CHAPTER ONE
MAGNETS AND MAGNETIC FIELDS
Chapter outlines
1.1 Molecular theory
1.1.1Domains and Domains walls
1.1.2 Suitability criteria
1.1.3 Coercivity mechanisms
1.2 Magnetic materials
1.2.1 Permanent magnets
1.2.2 Electromagnets
1.3 Magnetic storage
1.4 Magnetism and super conductivity
1.5 Magnetic shielding
1.1 MOLECULAR THEORY
The first accounts of magnetism date back to the ancient Greeks who also gave magnetism its
name. It derives from Magnesia, a Greek town and province in Asia Minor, the etymological
origin of the word “magnet” meaning “the stone from Magnesia.” This stone consisted of
magnetite (Fe3O4) and it was known that a piece of iron would become magnetized when rubbed
with it. More serious efforts to use the power hidden in magnetic materials were made only much
later. For instance, in the 18th century smaller pieces of magnetic materials were combined into a
larger magnet body that was found to have quite a substantial lifting power. Progress in
1
magnetism was made after Oersted discovered in 1820 that a magnetic field could be generated
with an electric current. Sturgeon successfully used this knowledge to produce the first
electromagnet in 1825. Although many famous scientists tackled the phenomenon of magnetism
from the theoretical side (Gauss, Maxwell, and Faraday) it is mainly 20th century physicists who
must take the credit for giving a proper description of magnetic materials and for laying the
foundations of modem technology
Theory
A popular theory of magnetism considers the molecular alignment of the material. This is called
webber‟s theory; this theory assumes that all magnetic substances are composed of tiny
molecular magnets. Any unmagnetised material has the magnetic forces of its molecular magnets
neutralized by adjacent molecular magnets, thereby eliminating any magnetic effect. A
magnetized material will have most of its molecular magnets lined up so that the north pole of
each molecular magnet points in one direction and the South Pole faces in the opposite direction,
thus a material with its molecules aligned will then have one effective North Pole and one
effective North Pole in the opposite direction
Figure 1.1: webber‟s theory (magnetizing by stroking)
Domain theory
A more modern theory of magnetism is based on the electron spin principle. From the study of
atomic structure it is known that all matter is composed of vast quantities of atoms, each atom
containing one or more orbital electrons. The electrons are considered to orbit in various shells
2
and subshells depending upon their distance from the nucleus. The structure of the atom has
previously been compared to the solar system, wherein the electrons orbiting the nucleus
correspond to the planets orbiting the sun. Along with its orbital motion about the sun, each
planet also revolves on its axis. It is believed that the electron also revolves on its axis as it orbits
the nucleus of an atom. It has been experimentally proven that an electron has a magnetic field
about it along with an electric field. The effectiveness of the magnetic field of an atom is
determined by the number of electrons spinning in each direction. If an atom has equal numbers
of electrons spinning in opposite directions, the magnetic fields surrounding the electrons cancel
one another, and the atom is unmagnetised. However, if more electrons spin in one direction than
another, the atom is magnetized. An atom with an atomic number of 26, such as iron, has 26
protons in the nucleus and 26 revolving electrons orbiting its nucleus. If 13 electrons are
spinning in a clockwise direction and 13 electrons are spinning in a counterclockwise direction,
the opposing magnetic fields will be neutralized. When more than 13 electrons spin in either
direction, the atom is magnetized. An example of a magnetized atom of iron is shown in figure
below.
Why poles are called north and south
A magnet suspended so that it can rotate freely horizontally will eventually settle down with one
pole facing north and the other south.
One pole is therefore called the „north seeking pole‟, usually shortened to just „north pole‟, and
the other pole is called the „South seeking pole‟, usually shortened to just „South pole‟. The
magnet has been orientated by the Earth‟s magnetic field.
Figure 1.2: Finding the North and South Pole of a magnet
**A compass is an application of this effect.
3
Law of magnets: Like poles repel unlike poles attract
A magnetic pole: is a region where the magnetic force is greatest
A magnetic field: is a region where the magnetic force is exerted
THE EARTH’S MAGNETIC FIELD
The earth‟s magnetic field is similar in shape to that around a bar magnet. It is thought to be
caused by electric currents flowing through the molten outer core of the Earth. At the present the
field pattern is like that with a magnetic SOUTH pole situated somewhere below northern
Greenland
Figure 1.3: Earth‟s magnetic field and field lines of force
1.1.1 DOMAINS AND DOMAIN WALLS
A magnetic domain is a region within a magnetic material which has uniform magnetization.
This means that the individual magnetic moments of the atoms are aligned with one another and
they point in the same direction, when cooled below a temperature called the Curie temperature
the magnetization of a ferromagnetic material divides into many small regions called magnetic
domains.
4
Magnetic moment is thus the quantity that determines the torque it will experience in an external
magnetic field. Magnetic moment is a vector having both magnitude and direction
Curie temperature (Curie point) is the critical point where a material‟s intrinsic magnetic
moments change direction. Magnetic moments are permanents dipole moments within the atom
which originates from electron‟s angular momentum and spin.
In magnetism, a domain wall is an interface separating magnetic domains. It is a transition
between different magnetic moments and usually undergoes an angular displacement of 90 or
180 degrees. The energy of a domain wall is simply the difference between the magnetic
moments before and after domain wall was created, this value is usually expressed as energy per
unit wall area.
Figure 1.4: magnetised atom
1.1.2 Suitability criteria
To consider how suitable magnetic material is able to allow and sustain the formation of
magnetic field within it we shall consider a magnetic property called magnetic permeability
5
Magnetic permeability (µ)
In electromagnetism, permeability is the measure of the ability of a material to support the
formation of a magnetic field within itself. In other words, it is the degree of magnetization that a
material obtains in response to an applied magnetic field. Magnetic permeability is typically
represented by the Greek letter μ. The term was coined in September 1885 by Oliver Heaviside.
The reciprocal of magnetic permeability is magnetic reluctivity.
In SI
units,
permeability is
measured
in henries per
meter
(H·m−1),
or newtons per
ampere squared (N·A−2). The permeability constant (μ0), also known as the magnetic constant or
the permeability of free space, is a measure of the amount of resistance encountered when
forming a magnetic field in a classical vacuum. The magnetic constant has the exact (defined)
value µ0 = 4π×10−7 H·m−1≈ 1.2566370614…×10−6 H·m−1 or N·A−2).
A closely related property of materials is magnetic susceptibility, which is a measure of the
magnetization of a material in addition to the magnetization of the space occupied by the
material.
Figure 1.5: Simplified comparison of permiabilities for ferromagnetic, paramagnetic, free space
and diamagnetic
1.1.3 Coercivity mechanisms
In material science, Coercivity (coercive field or coercive force) is a measure of a ferromagnetic
of ferroelectric material to withstand an external magnetic or electric field.
6
For ferromagnetic material the coercivity is the intensity of the applied magnetic field required to
reduce the magnetisation of that material to zero after the magnetisation of the sample has been
driven to saturation.
Thus coercivity measures the resistance of a ferromagnetic material to becoming demagnetized;
It is measured using a B-H Analyzer or magnetometer.
Ferromagnetic materials with high coercivity are called magnetically hard materials and are used
to make permanent magnets. Permanent magnets find application in electric motors, magnetic
recording media (e.g. hard drives, floppy disks and magnetic tape) and magnetic separation.
Material with low coercivity are said to be magnetically soft, they are used in transformer and
inductor cores, recording heads, microwave devices and magnetic shielding. The coercivity is a
horizontal intercept of the hysteresis loop.
1.2
MAGNETIC MATERIALS
Magnetic material is an object that produces a magnetic field. A magnetic material can either be
a permanent or electromagnetic. In general magnetic materials are classified into three groups
namely; Ferromagnetic, Paramagnetic and Diamagnetic materials.
DIAMAGNETIC
Diamagnetic materials have no permanent magnetic moment. However, moments are induced by
the influence of a magnetic field. The induced moments have a direction opposite to that of the
inducing field. Consequently the relative permeability is less than unity. The diamagnetic
susceptibility is negative. The diamagnetic susceptibility is very small, not more than a few parts
in a million, and usually independent of temperature and applied magnetic field. It is probably
present in all materials but usually masked by other larger effects. E.g. He and Be
 In electromagnetism,
the magnetic
susceptibility
is
a
dimensionless
proportionality constant that indicates the degree of magnetization of a material in
response to an applied magnetic field. A related term is magnetizability, the
proportion between magnetic moment and magnetic flux density. A closely related
7
parameter is the permeability, which expresses the total magnetization of material
and volume.
PARAMAGNETIC
Materials have a permanent magnetic moment but the interatomic spacing is so great that there is
negligible atomic interaction. The relative permeability is greater than unity so the susceptibility
is positive. Although the effect is greater than the diamagnetic effect, it is still very small, not
more than a few parts in a hundred thousand. E.g. Li and N
FERROMAGNETIC
Materials have a relative permeability greater than unity and generally very high. The
permeability is not constant but depends upon the degree of magnetization. Ferromagnetic
materials exhibit hysteresis, that is, the induction corresponding to a given magnetizing force
depends upon previous magnetic history. Furthermore, the intrinsic induction approaches a
limiting or saturation value as the magnetizing force is increased indefinitely. E.g. Steel and Iron
1.2.1 Permanent magnets
Permanent magnetic material is an object made from a material that is magnetised and creates its
own persistent magnetic field. The materials that can be magnetised and which are strongly
attracted to a magnetic, are called ferromagnetic these include iron, nickel, cobalt and lodestone.
1.2.2 Electromagnets
Electromagnetic material is one that is temporally magnetised by passing electric current through
a coil around it.
Ferromagnetic material can be divided into magnetically soft and hard materials, soft material
like annealed iron, which can be magnetised but do not tend to stay magnetised and a
magnetically hard are the ones from which permanent magnetic are made from after undergoing
a special process in a powerfully magnetic field, materials like ferrite and alnico are hard
magnetic materials. Electromagnets are normally made from “soft” magnetic materials like soft
iron core.
8
An electromagnet is made from a coil of wire that acts as a magnet when an electric current
passes through it but stops being a magnet when the current stops. Often the coil is wrapped
around the core of soft ferromagnetic material such as steel, which greatly enhances the magnetic
field produced by the coil
A magnet can be destroyed by three different methods namely: hammering, heating and electric
current.
Magnetic shapes
Because of the many uses of magnets, they are found in various shapes and sizes. However,
magnets usually come under one of three general classifications: bar magnets, horseshoe
magnets, or ring magnets. The bar magnet is most often used in schools and laboratories for
studying the properties and effects of magnetism. In the preceding material, the bar magnet
proved very helpful in demonstrating magnetic effects. Another type of magnet is the ring
magnet, which is used for computer memory cores. A common application for a temporary ring
magnet would be the shielding of electrical instruments. The shape of the magnet most
frequently used in electrical and electronic equipment is called the horseshoe magnet. A
horseshoe magnet is similar to a bar magnet but is bent in the shape of a horseshoe. The
horseshoe magnet provides much more magnetic strength than a bar magnet of the same size and
material because of the closeness of the magnetic poles. The magnetic strength from one pole to
the other is greatly increased due to the concentration of the magnetic field in a smaller area.
Electrical measuring devices quite frequently use horseshoe-type magnets
1.3.2 Magnetic storage
Magnetic storage (or magnetic recording) is the storage of data on a magnetized medium.
Magnetic storage uses different patterns of magnetization in a magnetizable material to store data
and is a form of non-volatile memory. The information is accessed using one or more read/write
heads.
Examples: Hard disks, floppy disks, VHS, recording tapes etc
Information is written to and read from the storage medium as it moves past devices called readand-write heads that operate very close (often tens of nanometers) over the magnetic surface.
9
The read-and-write head is used to detect and modify the magnetization of the material
immediately under it. There are two magnetic polarities, each of which is used to represent either
0 or 1.The magnetic surface is conceptually divided into many small sub-micrometer-sized
magnetic regions, referred to as magnetic domains, (although these are not magnetic domains in
a rigorous physical sense)
Magnetic recording classes
Analog recording (e.g. magnetic tape sound recording and VHS)
