NCERT Textbook - Magnetism and Matter Class 12 Notes | EduRev

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Class 12 : NCERT Textbook - Magnetism and Matter Class 12 Notes | EduRev

 Page 1


5.1  INTRODUCTION
Magnetic phenomena are universal in nature. Vast, distant galaxies, the
tiny invisible atoms, men and beasts all are permeated through and
through with a host of magnetic fields from a variety of sources. The earth
magnetism predates human evolution. The word magnet is derived from
the name of an island in Greece called magnesia where magnetic ore
deposits were found, as early as 600 BC. Shepherds on this island
complained that their wooden shoes (which had nails) at times stayed
struck to the ground. Their iron-tipped rods were similarly affected. This
attractive property of magnets made it difficult for them to move around.
The directional property of magnets was also known since ancient
times. A thin long piece of a magnet, when suspended freely, pointed in
the north-south direction. A similar effect was observed when it was placed
on a piece of cork which was then allowed to float in still water. The name
lodestone (or loadstone) given to a naturally occurring ore of iron-
magnetite means leading stone. The technological exploitation of this
property is generally credited to the Chinese. Chinese texts dating 400
BC mention the use of magnetic needles for navigation on ships. Caravans
crossing the Gobi desert also employed magnetic needles.
A Chinese legend narrates the tale of the victory of the emperor Huang-ti
about four thousand years ago, which he owed to his craftsmen (whom
Chapter Five
MAGNETISM AND
MATTER
Page 2


5.1  INTRODUCTION
Magnetic phenomena are universal in nature. Vast, distant galaxies, the
tiny invisible atoms, men and beasts all are permeated through and
through with a host of magnetic fields from a variety of sources. The earth
magnetism predates human evolution. The word magnet is derived from
the name of an island in Greece called magnesia where magnetic ore
deposits were found, as early as 600 BC. Shepherds on this island
complained that their wooden shoes (which had nails) at times stayed
struck to the ground. Their iron-tipped rods were similarly affected. This
attractive property of magnets made it difficult for them to move around.
The directional property of magnets was also known since ancient
times. A thin long piece of a magnet, when suspended freely, pointed in
the north-south direction. A similar effect was observed when it was placed
on a piece of cork which was then allowed to float in still water. The name
lodestone (or loadstone) given to a naturally occurring ore of iron-
magnetite means leading stone. The technological exploitation of this
property is generally credited to the Chinese. Chinese texts dating 400
BC mention the use of magnetic needles for navigation on ships. Caravans
crossing the Gobi desert also employed magnetic needles.
A Chinese legend narrates the tale of the victory of the emperor Huang-ti
about four thousand years ago, which he owed to his craftsmen (whom
Chapter Five
MAGNETISM AND
MATTER
Physics
174
nowadays you would call engineers). These 
built a chariot on which they placed a magnetic figure
with arms outstretched. Figure 5.1 is an artist
description of this chariot. The figure swiveled around
so that the finger of the statuette on it always pointed
south. With this chariot, Huang-ti
to attack the enemy from the rear in thick fog, and to
defeat them.
In the previous chapter we have learned that moving
charges or electric currents produce magnetic fields.
This discovery, which was made in the early part of the
nineteenth century is credited to Oersted, Ampere, Biot
and Savart, among others.
In the present chapter, we take a look at magnetism
as a subject in its own right.
Some of the commonly known ideas regarding
magnetism are:
(i) The earth behaves as a magnet with the magnetic
field pointing approximately from the geographic
south to the north.
(ii) When a bar magnet is freely suspended, it points in  the north-south
direction. The tip which points to the geographic north is called the
north pole and the tip which points to the geographic south is called
the south pole of the magnet.
(iii) There is a repulsive force when north poles ( or south poles ) of two
magnets are brought close together. Conversely, there is an attractive
force between the north pole of one magnet and the south pole of
the other.
