CHAPTER 2 | Problems for Chapter 2 57


14.Find the current through an electric heater with a resistance of 38when the p.d. of 240V is applied. Ans: 6.3A.

15.An electromagnet of resistance 12.4requires a current of 1.5A to operate it. Calculatethe required voltage. Ans: 18.6V.

16.A tungsten lamp has a resistance of 150 at 2850C. Determine its resistance at thetemperature of 20C. Temp. Coefficient of tungsten is 0.00492C at 0C. Ans: 11.

17.The resistance of a dynamo coil is 173at 16C. After working for 6hours on full-load,the resistance of the coil increases to 212. Find the temperature of the coil. Assume

temperature co-efficient of resistance of copper is 0.004261C at 0C. Ans: 72C.

18.Three resistors, 10 , 15 and 5 are connected in series across the 90 V battery.Calculate, a) the total resistance of the circuit, and b) the current through each resistor.

Ans: 30; 3A.

19.Resistors R1 = 3, R2 = 5and R3 = 4are connected in series across a p.d. of 12V.a) What is the total resistance of the circuit? b) Determine the current through each

resistor. c) Find the voltage drop across each resistor. Ans: 12; 1A; 3 V, 5 V and 4V.

20.A 60, a 90and an unknown resistor are connected in series to a battery of 130V. Ifthe current through the circuit is 0.67A, a) find the equivalent resistance of the circuit, and

b) determine the value of unknown resistor. Ans: 195; 44.

21.Three resistors of 15 are connected in parallel to a 3V battery. a) Find the effectiveresistance of the circuit. b) Determine the current passing through each resistor and hence

total current. Ans: 5; 0.2A in each resistor, 0.6A as total.

22.A 12 , a 15 and an unknown resistance are connected in parallel. The total currentpassing through the circuit is 12 A and the current across the 15branch is 4A. i) What isthe total voltage? ii) Calculate the total resistance. iii) Find the value of unknown

resistance. Ans: 60V; 5; 20.

23.Calculate the total current. Calculate the voltage drop across the 36resistor. Calculatethe voltage drop across the 16resistor. Calculate the current through the 48resistor.


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58 CHAPTER 2 | Problems for Chapter 2


24.Calculate the total current. Calculate the voltage drop across the 8resistor. Calculate thevoltage drop across the 12 resistor. Calculate the current passing through the 12resistor.

25.For the circuits shown below, find the voltages, .

26.Find the currents and voltages in the circuit shown below:


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CHAPTER 2 | Short Questions for Chapter 2 59


SHORT QUESTIONS FOR CHAPTER 2

1. Define Electric charge. List the properties of electric charge.2. State the Coulombs law of electrostatics and derive the relation to find force between two

charged particles.

3. What is electric field? Write the relation of electric field intensity.4. What are the properties of electric lines of forces?5. Define the following terms:

a. Electric Potentialb. Electric Powerc. Electric energyd. Resistance

6. Define Electric currentand ampere.7. What is meant by potential difference? Write its unit.8. Explain Electromotive force.9.

What is Resistor? Explain the different types of resistor.

10.List the characteristics of resistors in series circuit.11.Write a short note on Voltage and Current divider circuits.12.What are the characteristics of resistors in parallel circuit?13.Write the formula for the equivalent resistances of two resistors in series and parallel

circuits.

14.Mention the advantages of parallel circuits.15.State and explain the Kirchhoffs current and voltage laws.


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Chapter-2-Intro to Electricity and DC Electric Circuits

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