is based on the fact that remnant magnetization of a given material depends on the magnitude of
the applied field. The magnetic material is normally in the form of tape, with the tape in its blank
form being initially demagnetized. When recording, the tape runs at a constant speed, the writing
head magnetizes the tape with current proportional to the signal. A magnetization distribution is
achieved along the magnetic tape. Finally, the distribution of the magnetization can be read out,
reproducing the original signal. The magnetic tape is typically made by embedding magnetic
particles in a plastic binder on polyester film tape. The commonly used magnetic particles are
Iron oxide particles or Chromium oxide and metal particles with size of 0.5 micrometers. Analog
recording was very popular in audio and video recording. In the past 20 years, however, tape
recording has been gradually replaced by digital recording.
Digital recording
Instead of creating a magnetization distribution in analog recording, digital recording only needs
two stable magnetic states, which are the +Ms and -Ms on the hysteresis loop. Examples of
digital recording are floppy disks and hard disk drives (HDDs). Digital recording has also been
carried out on tapes. However, HDDs offer superior capacities at reasonable prices; at the time of
writing (2014), consumer-grade HDDs offer data storage at about 3 GB/$. Recording media on
HDDs use a stack of thin films to store information and a read/write head to read and write
information to and from the media; various developments have been carried out in the area of
used materials.
Magneto-optical recording
Magneto-optical recording writes/reads optically. When writing, the magnetic medium is heated
locally by a laser, which induces a rapid decrease of coercive field, then, a small magnetic field
10
can be used to switch the magnetization. The reading process is based on magneto-optical Kerr
effect. The magnetic medium is typically amorphous R-FeCo thin film (R being a rare earth
element). Magneto-optical recording is not very popular. One famous example is Minidisc
developed by Sony
Figure 1.6: Sony min disc
Domain propagation memory
Domain propagation memory is also called bubble memory. The basic idea is to control domain
wall motion in a magnetic medium that is free of microstructure. Bubble refers to a stable
cylindrical domain. Data is then recorded by the presence/absence of a bubble domain. Domain
propagation memory has high insensitivity to shock and vibration, so its application is usually in
space and aeronautics.
Access method
Magnetic storage media can be classified as either sequential access memory or random access
memory although in some cases the distinction is not perfectly clear
11
Current and Future use
Current
As of 2011, common uses of magnetic storage media are for computer data mass storage on hard
disks and the recording of analog audio and video works on analog tape. Since much of audio
and video production is moving to digital systems, the usage of hard disks is expected to increase
at the expense of analog tape. Digital tape and tape libraries are popular for the high capacity
data storage of archives and backups. Floppy disks see some marginal usage, particularly in
dealing with older computer systems and software. Magnetic storage is also widely used in some
specific applications, such as bank cheques (MICR) and credit/debit cards (mag stripes).
Future
A new type of magnetic storage, called Magneto Resistive Random Access Memory or
MRAM, is being produced that stores data in magnetic bits based on the tunnel magneto
resistance (TMR) effect. Its advantage is non-volatility, low power usage, and good shock
robustness. The 1st generation that was developed was produced by Everspin Technologies, and
utilized field induced writing; the 2nd generation is being developed through two approaches:
Thermal Assisted Switching (TAS) which is currently being developed by Crocus Technology,
and Spin Torque Transfer (STT) on which Crocus, Hynix, IBM, and several other companies are
working. However, with storage density and capacity orders of magnitude smaller than an HDD,
MRAM is useful in applications where moderate amounts of storage with a need for very
frequent updates are required, which flash memory cannot support due to its limited write
endurance.
1.6 MAGNETIC SHIELDING
There is no known INSULATOR for magnetic flux. If a nonmagnetic material is placed in a
magnetic field, there is no appreciable change in flux - that is, the flux penetrates the
nonmagnetic material. For example, a glass plate placed between the poles of a horseshoe
magnet will have no appreciable effect on the field although glass itself is a good insulator in an
12
electric circuit. If a magnetic material (for example, soft iron) is placed in a magnetic field, the
flux may be redirected to take advantage of the greater permeability of the magnetic material, as
shown in figure below. Permeability, as discussed earlier, is the quality of a substance which
determines the ease with which it can be magnetized
The sensitive mechanisms of electric instruments and meters can be influenced by stray magnetic
fields which will cause errors in their readings. Because instrument mechanisms cannot be
insulated against magnetic flux, it is necessary to employ some means of directing the flux
around the instrument. This is accomplished by placing a soft-iron case, called a MAGNETIC
SCREEN or SHIELD, about the instrument. Because the flux is established more readily through
the iron (even though the path is longer) than through the air inside the case, the instrument is
effectively shielded, as shown by the watch and soft-iron shield in figure below
Figure 1.7: Magnetic shielded watch
13
CHAPTER TWO
MAGNETIC FIELDS
Chapter outlines
2.1 Introduction
2.2 Some basic concepts and units
2.3 Magnetic circuit v Electric circuit
2.1 INTRODUCTION
The space surrounding a magnet where magnetic forces act is known as the magnetic field. A
pattern of this directional force can be obtained by performing an experiment with iron filings. A
piece of glass is placed over a bar magnet in figure 2.1 below, and the iron filings are then
sprinkled on the surface of the glass. The magnetizing force of the magnet will be felt through
the glass and each iron filing becomes a temporary magnet. If the glass is now tapped gently, the
iron particles will align themselves with the magnetic field surrounding the magnet just as the
compass needle did previously. The filings form a definite pattern, which is a visible
representation of the forces comprising the magnetic field. Examination of the arrangements of
iron filings in figure above will indicate that the magnetic field is very strong at the poles and
weakens as the distance from the poles increases. It is also apparent that the magnetic field
extends from one pole to the other, constituting a loop about the magnet
The glass does not prevent the penetration of magnetic flux, neither of huge thickness
Magnetic flux is only prevented by a process called MAGNETIC SHIELDING explained in
chapter one in this hand book
14
Figure 2.1: magnet drawing field lines with iron fillings
Magnetic lines of force
To further describe and work with magnet phenomena, lines are used to represent the force
existing in the area surrounding a magnet (refer figure 2.2(a)). These lines, called MAGNETIC
LINES OF FORCE, do not actually exist but are imaginary lines used to illustrate and describe
the pattern of the magnetic field. The magnetic lines of force are assumed to emanate from the
north pole of a magnet, pass through surrounding space, and enter the South Pole. The lines of
force then travel inside the magnet from the South Pole to the
North Pole, thus completing a
closed loop.
Figure 2.2 (a) magnetic lines of force
2.2(b) magnetic laws illustration
15
When two magnetic poles are brought close together, the mutual attraction or repulsion of the
poles produces a more complicated pattern than that of a single magnet. These magnetic lines of
force can be plotted by placing a compass at various points throughout the magnetic field, or
they can be roughly illustrated by the use of iron filings as before. A diagram of magnetic poles
placed close together is shown in figure below. Although magnetic lines of force are imaginary,
a simplified version of many magnetic phenomena can be explained by assuming the magnetic
lines to have certain real properties. The lines of force can be compared to rubber bands which
stretch outward when a force is exerted upon them and contract when the force is removed
The characteristics of magnetic lines of force can be described as follows
1. Magnetic lines of force are continuous and will always form closed loops.
2. Magnetic lines of force will never cross one another.
3. Parallel magnetic lines of force traveling in the same direction repel one another. Parallel
magnetic lines of force traveling in opposite directions tend to unite with each other and form
into single lines traveling in a direction determined by the magnetic poles creating the lines of
force.
4. Magnetic lines of force tend to shorten themselves. Therefore, the magnetic lines of force
existing between two unlike poles cause the poles to be pulled together.
5. Magnetic lines of force pass through all materials, both magnetic and nonmagnetic.
6. Magnetic lines of force always enter or leave a magnetic material at right angles to the surface.
Magnetic effects of magnetic lines of force
MAGNETIC FLUX (ɸ)
The total number of magnetic lines of force leaving or entering the pole of a magnet is called
MAGNETIC FLUX.
MAGNETIC FLUX DENSITY (B)
16
The number of flux lines per unit area is known as FLUX DENSITY.
MAGNETIC FIELD INTENSITY (H)
The intensity of a magnetic field is directly related to the magnetic force exerted by the field.
ATTRACTION/REPULSION, The intensity of attraction or repulsion between magnetic poles
may be described by a law almost identical to Coulomb's Law of Charged Bodies. The force
between two poles is directly proportional to the product of the pole strengths and inversely
proportional to the square of the distance between the poles.
This force is often called magnetizing force of a magnet.
Magnetic Induction
It has been previously stated that all substances that are attracted by a magnet are capable of
becoming magnetized. The fact that a material is attracted by a magnet indicates the material
must itself be a magnet at the time of attraction. With the knowledge of magnetic fields and
magnetic lines of force developed up to this point, it is simple to understand the manner in which
a material becomes magnetized when brought near a magnet. As an iron nail is brought close to a
bar magnet (fig. below, some flux lines emanating from the north pole of the magnet pass
through the iron nail in completing their magnetic path. Since magnetic lines of force travel
inside a magnet from the south pole to the north pole, the nail will be magnetized in such a
polarity that its south pole will be adjacent to the north pole of the bar magnet. There is now an
attraction between the two magnets.
Figure 2.3: Magnetising an iron nail by induction method
17
If another nail is brought in contact with the end of the first nail, it would be magnetized by
induction. This process could be repeated until the strength of the magnetic flux weakens as
distance from the bar magnet increases. However, as soon as the first iron nail is pulled away
from the bar magnet, all the nails will fall. The reason being that each nail becomes a temporary
magnet, and as soon as the magnetizing force is removed, their domains once again assume a
random distribution. Magnetic induction will always produce a pole polarity on the material
being magnetized opposite that of the adjacent pole of the magnetizing force. It is sometimes
possible to bring a weak north pole of a magnet near a strong magnet North Pole and note
attraction between the poles. The weak magnet, when placed within the magnetic field of the
strong magnet, has its magnetic polarity reversed by the field of the stronger magnet. Therefore,
it is attracted to the opposite pole. For this reason, you must keep a very weak magnet, such as a
compass needle, away from a strong magnet. Magnetism can be induced in a magnetic material
by several means. The magnetic material may be placed in the magnetic field, brought into
contact with a magnet, or stroked by a magnet. Stroking and contact both indicate actual contact
with the material but are considered in magnetic studies as magnetizing by INDUCTION
2.2 SOME BASIC CONCEPTS AND UNITS
The following table shows magnetic quantities there symbols and units (make sure you catch all
the staffs)
18
Table 2.1: Magnetic quantities with their symbols and units
19
CALCULATION OF MAGNETIC FIELD
Magnetic field can be calculated by the use of Maxwell‟s equation or ampere‟s law, the details of
which shall not be covered in this class
Consider three structures below
20
2.3 MAGNETIC CIRCUIT V ELECTRIC CIRCUIT
Let look on Ampere’s law first
Ampere’s law states that for any closed loop path, the sum of the length elements times the
magnetic field in the direction of the length element is equal to the permeability times the electric
current enclosed in the loop (illustration „ll be given in class)
In the electric case, the relation of field to source is quantified in Gauss‟s law which is a very
powerful tool to calculating electric field
Comparison between electrical circuit and magnetic circuit
From Amperes law we know that m.m.f (magneto motive force) around a closed path is equal to
the current enclosed by the closed path.
That is integration of H.dl = Ni
……… (1)
Where H = magnetic field intensity,
dl = small incremental length,
N = number of conductors and
i = current flowing in each conductors.
Now if H is constant then from equation (1)
Integration of H.dl = Ni
→
(H) integration of dl = Ni
→
Therefore
H. l = Ni
H = Ni/l
……… (2)
Now because flux ф = B.A = μH.A
(because B = μH)
………… (3)
Where B = magnetic flux density, H = magnetic field intensity, μ = permeability of medium and
A = area.
Now from equation (2) and equation (3)
21
Ф = μNil /A
→ Ф = Ni / (l / μA) ……………… (4)
Similarities between magnetic circuit and electric circuit