(iv) We cannot isolate the north, or south pole of a magnet. If a bar magnet
is broken into two halves, we get two similar bar magnets with
somewhat weaker properties. Unlike electric charges, isolated magnetic
north and south poles known as magnetic monopoles do not exist.
(v) It is possible to make magnets out of iron and its alloys.
We begin with a description of a bar magnet and its behaviour in an
external magnetic field. We describe Gauss
follow it up with an account of the earth
how materials can be classified on the basis of their magnetic properties.
We describe para-, dia-, and ferromagnetism. We conclude with a section
on electromagnets and permanent magnets.
5.2  THE BAR MAGNET
One of the earliest childhood memories of the famous physicist Albert
Einstein was that of a magnet gifted to him by a relative. Einstein was
fascinated, and played endlessly with it. He wondered how the magnet
could affect objects such as nails or pins placed away from it and not in
any way connected to it by a spring or string.
FIGURE 5.1 The arm of the statuette
mounted on the chariot always points
south. This is an artist
of the earliest known compasses,
thousands of years old.
Page 3


5.1  INTRODUCTION
Magnetic phenomena are universal in nature. Vast, distant galaxies, the
tiny invisible atoms, men and beasts all are permeated through and
through with a host of magnetic fields from a variety of sources. The earth
magnetism predates human evolution. The word magnet is derived from
the name of an island in Greece called magnesia where magnetic ore
deposits were found, as early as 600 BC. Shepherds on this island
complained that their wooden shoes (which had nails) at times stayed
struck to the ground. Their iron-tipped rods were similarly affected. This
attractive property of magnets made it difficult for them to move around.
The directional property of magnets was also known since ancient
times. A thin long piece of a magnet, when suspended freely, pointed in
the north-south direction. A similar effect was observed when it was placed
on a piece of cork which was then allowed to float in still water. The name
lodestone (or loadstone) given to a naturally occurring ore of iron-
magnetite means leading stone. The technological exploitation of this
property is generally credited to the Chinese. Chinese texts dating 400
BC mention the use of magnetic needles for navigation on ships. Caravans
crossing the Gobi desert also employed magnetic needles.
A Chinese legend narrates the tale of the victory of the emperor Huang-ti
about four thousand years ago, which he owed to his craftsmen (whom
Chapter Five
MAGNETISM AND
MATTER
Physics
174
nowadays you would call engineers). These 
built a chariot on which they placed a magnetic figure
with arms outstretched. Figure 5.1 is an artist
description of this chariot. The figure swiveled around
so that the finger of the statuette on it always pointed
south. With this chariot, Huang-ti
to attack the enemy from the rear in thick fog, and to
defeat them.
In the previous chapter we have learned that moving
charges or electric currents produce magnetic fields.
This discovery, which was made in the early part of the
nineteenth century is credited to Oersted, Ampere, Biot
and Savart, among others.
In the present chapter, we take a look at magnetism
as a subject in its own right.
Some of the commonly known ideas regarding
magnetism are:
(i) The earth behaves as a magnet with the magnetic
field pointing approximately from the geographic
south to the north.
(ii) When a bar magnet is freely suspended, it points in  the north-south
direction. The tip which points to the geographic north is called the
north pole and the tip which points to the geographic south is called
the south pole of the magnet.
(iii) There is a repulsive force when north poles ( or south poles ) of two
magnets are brought close together. Conversely, there is an attractive
force between the north pole of one magnet and the south pole of
the other.
(iv) We cannot isolate the north, or south pole of a magnet. If a bar magnet
is broken into two halves, we get two similar bar magnets with
somewhat weaker properties. Unlike electric charges, isolated magnetic
north and south poles known as magnetic monopoles do not exist.
(v) It is possible to make magnets out of iron and its alloys.
We begin with a description of a bar magnet and its behaviour in an
external magnetic field. We describe Gauss
follow it up with an account of the earth
how materials can be classified on the basis of their magnetic properties.
We describe para-, dia-, and ferromagnetism. We conclude with a section
on electromagnets and permanent magnets.