Magnetic circuit follows equation (4) that is Ni = (Ф) (l / μA) or m.m.f (magneto motive
force) = (Flux) (reluctance).
Electric circuit follows ohm‟s law that is E = I.R or e.m.f (electro motive force) =
(current) (Resistance)

From above point m.m.f in magnetic circuit is like as e.m.f in electrical circuit.

Flux in magnetic circuit is similar as current in electrical circuit.

Reluctance in magnetic circuit, S = (l / μA) is similar to resistance R = (ρl/A) in electric
circuit.

Permeance (= 1/reluctance) in magnetic circuit is equivalent to conductance (=1/resistance) in
electric circuit.
Differences between magnetic circuit and electric circuit

In magnetic circuit flux establishes but not flow like as current in magnetic circuit.

In magnetic circuit energy needed only to establish the flux but no consistent energy need to
maintain it whereas in electric circuit continuous energy needed to flow of current.

Resistance of an electric circuit is constant (for same temperature) and is independent of
current but reluctance of magnetic circuit is not constant because it depends on μ (=B/H)
which is not constant and depends on B/H.
22
CHAPTER THREE
INTRODUCTION TO ELECTROMAGNETISM
Chapter outlines
3.1 Introduction
3.2 Quantities and units
3.3 Magnetic field patterns around conductors
3.4 Electromagnetic induction and faraday‟s laws
3.1 INTROODUCTION
Electromagnetism or Electromagnetic force is the one of the four fundamental interactions in
nature:1. Strong interaction
Is the mechanism responsible for strong nuclear force
2. Weak interaction
Is the mechanism responsible for weak nuclear force
3. Gravitation
4. Electromagnetism
This force is described by electromagnetic fields and has innumerable physical instances
including the interaction of electrically charged particles and the interaction of unchanged
magnetic force fields with electrical conductors.
23
Originally electricity and magnetism were thought of as two separate forces, this view changed
however with the publication of James clerk Maxwell‟s 1873treatise on electricity and
magnetism in which the interactions of positive and negative changes were shown to be
regulated by one force. There are four main effects resulting from these interactions, all of which
have been clearly demonstrated by experiments.
(a) Electric charges attracts or repel one another with a force inversely proportion to the
square of the distance of separation (Fe∞1/d2)
Law of electrostatic: states that unlike charges attract, like charges repel one another.
Question: why a charged ebonite rod is able to attract piece of papers (uncharged)
(b) Magnetic poles attract or repel one another (Fm∞1/d2)
Law of magnetism: states that unlike poles attract, like poles repel
(c) An electric current in a wire creates a circular magnetic field around the wire, its
direction depends on that of current [Faraday laws to be covered later]
(d) A current is induced in a loop of wire when it is moved towards or Away from magnetic
fields, or a magnetic is moved towards or away from it, the direction of current depends
on that of movement.
N: B all these four facts have been well proved experimentally
3.2 QUANTITIES AND UNITS
Electromagnetic units are part of a system of electrical units based primarily upon the magnetic
properties of electric currents, the fundamental SI units being AMPERE
In the electromagnetic cgs system, electric current is a fundamental quantity defined via
Ampere’s law
24
Table 3.1: magnetic quantities and their units
Note: In electricity, a capacitor do store charges and its capacity of storing charges is measured
by its capacitance
Capacitor (-||-): Is a passive device in D.C circuit that stores electrical energy in the form of
charges.
In magnetism we have an inductor that stores magnetic energy in the form magnetic field and
returns it to the circuit whenever required
Inductor: Is a passive device that stores energy in its magnetic field, its capacity is measured by
its inductance.
MAGNETIC FIELD FROM ELECTRICITY
A very important point to note is that, both magnetic field vector and electric field vector
produce each other and are always moving perpendicular to each other in an electromagnetic
wave
25
Passing a steady current through a straight conductor does produce a magnetic field. The field
lines are circular, looping around the current according to the right hand rule. But when it is
passing through a helical coil such as the solenoid or even an inductor, a stead magnetic field is
produced along the axis of the coil hence both magnetic and electric fields are related to each
other. WHY and HOW this is possible?
Answer: This is an experimental fact and NO amount of philosophy can trump that
A static distribution of charges produces an electric field, charges in motion (an electric current)
produces a magnetic field.
Electric current is an example of charges in motion
Right hand grip rule
Grip the wire with the RIGHT hand. If the thumb is placed in the direction of the electric current,
then the gripping fingers show the direction of the circular magnetic field.
(a)
26
(b)
Figure 3.1: Demonstration of right hand rule (a) current carrying conductor
(b) Solenoid conductor
Arranging wire in a coil and running a current through , produces a magnetic field that looks a
lot like a bar magnet.
-called an electromagnet
-Putting a real magnet inside, can shove (push) the magnet back and forth depending on
current direction called a solenoid
This effect helps us to make a very good helper of our today security systems called
ELECTROMAGNET
Figure 3.2: Solenoid
27
Defn
Electromagnet is a soft metal core (or iron core) made into magnet by the passage of electric
current through a coil surrounding it.
Practical: LAB1
Aim: To make an electromagnet
Materials and Procedures
It is fairly easy to build an electromagnet. All you need to do is wrap some insulated copper wire
around an iron core. If you attach a battery to the wire, an electric current will begin to flow and
the iron core will become magnetized. When the battery is disconnected, the iron core will lose
its magnetism. Follow these steps if you would like to build the electromagnet described in our
Magnets and Electromagnets experiment:
Step 1 - Gather the Materials
To build the electromagnet described in our Magnets and Electromagnets experiment, you will
need:
One iron nail fifteen centimeters (6 in) long
Three meters (10 ft) of 22 gauge insulated, stranded copper wire
One or more D-cell batteries
A pair of wire strippers
Step 2 - Remove some Insulation
Some of the copper wire needs to be exposed so that the battery can make a good electrical
connection. Use a pair of wire strippers to remove a few centimeters of insulation from each end
of the wire.
28
Step 3 - Wrap the Wire around the Nail
Neatly wrap the wire around the nail. The more wire you wrap around the nail, the stronger your
electromagnet will be. Make certain that you leave enough of the wire unwound so that you can
attach the battery.
When you wrap the wire around the nail, make certain that you wrap the wire all in one
direction. You need to do this because the direction of a magnet field depends on the direction of
the electric current creating it. The movement of electric charges creates a magnetic field. If you
could see the magnetic field around a wire that has electricity flowing through it, it would look
like a series of circles around the wire. If an electric current is flowing directly towards you, the
magnetic field created by it circles around the wire in a counter-clockwise direction. If the
direction of the electric current is reversed, the magnetic field reverses also and circles the wire
in a clockwise direction. If you wrap some of the wire around the nail in one direction and some
of the wire in the other direction, the magnetic fields from the different sections fight each other
and cancel out, reducing the strength of your magnet.
29
Step 4 - Connect the Battery
Attach one end of the wire to the positive terminal of the battery and the other end of the wire to
the negative terminal of the battery. If all has gone well, your electromagnet is now working!
Don't worry about which end of the wire you attach to the positive terminal of the battery and
which one you attach to the negative terminal. Your magnet will work just as well either way.
What will change is your magnet's polarity. One end of your magnet will be its north pole and
the other end will be its south pole. Reversing the way the battery is connected will reverse the
poles of your electromagnet.
Hints to Make Your Electromagnet Stronger
The more turns of wire your magnet has, the better. Keep in mind that the further the wire is
from the core, the less effective it will be.
The more current that passes through the wire, the better, Caution! Too much current can be
dangerous! As electricity passes through a wire, some energy is lost as heat. The more current
that flows through a wire, the more heat is generated. If you double the current passing through a
wire, the heat generated will increase 4 times! If you triple the current passing through a wire,
the heat generated will increase 9 times! Things can quickly become too hot to handle.
Try experimenting with different cores. A thicker core might make a more powerful magnet. Just
make certain that the material you choose can be magnetized. You can test your core with a
permanent magnet. If a permanent magnet is not attracted to your core, it will not make a good
electromagnet. An aluminum bar, for example, is not a good choice for your magnet's core.
30
3.3 MAGNETIC FIELD PATTERNS AROUND CONDUCTORS
1. Straight wire
The magnetic field consists of concentric circles centered on the wire, the magnetic field
is strongest near the wire and this is shown by the field lines being closet together near to
the wire and the strength of the field increases if the electric current is increased
Figure 3.3: Magnetic field around a conductor when you look at the conductor from one
end. (a) Current flows out of the page and the magnetic field is counter-clockwise.
(b) Current flows into the page and the magnetic field is clockwise.
Direction of magnetic field is given by right hand grip rule which states that for straight
conductor carrying current, if you grip the wire with RIGHT hand, the thumb pointing in
the direction of current, then the griping fingers points the direction of fields.
2. Flat circular coil
If you make a loop of current carrying conductor, then the direction of the magnetic field
is obtained by applying the Right Hand Rule to different points in the loop. Notice that
there is a variation on the Right Hand Rule. If you make the fingers of your right hand
follow the direction of the current in the loop, your thumb will point in the direction
where the field lines emerge. This is similar to the North Pole (where the field lines
emerge from a bar magnet) and shows you which side of the loop would attract a bar
magnet's North Pole.
31
Figure 3.4: Magnetic field patterns around a circular coil
3. Solenoid
A solenoid is a coil of wire carrying an electric current, The magnetic field is similar to
that around a bar magnet
The strength of the field increases with
 The electric current
 The number of turns in the coil
Figure 3.4: Magnetic field around a solenoid
32
Right hand grip rule (for poles)
Grip the coil with the right hand. The griping fingers points the direction of current and the
thumb points towards NORTH POLE
Interesting Fact:
When lightning strikes a ship or an aeroplane, it can damage or otherwise change its magnetic
compass. There have been recorded instances of a lightning strike changing the polarity of the
compass so the needle points south instead of north.
Tutorial
Problem 1:
Give evidence for the existence of a magnetic field near a current carrying wire.
Answer 1:
If you hold a compass near a wire through which current is flowing, the needle on the compass
will be deflected. Since compasses work by pointing along magnetic field lines, this means that
there must be a magnetic field near the wire through which the current is flowing. If the current
stops flowing the compass returns to its original direction, If the current starts to flow again then
the deflection happens again.
Problem 2:
Describe how you would use your right hand to determine the direction of a magnetic field
around a current carrying conductor.
Answer 2:
33
We use the right hand rule which says that the magnetic field lines produced by a currentcarrying wire will be oriented in the same direction as the curled fingers of a person's right hand
(in the “hitchhiking” position), with the thumb pointing in the direction of the current flow:
Problem 3
Use the Right Hand Rule to find the direction of the magnetic fields at each of the points labeled
A - H in the following diagrams.
Problem 4
Locate north and south poles
Uses of electromagnets
1. Scrap yard crane
The iron core of the electromagnet is a SOFT magnetic material, when current flows the
iron becomes strongly magnetised and so picks up the scrap iron and steel. When current
is turned off, the iron loses its magnetisation and so releases the scrap
2. The electric bell
Draw and explain the operation of an electric bell
34
3. The relay switch (will be explained in class)
4. Circuit breaker (will be explained in class)
3.4 ELECTROMAGNETIC INDUCTION AND FARADAY’S LAWS
Consider the magnetic field below
The below equations can be derived from the surface above (derivation is from Maxwell‟s
equation and the study of those equations is beyond this level of study)
Signifies that the quantities are vectors
This is called faraday‟s law of electromagnetic induction
Faraday's law states that the EMF is also given by the rate of change of the magnetic flux:
35
,
the electromotive force (EMF) and ΦB is is the magnetic flux. The direction of the
Where
electromotive force is given by Lenz's law.
For a tightly wound coil of wire, composed of N identical turns, each with the same ΦB,
Faraday's law of induction states that
Where N is the number of turns of wire and ΦB is the magnetic flux through
a single loop.
Faraday’s law of electromagnetic induction
States that: Whenever magnetic flux linkage change, an e.m.f is induced in the circuit whose
direction is always opposite to the change producing it
OR
Changing magnetic flux through a loop induces a current in that loop.
Electric field can be created from charges and constant magnetic fields created by moving
charges.
Now we investigate the effects of time varying magnetic fields on loops and we will find electric
fields are induced in loops, which create EMFs and cause current to flow
Electric generators are based on the physics of electromagnetic induction and faraday‟s law
Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field
will interact with an electric circuit to produce an electromotive force (EMF) a phenomenon
called electromagnetic
induction.
It
is
the
fundamental
operating
principle
of transformers, inductors, and many types of electrical motors, generators and solenoids.
36
The Maxwell–Faraday equation is a generalization of Faraday's law, and forms one of
Maxwell's equations
Demonstration
Figure 3.5: Demonstration of faraday‟s law of electromagnetic induction
Practical: LAB2
AIM: Investigating the direction of induced current in loop while changing magnetic field
Materials and procedures
Bar magnet, Ammeter, a piece of copper wire
Arrange the apparatus as per figure 3.5 above and observe the direction of current generated
37
LENZ’S LAW
Lenz's Law: the magnetic field of any induced current opposes the change that induces it.
 Convenient method to determine I direction
Example if an external magnetic field on a loop is increasing; the induced current creates a field
opposite that reduces the net field
Example if an external magnetic field on a loop is decreasing; the induced current creates a field
parallel to the change that tends to increase the net field.
Lenz‟s Law is a consequence of conservation of energy
38
Tutorial
4. In a physics laboratory experiment, a coil with 200 turns enclosing an
area of 122cm^2 is rotated in a time interval of 0.04s from a position where
its plane is perpendicular to the earth's magnetic field to one where its
plane is parallel to the field. The earth's magnetic field at the lab location is
6x10-5T.
a. What is the total magnetic flux through the coil before it is rotated?
b. What is the total magnetic flux through the coil after it is rotated?
c. What is the average e.m.f induced in the coil?
39
CHAPTER FOUR
MAGNETIC CIRCUIT
Chapter Outline
4.1 Magnetic Materials
4.1.1 Diamagnetism
4.1.2 Paramagnetism
4.1.3 Ferromagnetism
4.2 Hysteresis
4.3 Eddy Current Lost
4.4 The Concept of Magnetic Circuit
4.4.1 Magnetic Circuit with and Air Gap
4.1 MAGNETIC MATERIALS
In previous chapter, we knew that magnetic field can be induced by the free charges that flow
in a current-carrying wire loop and the direction of the induced magnetic field is described by
the right-hand rule. On the atomic scale, all materials contain spinning electrons that circulate
in orbits, and these electrons can also produce magnetic fields if each of theirs magnetic
moments is properly oriented. Thus, a resultant magnetic moment in a macroscopic substance
can be observed and such a substance is then said to be magnetized and this type of substance
is called magnetic material.
Defn: Magnetic material is a material in which the magnetic field moments of the spinning
electrons in it are properly oriented.
A magnetic material is said to be linear, isotropic, or homogenous if it magnetic properties (i.e.
r and m) is linear over a specified range of field, independent of the direction of field, or does
not vary throughout the whole medium of the material, respectively. Magnetic materials also
classified as soft and hard materials. Soft materials are normally used as the magnetic core
materials for inductors, transformers, and actuators in which the magnetic fields vary frequently.
40
Hard materials or sometime called as permanent magnets are used to generate static magnetic
fields in electric motors
The magnetization in a material substance is associated with atomic current loops generated by
two principal mechanisms: (1) orbital motions of the electrons around the nucleus and similar
motions of the protons around each other in the nucleus and (2) spinning motions of the electrons
around its own axis. The magnetic moment of an electron is due to the combination of its orbital
motion around nucleus and spinning motions around its own axis. Similarly, the magnetic
moment of the nucleus also consist of the orbital and spin magnetic moments, which are much
smaller than that of the electron. This is because the mass of the nucleus is larger than the mass
of electron. Thus, the total magnetic moment of an atom is usually assumed to be calculated by
the vector sum of the magnetic dipole moments of its electrons.
If m is the average magnetic dipole moment per atom, and if N is the number of atoms per unit
volume, the magnetization per unit volume, M is defined as M Nm
The unit of M is given as amperes/meter.
The magnitude of the individual magnetic moment m of a loop area A is calculated as
m = current I loop area A. (4.2)
The direction of m is normal to the plane of the loop in accordance with the right-hand rule
41
The orbital magnetic moment m0 of an electron can be calculated using the classical model of
atom. An electron with charge of –e moving with a constant velocity u in a circular orbit of
radius r [figure 4.1(a)] completes one revolution in time T = 2r/u. This circular motion of the
electron constitutes a tiny current loop with current I given by
Where L m ur e e is the angular momentum of the electron and me is its mass. The value of Le
is quantised and is some integer multiple of h h / 2( 0,h,2h,.... e L ), where h is Plank‟s
constant
Relationship of magnetisation vector M, magnetic flux density B, and permeability The
magnetic flux density corresponding to M is Bm = 0M. In the presence of an applied magnetic
field H, the total magnetic flux density in the magnetic material is given as

Where the first term represents the contribution of the external field and the second term
represents the contribution of the magnetisation of the material. For diamagnetic and
paramagnetic materials at a given temperature, the magnetisation can be written as
42
Where m is a dimensionless quantity called the magnetic susceptibility of the material, however,
for ferromagnetic material, Eq. (4.8) is nonlinear and also depends on the “history” of the
material. Substitute Eq. (4.8) into Eq. (4.7), one gets
The relative permeability and the magnetic behavior of a material can be used as the basis
guideline for classifying materials as diamagnetic, paramagnetic, or ferromagnetic
When a diamagnetic material is exposed to a strong magnetic field, electrons in this material
rearrange their orbits (orbital motion of electrons) and creating small persistent currents which
oppose the external magnetic field. Hence, diamagnetic materials have a very weak and negative
susceptibility to external magnetic fields, and also slightly repelled by the magnetic field. In
addition, diamagnetic material does not retain the magnetic properties when the external field is
removed and does not have permanent net magnetic moment per atom since its orbital are fullyfilled. For example, Neon (1s22s22p6) has 10 electrons in an atom, thus all electrons are paired-up
and do not have permanent magnet as shown in the orbital diagram.
43
Most elements in the periodic table, including copper, silver, gold, mercury, bismuth, and carbon
graphite are diamagnetic
Some materials behave like superconductor at very low temperature and they are perfect
diamagnetic materials which have m = -1 or r = 0 and B = 0 (no magnetic field could be
established inside superconductor materials since they expel all the magnetic fields that applied
on it)
The most popular application of diamagnetic materials is magnetic levitation, where an
object can be made to float in the air above a strong magnet.
4.4.2 Paramagnetism
Paramagnetic materials are weakly attracted to magnets and have a small positive susceptibility
to magnetic fields. Paramagnetic properties are due to the presence of some unpaired electrons
that produce the net spin magnetic moments which tend to align themselves in the direction of
the external magnetic field. They do not retain the magnetic properties when the external field is
removed. For example, aluminium (1s22s22p63s23p1) has 13 electrons in an atom, thus the
44
unpaired electrons produce net spin magnetic moments and have weak permanent magnetic
moment as shown in the orbital diagram
Paramagnetic materials are sensitive to temperature and materials like aluminum, uranium and
platinum become more magnetic when theirs temperature reduce. Other paramagnetic materials
include magnesium, titanium, tungsten, molybdenum, and lithium. Paramagnetic materials are
typically considered nonmagnetic since they have very small positive susceptibility (of the order
10-5) as compared to ferromagnetic materials
4.4.3 Ferromagnetism
Ferromagnetic materials have a large and positive susceptibility to an external magnetic field.
They exhibit a strong attraction to magnetic fields and are able to retain their magnetic properties
after the external field has been removed. Iron, nickel, and cobalt are examples of ferromagnetic
materials and usually used to fabricate permanent magnets due to the ability to retain their
magnetism properties for long time. Ferromagnetic materials have some unpaired electrons so
their atoms have a net magnetic moment. They get their strong magnetic properties due to the
presence of magnetic domains within which the magnetic moments of all its atoms (1019) are
aligned parallel to each other.
45
When a ferromagnetic material is in the unmagnetised state, the domains are nearly randomly
organised as shown in figure 4.5(a) and the net magnetic field for the part as a whole is zero.
When a magnetising force is applied, the domains become aligned as shown in figure 4.5(b) to
produce a strong magnetic field within the part. The boundaries between the adjacent domains
consist of thin transition regions called as domain walls. In addition, strong magnetic
ferromagnetic materials like nickel or steel lose all their magnetic properties if they are heated to
a critical temperature. This is because the magnetized domains will organize themselves
randomly after theirs atoms are being heated. The temperature at which a ferromagnetic material
loses its magnetism is called the Curie temperature and it is different for every metal. For
example, the Curie temperature for nickel is about 350°C. Table 4.1 summarizes the
characteristics of magnetic materials
4.2 HYSTERESIS
The magnetisation behaviour of the ferromagnetic materials is described by the B-H
magnetisation curve (hysteresis loop) as shown in figure 4.6.
46
Figure 4.6: Hysteresis loop
The loop is generated by measuring the magnetic flux B of a ferromagnetic material
while the magnetising force H is changed. A ferromagnetic material that has never been
previously magnetised or has been thoroughly demagnetised will follow the dashed line as H is
increased. As the line demonstrates, the greater the amount of current applied (H+), the stronger
the magnetic field in the component (B+). At point “a” almost all of the magnetic domains are
aligned and an additional increase in the magnetising force will produce very little increase in
magnetic flux. The material has reached the point of magnetic saturation. When H is reduced
down to zero, the curve will move from point "a" to point “b”. At this point, it can be seen that
some magnetic flux remains in the material even though the magnetising force is zero. This is
referred to as the point of retentivity on the graph and indicates the remanence or level of
residual magnetism in the material. (Some of the magnetic domains remain aligned but some
have lost their alignment.) As the magnetising force is reversed, the curve moves to point “c”,
where the flux has been reduced to zero. This is called the point of coercivity on the curve. (The
reversed magnetising force has flipped enough of the domains so that the net flux within the
material is zero.) The force required to remove the residual magnetism from the material, is
called the coercive force or coercivity of the material.
47
As the magnetising force is increased in the negative direction, the material will again
become magnetically saturated but in the opposite direction (point “d”). Reducing H to zero
brings the curve to point “e”. It will have a level of residual magnetism equal to that achieved
in the other direction. Increasing H back in the positive direction will return B to zero. Notice
that the curve did not return to the origin of the graph because some force is required to remove
the residual magnetism. The curve will take a different path from point “f” back to the saturation
point where it completes s the loop. The complete close loop abcdefa is called as a hysteresis
loop. Hard magnetic materials have wider hysteresis loops as compared to that of soft magnetic
materials as shown in figure 4.7.
From the hysteresis loop, a number of primary magnetic properties of a material can be
determined.
Retentivity - it is a material's ability to retain a certain amount of residual magnetic field when
the magnetising force is removed after achieving saturation. (The value of B at point “b” on the
hysteresis curve)
Residual flux density - the magnetic flux density that remains in a material when the
magnetising force is zero, note that residual flux density and retentivity are the same when the
material has been magnetised to the saturation point. However, the level of residual flux density
may be lower than the retentivity value when the magnetising force did not reach the saturation
level.
48
Coercive force - The amount of reverse magnetic field which must be applied to a magnetic
material to make the magnetic flux return to zero (The value of H at point “c” on the hysteresis
curve)
Permeability,_- A property of a material that describes the ease with which a magnetic flux
is established in the component.
Reluctance - Is the opposition that a ferromagnetic material shows to the establishment of a
magnetic field. Reluctance is analogous to the resistance in an electrical circuit
Energy dissipated in a hysteresis loop
Consider the solenoid that shown in figure 4.8(a), when the current I is increasing, the
electromotive force (e.m.f) induced in the winding opposes the increase in current according to
Lenz‟s law, the extra power spent by the source is
49
Where A is the cross sectional area of the core, N is the number of turns, B is the magnetic flux
density induced in the core, and is the magnetic flux generated. From the relation of NI = HL,
Eq. (4.13) can be written as
Where L is the circumference of the core and V = AL is the volume. Thus
Steinmetz hysteresis law
The power of the hysteresis loss is empirically given by Steinmetz hysteresis law,
Where f is the frequency of excitation, kh is a constant determined by the nature of the magnetic
material, Bp is the peak value of the magnetic flux density, and x is the Steinmetz coefficient
ranging from 1.5 to 2.5.
4.3 EDDY CURRENT LOST
When a changing magnetic field cuts through a sample of metal or magnetic materials that is
not connected to a circuit, by Faraday‟s law, a circulating current is induced. This current is
known as eddy current, it is localised within that material and has a flow pattern as shown in
figure 4.9.
50
This circulating current creates a magnetic field that opposes the external magnetic
field. The direction of the eddy current is described by Lenz‟s law. The stronger of the external
magnetic field or the greater of the electrical conductivity of the material, the eddy current that is
developed will be stronger and also yields stronger opposing force.
Eddy current creates losses through Joule heating, and it reduces the efficiency of device
that operates under alternating magnetic field condition such as iron core of transformers and
alternating current motors. This power loss is known as eddy current loss due to the induced
eddy current in the metal or magnetic materials.
In order to reduce the eddy current loss, the resistivity of the material is increased by