5.2  THE BAR MAGNET
One of the earliest childhood memories of the famous physicist Albert
Einstein was that of a magnet gifted to him by a relative. Einstein was
fascinated, and played endlessly with it. He wondered how the magnet
could affect objects such as nails or pins placed away from it and not in
any way connected to it by a spring or string.
FIGURE 5.1 The arm of the statuette
mounted on the chariot always points
south. This is an artist
of the earliest known compasses,
thousands of years old.
Magnetism and
Matter
175
We begin our study by examining iron filings sprinkled on a sheet of
glass placed over a short bar magnet. The arrangement of iron filings is
shown in Fig. 5.2.
 The pattern of iron filings suggests that the magnet has two poles
similar to the positive and negative charge of an electric dipole. As
mentioned in the introductory section, one pole is designated the North
pole and the other, the South pole. When suspended freely, these poles
point approximately towards the geographic north and south poles,
respectively. A similar pattern of iron filings is observed around a current
carrying solenoid.
5.2.1  The magnetic field lines
The pattern of iron filings permits us to plot the magnetic field lines*. This is
shown both for the bar-magnet and the current-carrying solenoid in
Fig. 5.3. For comparison refer to the Chapter 1, Figure 1.17(d). Electric field
lines of an electric dipole are also displayed in Fig. 5.3(c). The magnetic field
lines are a visual and intuitive realisation of the magnetic field. Their
properties are:
(i) The magnetic field lines of a magnet (or a solenoid) form continuous
closed loops. This is unlike the electric dipole where these field lines
begin from a positive charge and end on the negative charge or escape
to infinity.
(ii) The tangent to the field line at a given point represents the direction of
the net magnetic field B at that point.
FIGURE 5.2 The
arrangement of iron
filings surrounding a
bar magnet. The
pattern mimics
magnetic field lines.
The pattern suggests
that the bar magnet
is a magnetic dipole.
* In some textbooks the magnetic field lines are called magnetic lines of force.
This nomenclature is avoided since it can be confusing. Unlike electrostatics
the field lines in magnetism do not indicate the direction of the force on a
(moving) charge.
FIGURE 5.3 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and
(c) electric dipole. At large distances, the field lines are very similar. The curves
labelled  i  and ii are closed Gaussian surfaces.
Page 4


5.1  INTRODUCTION
Magnetic phenomena are universal in nature. Vast, distant galaxies, the
tiny invisible atoms, men and beasts all are permeated through and
through with a host of magnetic fields from a variety of sources. The earth
magnetism predates human evolution. The word magnet is derived from
the name of an island in Greece called magnesia where magnetic ore
deposits were found, as early as 600 BC. Shepherds on this island
complained that their wooden shoes (which had nails) at times stayed
struck to the ground. Their iron-tipped rods were similarly affected. This
attractive property of magnets made it difficult for them to move around.
The directional property of magnets was also known since ancient
times. A thin long piece of a magnet, when suspended freely, pointed in
the north-south direction. A similar effect was observed when it was placed
on a piece of cork which was then allowed to float in still water. The name
lodestone (or loadstone) given to a naturally occurring ore of iron-
magnetite means leading stone. The technological exploitation of this
property is generally credited to the Chinese. Chinese texts dating 400
BC mention the use of magnetic needles for navigation on ships. Caravans
crossing the Gobi desert also employed magnetic needles.
A Chinese legend narrates the tale of the victory of the emperor Huang-ti
about four thousand years ago, which he owed to his craftsmen (whom
Chapter Five
MAGNETISM AND
MATTER
Physics
174
nowadays you would call engineers). These 
built a chariot on which they placed a magnetic figure
with arms outstretched. Figure 5.1 is an artist
description of this chariot. The figure swiveled around
so that the finger of the statuette on it always pointed
south. With this chariot, Huang-ti
to attack the enemy from the rear in thick fog, and to
defeat them.
In the previous chapter we have learned that moving
charges or electric currents produce magnetic fields.
This discovery, which was made in the early part of the
nineteenth century is credited to Oersted, Ampere, Biot
and Savart, among others.