adding silicon in the metal or ferromagnetic materials. Another effective way to achieve low
eddy current loss is by using lamination of electrical metal sheets. These metal sheets are coated
with insulator which breaks the eddy currents path as illustrated in the diagram below
Figure 4.10: Eddy currents in a laminated toroidal core
The power due to the eddy current loss is given as
Where f is the frequency of excitation, ke is a constant determined by the nature of the metal or
magnetic material, Bp is the peak value of the magnetic flux density, and d is the thickness of the
lamination. This formula is obtained under the assumption of global eddy current as shown in
figure 4.9. This is incorrect for ferromagnetic materials due to the magnetic domains. When the
excitation field varies, the domain walls move accordingly and local eddy currents are induced
51
by the fluctuating of the local flux density caused by the domain wall motion as shown in figure
4.11.
Figure 4.11: Eddy currents for domain wall model.
4.4 THE CONCEPT OF MAGNETIC CIRCUIT
Figure 4.12: Ferromagnetic toroid with concentrated windings. The lines of force
shown are in the plane of the toroid. They apply only when there is no iron present
Figure 4.12 shows a ferromagnetic core around which is wound a coil of N turns carrying a
current I. The magnetic flux through some cross section of the core is to be determined. In the
absence of ferromagnetic material, the lines of B are as shown in figure 4.12 and it is of the same
order of magnitude at all points within the ferromagnetic material. This is because the magnetic
induction due to the current I magnetises the core in the region near the coil, and the
magnetisation produces Amperian currents that both increase B and extend it along the core. This
52
further increases and extends the magnetisation and until the lines of B extend all around the
core. Some of the lines of B escape into the air and then return to the core to pass again through
the coil. This constitutes the leakage flux that may, or may not, be negligible. For example, if the
toroid is made up of a long thin iron wire, the flux at P is negligible compared to that near the
coil.
Suppose the cross section of the toroid is large enough to render the leakage flux. Then, applying
Ampère‟s circuital law to a circular path of radius r going all around inside the toroid,
Then, taking R1 to be the radius corresponding to the average value of B, and R2 to be the inner
radius of the toroid, the flux through the core is
This equation shows that the magnetic flux is given by the magnetomotance or another name as
magneticmotive force (m.m.f.), NI multiplied by the factor
This is called the permeance of the magnetic circuit. The inverse of the permeance is called as
the reluctance. Hence,
Magnetic flux = Permeance Magnetomotive force
= Magnetomotive force/Reluctance.
The above formula analogies with Ohm‟s law of electric circuit: if an electromotive force were
induced in the core, the current would be
53
Some corresponding quantities in electric and magnetic circuit are listed as below.
The differences between electric and magnetic circuits are as below:
The path of the magnetic flux flows is perpendicular to the current flows in the circuit. In other
words, the directions of B and J are perpendicular.
For a given temperature, electric resistance is constant and does not depend on current density.
However, the magnetic reluctance depends on magnetic field and flux intensity since the
permeability is not constant.
Current flowing in a electric circuit involves dissipation of energy, but for magnetic circuit,
energy is needed to generate magnetic flux.
54
4.4.1 Magnetic Circuit with and Air Gap
Figure 4.13: A simple magnetic circuit with an air gap.
Figure 4.13 shows a simple magnetic circuit with an air gap of length lg cut in the middle of a
leg. The winding provides NI ampere-turn. The spreading of the magnetic flux lines outside the
common area of the core for the air gap is known as fringing field [figure 4.14(a)]. For
simplicity, this effect is negligible and the flux distribution is assumed to be as in figure 4.14(b).
It can be shown that the magnetic flux generated in the air gap is equal to the Magnetomotive
force NI divided by the sum of the reluctances of the core and of the air gap.
(a)
(b)
Figure 4.14: Air gaps (a) with fringing and (b) ideal.
Suppose the leakage flux is negligible. Applying the Ampère‟s circuital law, one gets
55
where the subscript c refers to the core and g to the air gap. The path length Lc in the core can be
taken to be the length measured along the centre of the cross section of the core. Hc and Hg can be
written in terms of the magnetic flux as
According to Gauss‟s law of magnetism, the net outward flux of B through any closed
surface must be equal to zero. Hence, the flux of B must be the same over any cross section of
the magnetic circuit and
where Ac and Ag are the cross sections of the core and of the air gap, respectively. Combining Eq.
(4.28) and Eq. (4.29b) gives
The denominator of Eq. (4.31) gives the reluctances of the core and air gap in series. Since the
leakage flux is neglected, this equation gives the upper limit for .
56
Magnetic circuit equivalent and electric circuit analogy for the electromagnet circuit
can be rewritten as
The above magnetic circuit with an air gap can be represented in a magnetic circuit diagram as
shown in figure 4.15(a) and it is analogous to a series electric circuit in
figure 4.15(b). Further if HcLc and HgLg are regarded as the m.m.f. drops (analogy to voltage
drops in electric circuit) across the reluctance of the core and air gap respectively, Eq. (4.32)
derived from Ampère‟s circuital law can be interpreted as an analogue to the Kirchhoff’s voltage
law (KVL) in electric circuit theory
Eq. (4.33) states that the algebraic sum of the rises and drops of the Magnetomotive force around
a closed loop of a magnetic circuit is equal to zero. In other words, the sum of the
Magnetomotive force rises equals the sum of the Magnetomotive drops around a closed loop.
57
Figure 4.15: (a) Magnetic circuit equivalent and (b) electric circuit analogy for the electromagnet
If c and g are regarded as the “current entering/leaving” a junction in the magnetic circuit, Eq.
(4.29a) derived Gauss‟s law of magnetism can be interpreted as an analogue to the Kirchhoff’s
current law (KCL) in electric circuit theory
Eq. (4.34) states that the algebraic sum of the fluxes entering or leaving a junction of a magnetic
circuit is equal to zero. In other words, the sum of the fluxes entering a junction is equal to the
sum of the fluxes leaving a junction
ANALOGY BETWEEN ELECTRIC AND MAGNETIC CIRCUIT
As you might have guessed already, the relationship between field force, field flux, and
reluctance is much the same as that between the electrical quantities of electromotive force (E),
current (I), and resistance (R), This provides something akin to an Ohm's Law for magnetic
circuits
And, given that permeability is inversely analogous to specific resistance, the equation for
finding the reluctance of a magnetic material is very similar to that for finding the resistance of a
conductor
58
The major caveat here is that the reluctance of a material to magnetic flux actually changes with
the concentration of flux going through it. This makes the "Ohm's Law" for magnetic circuits
nonlinear and far more difficult to work with than the electrical version of Ohm's Law. It would
be analogous to having a resistor that changed resistance as the current through it varied (a
circuit composed of varistors instead of resistors)
In either case, a longer piece of material provides a greater opposition, all other factors being
equal. Also, a larger cross-sectional area makes for less opposition, all other factors being equal
59
Tutorial
 Some examples of magnetic circuit calculation
Example 4.1
Find the value of I required to establish a magnetic flux of = 0.75 10-4 Wb in the series
magnetic circuit as shown in figure 4.17. Calculate the force exerted on the armature (moving
part) when the flux is established. The relative permeability for the steel is r = 1424.
Figure 4.17: Magnetic circuit with an air gap.
Solution:
The above device can be analysed by its magnetic circuit equivalent and its electric circuit
analogy as shown in figure 4.18.
60
Figure 4.18: (a) Magnetic circuit equivalent and (b) electric circuit analogy for the
electromagnetic device in figure 4.17.
From the Gauss law (analogy to KCL in electric circuit), the flux density for each section is
61
Example 4.2
Determine the value of I required to establish a magnetic flux of = 1.54 10-4 Wb in the
section of the core indicated in figure 4.19. The relative permeability for the steel at region bcde,
be, and efab are 2 = 4972, 1 = 4821, and T = 2426, respectively
Figure 4.19: Electromagnetic device in a series-parallel configuration.
Solution:
The above device can be analysed by its magnetic circuit equivalent and its electric circuit
analogy as shown in figure 4.20
62
Figure 4.20: (a) Magnetic circuit equivalent and (b) electric circuit analogy for the
electromagnetic device in example 4.19.
63
64
Example 4.3
The core of figure 4.21 is made of cast steel. Calculate the current I that needed to establish a
flux of g = 6 10-3 Wb at the air gap if fringing field is neglected
Figure 4.21: Series-parallel magnetic circuit made of cast steel core.
Solution
Figure 4.22(a): Magnetic circuit equivalent.
65
Figure 4.22(b): Electric circuit analogy.
66
CHAPTER FIVE
ELECTROSTATIC IN DC CIRCUITS
ELECTROSTATICS: Study of Electricity in which electric charges are static i.e. not moving, is
called electrostatics
Figure 5.1: Conceptual image of a man holding a bottle with lightning captured in
ELECTRIC CHARGE
Electric charge is characteristic developed in particle of material due to which it exerts force on
other such particles. It automatically accompanies the particle wherever it goes.
• Charge cannot exist without material carrying it
• It is possible to develop the charge by rubbing two solids having friction.
• Giving a body charges is called electrification.
• Electrification due to friction is called frictional electricity.
Since these charges are not flowing it is also called static electricity.
There are two types of charges