In the present chapter, we take a look at magnetism
as a subject in its own right.
Some of the commonly known ideas regarding
magnetism are:
(i) The earth behaves as a magnet with the magnetic
field pointing approximately from the geographic
south to the north.
(ii) When a bar magnet is freely suspended, it points in  the north-south
direction. The tip which points to the geographic north is called the
north pole and the tip which points to the geographic south is called
the south pole of the magnet.
(iii) There is a repulsive force when north poles ( or south poles ) of two
magnets are brought close together. Conversely, there is an attractive
force between the north pole of one magnet and the south pole of
the other.
(iv) We cannot isolate the north, or south pole of a magnet. If a bar magnet
is broken into two halves, we get two similar bar magnets with
somewhat weaker properties. Unlike electric charges, isolated magnetic
north and south poles known as magnetic monopoles do not exist.
(v) It is possible to make magnets out of iron and its alloys.
We begin with a description of a bar magnet and its behaviour in an
external magnetic field. We describe Gauss
follow it up with an account of the earth
how materials can be classified on the basis of their magnetic properties.
We describe para-, dia-, and ferromagnetism. We conclude with a section
on electromagnets and permanent magnets.
5.2  THE BAR MAGNET
One of the earliest childhood memories of the famous physicist Albert
Einstein was that of a magnet gifted to him by a relative. Einstein was
fascinated, and played endlessly with it. He wondered how the magnet
could affect objects such as nails or pins placed away from it and not in
any way connected to it by a spring or string.
FIGURE 5.1 The arm of the statuette
mounted on the chariot always points
south. This is an artist
of the earliest known compasses,
thousands of years old.
Magnetism and
Matter
175
We begin our study by examining iron filings sprinkled on a sheet of
glass placed over a short bar magnet. The arrangement of iron filings is
shown in Fig. 5.2.
 The pattern of iron filings suggests that the magnet has two poles
similar to the positive and negative charge of an electric dipole. As
mentioned in the introductory section, one pole is designated the North
pole and the other, the South pole. When suspended freely, these poles
point approximately towards the geographic north and south poles,
respectively. A similar pattern of iron filings is observed around a current
carrying solenoid.
5.2.1  The magnetic field lines
The pattern of iron filings permits us to plot the magnetic field lines*. This is
shown both for the bar-magnet and the current-carrying solenoid in
Fig. 5.3. For comparison refer to the Chapter 1, Figure 1.17(d). Electric field
lines of an electric dipole are also displayed in Fig. 5.3(c). The magnetic field
lines are a visual and intuitive realisation of the magnetic field. Their
properties are:
(i) The magnetic field lines of a magnet (or a solenoid) form continuous
closed loops. This is unlike the electric dipole where these field lines
begin from a positive charge and end on the negative charge or escape
to infinity.
(ii) The tangent to the field line at a given point represents the direction of
the net magnetic field B at that point.
FIGURE 5.2 The
arrangement of iron
filings surrounding a
bar magnet. The
pattern mimics
magnetic field lines.
The pattern suggests
that the bar magnet
is a magnetic dipole.
* In some textbooks the magnetic field lines are called magnetic lines of force.
This nomenclature is avoided since it can be confusing. Unlike electrostatics
the field lines in magnetism do not indicate the direction of the force on a
(moving) charge.
FIGURE 5.3 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and
(c) electric dipole. At large distances, the field lines are very similar. The curves
labelled  i  and ii are closed Gaussian surfaces.
Physics
176
(iii) The larger the number of field lines crossing per unit area, the stronger
is the magnitude of the magnetic field B. In Fig. 5.3(a), B is larger
around region  ii  than in region  i  .
(iv) The magnetic field lines do not intersect, for if they did, the direction
of the magnetic field would not be unique at the point of intersection.
One can plot the magnetic field lines in a variety of ways. One way is
to place a small magnetic compass needle at various positions and note
its orientation. This gives us an idea of the magnetic field direction at
various points in space.