+ve and –ve.

Similar charges repel each other,

Opposite charges attract each other.
67
 Benjamin Franklin made this nomenclature of
charges being +ve and –ve for mathematical calculations because adding them together
cancel each other.
 Any particle has vast amount of charges. The number of positive and negative charges
are equal, hence matter is basically neutral.
 Inequality of charges give the material a net charge which is equal to the difference of the
two type of charges
BASIC LAW OF ELECTRICAL CHARGES
States that: Opposite charges attract and like charges repel
Figure 5.2: Demonstration of fundamental electrostatic law
ELECTROSTATIC SERIES
The electrostatic series is a list of substances in order of increasing tendency to gain electrons.
When 2 substances from the lit are rubbed together, the top one on the list gets a (+) charge and
the bottom one gets a (-) charge.
 If two substances are rubbed together the former in series acquires the positive charge
and later, the –ve.
68
POSITIVE
Air
Acetate
Glass
Nylon
Wool
Fur/Hair
Ca, Mg, Pb
Silk
Al, Zn
Paper
Cotton
Steel
Wood
Paraffin wax
Ebonite
Plastic (polyethylene)
Cu, C, Ni
Brass, Ag
Rubber
Pt, Au
S
Acetate
Polyester
Styrene
Polyethylene
Vinyl
Silicon
Teflon
NEGATIVE
Table5.1: Electrostatic series for few materials
GROUNDING
A charged object comes into contact with a large reservoir of charge. The charged object
becomes neutral.
e.g.: Your hand is charged. You touch a door knob; you lose charge to the door.
69
e.g.: A balloon sticks to your hair. You touch the balloon to the floor. The balloon no longer
sticks.
Qn: A cat plays in your hair. Which one becomes positive?
A) Cat
B) Hair <-- positive
C) Neither
D) Don't know
How can I make two balloons attract each other?
- Rub one balloon on hair, balloon becomes negative.
- Rub the other balloon on polyester, it becomes positive.
- Since the charges are opposite, they balloons will attract each other.
ELECTRON THEORY OF ELECTRIFICATION

Nucleus of atom is positively charged.

The electron revolving around it is negatively charged, they are equal in numbers, and
hence atom is electrically neutral.

With friction there is transfer of electrons, hence net charge is developed in the particles.
It also explains that the charges are compulsorily developed in pairs equally. +ve in one
body and –ve in second
-It establishes conservation of charges in the universe.
-The loss of electrons develops +ve charge. While excess of electrons
develop –ve charge

A proton is 1837 times heavier than electron hence it cannot be transferred. Transferring
lighter electron is easier. Therefore for electrification of matter, only electrons are
active and responsible.
Charge and Mass relation
• Charge cannot exist without matter; one carrier of charge is electron which has mass as well.
70
• Hence if there is charge transfer, mass is also transferred.
• Logically, negatively charged body is heavier than positively charged body.
Conductors, Insulators and Semiconductors
• Conductors: Material in which electrons can move easily and freely.
Ex. Metals, Tap water, human body. Brass rod in our hand, if charged by rubbing the charge will
move easily to earth. Hence Brass is conductor. The flow of this excess charge is called
discharging
Insulator: Material in which charge cannot move freely. E.g. Glass, pure water, plastic etc
Electrons can be forced to move across an insulator by applying strong force (called electric
field.) Then this acts like a conductor.
Dielectric strength
The maximum electric field an insulator can withstand without becoming a conductor is called
its dielectric strength.
Semiconductor
Is a material which under little stimulation (heat or Elect. Field) converts from insulator to a
conductor. E.g. Silicon, germanium
Superconductor
Is that material which presents no resistance to the movement of the charge through it. The
resistance is precisely zero.
ELECTROSTATIC INDUCTION
• Phenomenon of polarization of charges in a body, when a charged body is present near it, is
called electrostatic induction. In this process bodies are charged without touching them.
71
Figure 5.3: Charging by induction
A charged object will induce a charge on a nearby conductor. In this example, a negatively
charged rod pushes some of the negatively charged electrons to the far side of a nearby copper
sphere because like charges repel each other. The positive charges that remain on the near side of
the sphere are attracted to the rod.
If the sphere is grounded so that the electrons can escape altogether, the charge on the sphere
will remain if the rod is removed.
NOTE
There are three way of electrifying a body, namely
One.
By friction (rubbing two bodies)
Two.
By conduction (will be explained on the blackboard and by experiment)
Three.
By induction (will be demonstrated in class by experiment)
Basic properties of Electric charge
• Additivity of Electric charges
• Quantization of Electric charge
• Conservation of Electric Charge
Additivity of Charges
Charges can be added by simple rules of algebra. Addition of positive and negative charge makes
Zero charge
Quantization of Electric charge
72
Principle: Electric charge is not a continuous quantity, but is an integral multiple of
minimum charge (e)
Reason of quantization:
• Minimum charge e exists on an electron.
• The material which is transferred during
electrification is an electron, in integral numbers.
• Hence charge transferred has to be integral multiple of e.
• Charge on an electron (-e) and charge on a
proton (+e) are equal and opposite, and are the minimum.
This minimum charge is 1.6 x 10-19 coulomb.
one electron has charge - 1.6 x 10-19 C
One proton has charge + 1.6 x 10-19 C
• Charge on a body Q is given by
Q = + ne
Where n is a whole number 1,2,3….. and e = 1.6 x 10-19
since e is smallest value of charge, it is called elementary Charge or Fundamental charge
QUARK
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to
form composite particles called hadrons, the most stable of which are protons and neutrons, the
components of atomic nuclei, Due to a phenomenon known as color confinement, quarks are
never directly observed or found in isolation; they can be found only within hadrons, such as
baryons (of which protons and neutrons are examples), and mesons, For this reason, much of
what is known about quarks has been drawn from observations of the hadrons themselves
But because free quarks do not exist and their sum is always an integral number, it does not
violet the quantization rules
e.g a proton may compose of 2 up quarks and 1 down quarks (1/3e for this case)
73
CONSERVATION OF CHARGES
Like conservation of energy, and Momentum, the electric charges also follow the rules of
conservation.
1. Isolated (Individual) Electric charge can neither be created nor destroyed, it can only be
transferred
2. Charges in pair can be created or destroyed.
Example for 1
At Nuclear level : Decay of U-238
238
234
U
Th + 4 He (Radio active decay)
Atomic number Z of radioactive material U-238 is 92.
Hence it has 92 protons hence charge is 92e. Thorium
has Z= 90, hence charge is 90e, alpha particles have
charge 2e. Therefore charges before decay are 92 and
after decay are 90+2=92
Example for 2
(a) Annihilation (destruction in pair)
In a nuclear process an electron -e and its antiparticle
positron +e undergo annihilation process in which they
transform into two gamma rays (high energy light)
e- + e
y+y
(b) Pair production
Is converse of annihilation, charge is also conserved when a gamma ray transforms into an
electron and a positron
y
e- + e+ (pair production)
74
ELECTRIC FORCE (COULOMBIC FORCES) - COULUMB’S LAW
Coulomb‟s states that force of Interaction between two stationery point charges directly
proportional to the product of the charges, inversely proportional to the square of the distance
between them and acts along the straight line joining the two charges.
Mathematically, if two charges q1 and q2 are placed at distance r then
C is called Coulomb's constant and its value is
But C depends on the medium in which the charges are placed
e0 is permittivity of free space or vacuum and its value is e0 = 8.85 x 10-12 coul2 / N x m2
If point charges are immersed in a dielectric medium then e0 is replaced by e a quantitycharacteristic of the matter involved In such case. For vacuum e = e0
Permittivity, Relative Permittivity and Dielectric Constant
Permittivity is a measure of the property of the medium surrounding electric charge which
determines the forces between the charges. Its value is known as Absolute permittivity of that
Medium the More is Permittivity of medium, less is coulombs Force. For water, permittivity is
80 times than that of vacuum; hence force between two charges in water will be 1/80 times force
in vacuum (or air.)
Relative Permittivity(er): It is a dimension-less characteristic constant, which express absolute
75
permittivity of a medium w.r.t. permittivity of vacuum or air. It is also called
Dielectric constant (K) K= er = e/e0
Unit of charge: - In S.I. System of units, the unit of charge is Coulomb.
One coulomb is defined as that charge, which, when placed at a distance of 1 m in air or vacuum
from an equal and similar charge, repel it with a force of 9 x 109 Newton
• Charge on one electron is 1.6019x10-19 coul. Hence one coulomb is equivalent to a charge of
6.243 x 10 18 electrons.
Is electric charge a fundamental quantity?
No, In S.I. System, the fundamental quantity is Electric current and its unit is Ampere.
Therefore coulomb is defined in its terms as under:
Coulomb is that quantity of charge which passes across any section of a conductor per second
when current of one ampere flows through it, i.e. 1 coulomb=1 Ampere x 1 sec
Comparison of Electrostatic and Gravitational Force
1. Identical Properties :

Both the forces are central forces, i.e., they act along the line joining the centers of two
charged bodies.

Both the forces obey inverse square law, F

Both are conservative forces, i.e. the work done by them is independent of the path
followed.

Both the forces are effective even in free space.
2. Non identical properties:

Gravitational forces are always attractive in nature while electrostatic forces may be
attractive or repulsive.

Gravitational constant of proportionality does not depend upon medium; the electrical
constant of proportionality depends upon medium.
76