5.2.2  Bar magnet as an equivalent solenoid
In the previous chapter, we have explained how a current loop acts as a
magnetic dipole (Section 4.10). We mentioned Ampere
all magnetic phenomena can be explained in terms of circulating currents.
Recall that the magnetic dipole moment m
associated with a current loop was defined
to be m = NIA where N is the number of
turns in the loop, I the current and A the
area vector (Eq. 4.30).
The resemblance of magnetic field lines
for a bar magnet and a solenoid suggest that
a bar magnet may be thought of as a large
number of circulating currents in analogy
with a solenoid. Cutting a bar magnet in half
is like cutting a solenoid. We get two smaller
solenoids with weaker magnetic properties.
The field lines remain continuous, emerging
from one face of the solenoid and entering
into the other face. One can test this analogy
by moving a small compass needle in the
neighbourhood of a bar magnet and a
current-carrying finite solenoid and noting
that the deflections of the needle are similar
in both cases.
To make this analogy more firm we
calculate the axial field of a finite solenoid
depicted in Fig. 5.4 (a). We shall demonstrate
that at large distances this axial field
resembles that of a bar magnet.
Let the solenoid of Fig. 5.4(a) consists of
n turns per unit length. Let its length be 2l
and radius a. We can evaluate the axial field
at a point P, at a distance r from the centre O
of the solenoid. To do this, consider a circular element of thickness dx of
the solenoid at a distance x from its centre. It consists of n d x turns. Let
I be the current in the solenoid. In Section 4.6 of the previous chapter we
have calculated the magnetic field on the axis of a circular current loop.
From Eq. (4.13), the magnitude of the field at point P due to the circular
element is
FIGURE 5.4 Calculation of (a) The axial field of a
finite solenoid in order to demonstrate its similarity
to that of a bar magnet. (b) A magnetic needle
in a uniform magnetic field B. The
arrangement may be used to
determine either B or the magnetic
moment m of the needle.
Page 5


5.1  INTRODUCTION
Magnetic phenomena are universal in nature. Vast, distant galaxies, the
tiny invisible atoms, men and beasts all are permeated through and
through with a host of magnetic fields from a variety of sources. The earth
magnetism predates human evolution. The word magnet is derived from
the name of an island in Greece called magnesia where magnetic ore
deposits were found, as early as 600 BC. Shepherds on this island
complained that their wooden shoes (which had nails) at times stayed
struck to the ground. Their iron-tipped rods were similarly affected. This
attractive property of magnets made it difficult for them to move around.
The directional property of magnets was also known since ancient
times. A thin long piece of a magnet, when suspended freely, pointed in
the north-south direction. A similar effect was observed when it was placed
on a piece of cork which was then allowed to float in still water. The name
lodestone (or loadstone) given to a naturally occurring ore of iron-
magnetite means leading stone. The technological exploitation of this
property is generally credited to the Chinese. Chinese texts dating 400
BC mention the use of magnetic needles for navigation on ships. Caravans
crossing the Gobi desert also employed magnetic needles.
A Chinese legend narrates the tale of the victory of the emperor Huang-ti
about four thousand years ago, which he owed to his craftsmen (whom
Chapter Five
MAGNETISM AND
MATTER
Physics
174
nowadays you would call engineers). These 
built a chariot on which they placed a magnetic figure
with arms outstretched. Figure 5.1 is an artist
description of this chariot. The figure swiveled around
so that the finger of the statuette on it always pointed
south. With this chariot, Huang-ti
to attack the enemy from the rear in thick fog, and to
defeat them.
In the previous chapter we have learned that moving
charges or electric currents produce magnetic fields.
This discovery, which was made in the early part of the
nineteenth century is credited to Oersted, Ampere, Biot
and Savart, among others.
In the present chapter, we take a look at magnetism
as a subject in its own right.
Some of the commonly known ideas regarding
magnetism are:
(i) The earth behaves as a magnet with the magnetic
field pointing approximately from the geographic
south to the north.