Electrostatic forces are extremely large as compared to gravitational forces
Qn. Compare electrostatic and gravitational force between one electron and one proton system
Tutorial questions
Qn ONE
A charge q is placed at the center of the line joining two equal charges Q. Show that the system
of three charges will be in equilibrium if q = Q/4.
Qn TWO
Two particles having charges 8q and –2q are fixed at a distance L. where, in the line joining the
two charges, a
Proton is placed so that it is in equilibrium (the net force is zero). Is that equilibrium stable or
unstable?
Qn THREE
Above two charged particles are free to move. At one point, however a third charged particle can
be placed such that all three particles are in equilibrium.
(a) Is that point to the left of the first two particles, to the right, or between them?
(b) Should the third particle be positive or negative charged?
(c) Is the equilibrium stable or unstable?
77
Qn FOUR
Two similar helium-filled spherical balloons tied to a 5 g weight with strings and each carrying
a charge q float in equilibrium as shown. Find
(a) The magnitude of q, assuming that the charge on each balloon is at its centre and
(b) The volume of each balloon.
Assume that the density of air =1.29 kg m-3, and the density of helium in the balloon is= 0.2 kg
m-3. Neglect the weight of the unfilled balloons.
Ans: q = 5.5 x 10-7 V = 2.3 x 10-3
ELECTRIC FIELD
ELECTRIC FIELD-is the environment created by an electric charge (source charge) in the space
around it,
such that if any other electric charges(test charges)is present in this space, it will come to know
of its
presence and exert a force on it.
INTENSITY (OR STRENGTH) OF ELECTRIC FIELD AT A LOCATION
Is the force exerted on a unit charge placed at that location denoted by E
78
:If intensity of electric field at a location is E and a charge „q‟ is placed ,then force experienced
by this charges F is
ELECTRIC LINE OF FORCE
The idea of Lines of Force was given by Michel Faraday; these are imaginary lines which give
visual idea of electric field, its magnitude, and direction. A line of force is continuous curve the
tangent to which at a point gives the direction of Electric field, and its concentration gives the
strength of Field
Properties of Electric Lines of Force
1.start from positive charge and end at negative.
2. Electric Lines of forces are imaginary but Electric field they represent is real.
3. The tangent drawn at any point on the line of force gives the direction of force acting on a
positive charge at that point.
4. In SI system, the number of electric lines originating or terminating on charge q is q/ε .
That means lines associated with unit charge are 1/ ε
5. Two lines of force never cross each other, because if they do so then at the point of
intersection, intensity will have two directions which is absurd.
6. Electric Lines of force can never be a closed loop since they do not start and end at the same
point. The lines are discontinuous, start from + and terminate at –
7. The electric line of force do not pass through a conductor as electric field inside a conductor is
zero
8. Lines of force have tendency to contract longitudinally like a stretched string, producing
attraction between opposite charges and edge effect.
9. Electric Lines of force start and end Normal to the surface of conductor.
79
10. Crowded lines represent strong field while distant lines represent weak field. Equidistant
parallel lines represent uniform field. Non-straight or non- parallel represent non-uniform field.
In the diagram a is uniform while b, c, and d are non-uniform fields.
Field Lines due to some charge configurations.
1. Single positive or negative charge
2. Two equal and opposite charges
80
3. Lines of force due to Two positive charges
DISTRIBUTION OF CHARGE
Electric charge on a body may be concentrated at a point, and then it is called a „point charge‟. If
it is distributed all over, then it is called distribution of charge. Depending on shape of it is given
different
Names
We have generally three distributions
1. Length distribution (charge density is lambda)
2. Area distribution
3. Volume distribution
DIPOLE
Dipole is a system of two equal and opposite charges at finite & fixed distance. Example:
molecule of electrolytic compounds. Example HCl, H2O.
81
CAPACITOR
It is a device to store charge and in turn store the electrical energy.
A capacitor (originally known as a condenser) is a passive two-terminal electrical component
used to store energy electrostatically in an electric field. The forms of practical capacitors vary
widely, but all contain at least two electrical conductors (plates) separated by a dielectric (i.e.
insulator). The conductors can be thin films of metal, aluminum foil or disks, etc. The
"nonconducting" dielectric acts to increase the capacitor's charge capacity. A dielectric can be
glass, ceramic, plastic film, air, paper, mica, etc. Capacitors are widely used as parts of electrical
circuits in many common electrical devices. Unlike a resistor, an ideal capacitor does not
dissipate energy. Instead, a capacitor stores energy in the form of an electrostatic field between
its plates.
When there is a potential difference across the conductors (e.g., when a capacitor is attached
across a battery), an electric field develops across the dielectric, causing positive charge +q to
collect on one plate and negative charge –q to collect on the other plate. If a battery has been
attached to a capacitor for a sufficient amount of time, no current can flow through the capacitor.
However, if a time-varying voltage is applied across the leads of the capacitor, a displacement
current can flow.
Figure 5.6:Circuit symbols of capacitors in electronics
82
Any conductor can store charge to some extent. But we cannot give infinite charge to a
conductor. When charge is given to a conductor its potential increases. But charge cannot escape
the conductor because of air, or medium around conductor is di-electric. When due to increasing
charge the potential increase to such extent that air touching the conductor starts getting ionized
and hence charge gets leaked. No more charge can be stored and no more potential increase. This
is limit of charging a conductor. The electric field which can ionize air is 3 x 10 9 Vm-1.
CAPACITANCE OF A CONDUCTOR
Term capacitance of a conductor is the ratio of charge to it by rise in its Potential
C=q/V
In this relation if V=1 then C= q. Therefore,
Capacitance of a conductor is equal to the charge which can change its potential by one volt.
Unit of capacitance: Unit of capacitance is farad, (symbol F).
One farad is capacitance of such a conductor whose potential increase by one volt when charge
of one coulomb is given to it.
One coulomb is a very large unit. The practical smaller units are
i. Micro farad (μF) = 10-6F.(used in electrical circuits)
Ii Pico farad (pF) = 10-12 used in electronics circuits
PARALLEL PLATE CAPACITOR
Since single conductor capacitor does not have large capacitance, parallel plate capacitors are
constructed.
Principle: Principle of a parallel plate capacitor is that an uncharged plate brought near a charged
plate decrease the potential of charged plate and hence its capacitance (C =q/v) increase. Now it
can take more charge. Now if uncharged conductor is earthed, the potential of charged plate
83
further decreases and capacitance further increases. This arrangement of two parallel plates is
called parallel plate capacitor.
Expression for capacitance
Charge q is given to a plate Of area „A‟.
Another plate is kept at a distance „d‟.
After induction an
Electric field E is set-up between the plates. Here
If a dielectric of dielectric constant K is inserted between the plates, then capacitance increase by
factor K and become
Note : The capacitance depends only on its configuration i.e. plate area and distance, and on the
medium between them. The other examples of parallel plate capacitors is
84
COMBINATION OF CAPACITORS
Capacitors can be combined in two ways.
1. Series and
2. Parallel
SERIES COMBINATION
If capacitors are connected in such a way that we can proceed from one point to other by only
one path passing through all capacitors then all these capacitors are said to be in series.
Here
Figure 5.7: Circuit showing three capacitors connected in series
Here three capacitors are connected in series and are connected across a battery of P.D. „V‟.
The charge q given by battery deposits at first plate of first capacitor. Due to induction it attract
–q on the opposite plate. The pairing +ve q charges are repelled to first plate of Second capacitor
which in turn induce -q on the opposite plate. Same action is repeated to all the capacitors and in
this way all capacitors get q charge. As a result ; the charge given by battery q, every capacitor
gets charge q.
The Potential Difference V of battery is sum of potentials across all capacitors. Therefore
V = v1 + v2 + v3
85
Equivalent Capacitance : The equivalent capacitance across the combination can be calculated
as
Ce = q/V
(Give final expression)
The equivalent capacitance in series decreases and become smaller then smallest member. In
series q is same. Therefore by q=cv, we have
c1v1 = c2v2 = c3v3
larger c has smaller v, and smaller c has larger v across it.
If n capacitor of capacitance c are joint in series then equivalent capacitance
Ce = c/n
PARALLEL COMBINATION
If capacitors are connected in such a way that there are many paths to go from one point to other
All these paths are parallel and capacitance of each path is said to be connected in parallel.
Figure 5.8: Circuit showing three capacitors connected in parallel
86
Here three capacitors are connected in parallel and are connected across a battery of P.D. „V‟.
The potential difference across each capacitor is equal and it is same as P.D. across Battery.
The charge given by source is divided and each capacitor gets some charge. The total charge
q = q1 + q2 + q3
Each capacitor has charge q1=c1v1, q2=c2v2, q3=c3v3
Equivalent Capacitance :
We know that q = q1 + q2 + q3
divide by v we get
C = c1+c2+c3
The equivalent capacitance in parallel increases, and it is more than largest in parallel.
ENERGY STORED IN A CAPACITOR
When charge is added to a capacitor then charge already present on the plate repels any new
incoming charge. Hence a new charge has to be sent by applying force and doing work on it. All
this work done on charges becomes energy stored in the capacitor. At any instant work done
dw = V.dq, or dw =dV.q Therefore work done in charging the capacitor from charge 0 to q
This work done convert into electrical Potential
Energy stored in the capacitor
87
This energy is stored in the form of Electric field between the plates.
CONNECTING TWO CHARGED CAPACITORS
When two conductors are connected the charges flow from higher potential plate to lower
potential plate till they reach a common potential
Common Potential
A capacitor of capacitance c1 and potential v1 is connected to another capacitor of capacitance
c2 and potential v2. The charge flow from higher potential to lower potential and it reach an in
between value V such that
Loss of Energy on connecting two capacitors
A capacitor of capacitance c1 and potential v1 is connected to another capacitor of capacitance
c2 and potential v2. The charge flow from higher potential to lower potential and in this process
it loses some energy as charge has to do some work while passing through connecting wire. The
energy is lost in form of heat of connecting wire. Expression for energy lost: In the above two
capacitors the energy contained in the two before connection,
Common Potential after connection
Combined capacitance
88
CHARGING AND DISCHARGING CAPACITORS: RC CIRCUITS
One last thing we would like to know about capacitors in circuits is exactly how capacitors
charge and discharge. That is, we want to know the charge on a capacitor and the current flowing
through a circuit as a function of time. Let us begin with the very simple circuit shown in Fig.
5(a). In this circuit the capacitor is originally connected to a battery of voltage V0 and charged to
an initial charge Q0 = C V0. At time t = 0, the battery is connected to the resistor, as shown in the
figure. Of course, current will flow from the positive plate of the capacitor to the negative plate
of the capacitor. The resistor serves to limit the amount of current that can flow. The larger the
capacitance and the larger the resistance, the longer time it will take the capacitor to discharge.
The charge as a function of time is shown in Fig. 5(b).
Figure 5.9: (a) A simple circuit to discharge a capacitor.
(b) Charge on the capacitor as a function of time
Now let analyze this same circuit quantitatively. To do this, we first make a voltage diagram of
the circuit.
89
Figure 5.10: A voltage diagram for the circuit in Fig. 5
Taking I to be positive, we can then write an equation for the voltage in the diagram
Since the current comes from the charge on the capacitor flowing through the circuit, we know
that Q and I must be related. Current is the amount of charge q flowing past a given point in the
circuit during a small time t. The change in charge on the capacitor during this same time is dQ =
–d q. The minus sign is a consequence of the fact that when current is positive, the charge on the
capacitor is getting smaller so Q is negative. This then means
This equation tells us that the current in the circuit, I, comes from discharging the capacitor.
Since Q is the charge on the capacitor and Q is decreasing, I is a positive quantity.
We can now substitute this expression
90
This is a differential equation (an equation involving derivatives) that can then be solved for Q(t)
subject to the initial condition that Q(0) Q0 CV0 . Normally, we will just give you the solutions
to differential equations; however, this equation is simple enough we can solve it with
elementary calculus:
We have written the constant of integration as ln K for convenience
This solution tells us that the charge is initially CV0 and that the charge decays exponentially.
Since the argument of an exponential function must be dimensionless, RC must have units of
time. In SI units, therefore, 1 × 1 F = 1s. We use the Greek letter (tau) to represent this quantity,
t= RC, and call this the “time constant” of the exponential decay. It is useful to look at the
meaning of the time constant. Clearly if t is large, the exponent is small, and it takes a long time
for the capacitor to discharge. Note that this is consistent with our original analysis of the circuit,
wherein we suggested that if the capacitance is large and the resistance is large, it would take a
long time to discharge the capacitor. More quantitatively we can calculate the time it takes for
the capacitor to discharge to various fractions of its original charge. Let f be the ratio of the
charge at time t to the initial charge.
91
These results are shown graphically in Fig. 6.14. Fig. 6.15 shows the discharge curve of the same
capacitor with values of resistance adjusted so that t= 1 s, 2 s, and 3 s. Note that the time
constant is the time required for the charge on the capacitor to drop to 1 / e = 0.3679 of its
original value.
Figure 5.11:The exponential decay of a discharging capacitor
92
Figure 5.12: Q(t) for discharging RC circuits with time constants of J = 1 s, 2 s, and 3 s.
CHARGING A CAPACITOR (RC CIRCUIT)
Consider a charging capacitor in a circuit below
Figure 5.13: A circuit to charge a capacitor
The last circuit we wish to consider is one in which a capacitor is charged, as shown in Fig.9
Note how a switch is represented in the circuit diagram. In this case, we take the initial charge on
the capacitor to be zero. After a long period of time the capacitor completely charges and current
ceases to flow in the circuit. When the capacitor is fully charged, the voltage on the capacitor
becomes V0. The final charge on the capacitor is then Qf = C V0.
93
Let us draw a voltage diagram of this circuit at some time during the charging process
Figure 10: A voltage diagram for the circuit in Fig.9
We proceed much as we did in the previous case, except that this time we recognize that the
current flowing in the circuit causes the capacitor to charge. Since the capacitor is charging
dQ / dt > 0, and
This differential equation is not quite so simple. We will just give the solution for the initial
conditions described above.
94
The current is an exponentially decreasing function with a time constant τ= RC. This is the same
time constant we found for the charging capacitor. The charge, however, increases toward its
final value with the same time constant. That is, the time constant is the time required for the
current to reach 1 – 1 / e = 0.6321 of its final value
Figure 11: Q(t) for charging RC circuits with time constants of 1 s, 2 s, and 3 s.
Things to remember:
• Capacitors charge and discharge with a time constant RC.
• The time constant is the time it takes a decaying exponential function to fall to 1 / e of the
function‟s initial value if decreasing or to rise to 1 – 1 / e of its final value if increasing.
• In circuits where capacitors are charged and discharged, the charges and currents are of one of
two forms:
95
REFERENCES
[1] M. N. O. Sadiku, (2006), Elements of Electromagnetics-Oxford
University Press,
[2] W. H. Hayt, (2007), Engineering Electromagnetics, McGraw
Hill,
[3] D.V. Prasad, (2003), “Electromagnetic Fields Waves and Antennas, “Khanna Publishers,
Delhi, India [4] J. P. Tewari, (2003) “Engineering Electromagnetics, “Khanna Publishers, Delhi
India
96