(ii) When a bar magnet is freely suspended, it points in  the north-south
direction. The tip which points to the geographic north is called the
north pole and the tip which points to the geographic south is called
the south pole of the magnet.
(iii) There is a repulsive force when north poles ( or south poles ) of two
magnets are brought close together. Conversely, there is an attractive
force between the north pole of one magnet and the south pole of
the other.
(iv) We cannot isolate the north, or south pole of a magnet. If a bar magnet
is broken into two halves, we get two similar bar magnets with
somewhat weaker properties. Unlike electric charges, isolated magnetic
north and south poles known as magnetic monopoles do not exist.
(v) It is possible to make magnets out of iron and its alloys.
We begin with a description of a bar magnet and its behaviour in an
external magnetic field. We describe Gauss
follow it up with an account of the earth
how materials can be classified on the basis of their magnetic properties.
We describe para-, dia-, and ferromagnetism. We conclude with a section
on electromagnets and permanent magnets.
5.2  THE BAR MAGNET
One of the earliest childhood memories of the famous physicist Albert
Einstein was that of a magnet gifted to him by a relative. Einstein was
fascinated, and played endlessly with it. He wondered how the magnet
could affect objects such as nails or pins placed away from it and not in
any way connected to it by a spring or string.
FIGURE 5.1 The arm of the statuette
mounted on the chariot always points
south. This is an artist
of the earliest known compasses,
thousands of years old.
Magnetism and
Matter
175
We begin our study by examining iron filings sprinkled on a sheet of
glass placed over a short bar magnet. The arrangement of iron filings is
shown in Fig. 5.2.
 The pattern of iron filings suggests that the magnet has two poles
similar to the positive and negative charge of an electric dipole. As
mentioned in the introductory section, one pole is designated the North
pole and the other, the South pole. When suspended freely, these poles
point approximately towards the geographic north and south poles,
respectively. A similar pattern of iron filings is observed around a current
carrying solenoid.
5.2.1  The magnetic field lines
The pattern of iron filings permits us to plot the magnetic field lines*. This is
shown both for the bar-magnet and the current-carrying solenoid in
Fig. 5.3. For comparison refer to the Chapter 1, Figure 1.17(d). Electric field
lines of an electric dipole are also displayed in Fig. 5.3(c). The magnetic field
lines are a visual and intuitive realisation of the magnetic field. Their
properties are:
(i) The magnetic field lines of a magnet (or a solenoid) form continuous
closed loops. This is unlike the electric dipole where these field lines
begin from a positive charge and end on the negative charge or escape
to infinity.
(ii) The tangent to the field line at a given point represents the direction of
the net magnetic field B at that point.
FIGURE 5.2 The
arrangement of iron
filings surrounding a
bar magnet. The
pattern mimics
magnetic field lines.
The pattern suggests
that the bar magnet
is a magnetic dipole.
* In some textbooks the magnetic field lines are called magnetic lines of force.
This nomenclature is avoided since it can be confusing. Unlike electrostatics
the field lines in magnetism do not indicate the direction of the force on a
(moving) charge.
FIGURE 5.3 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and
(c) electric dipole. At large distances, the field lines are very similar. The curves
labelled  i  and ii are closed Gaussian surfaces.
Physics
176
(iii) The larger the number of field lines crossing per unit area, the stronger
is the magnitude of the magnetic field B. In Fig. 5.3(a), B is larger
around region  ii  than in region  i  .
(iv) The magnetic field lines do not intersect, for if they did, the direction
of the magnetic field would not be unique at the point of intersection.
One can plot the magnetic field lines in a variety of ways. One way is
to place a small magnetic compass needle at various positions and note
its orientation. This gives us an idea of the magnetic field direction at
various points in space.
5.2.2  Bar magnet as an equivalent solenoid
In the previous chapter, we have explained how a current loop acts as a
magnetic dipole (Section 4.10). We mentioned Ampere
all magnetic phenomena can be explained in terms of circulating currents.
Recall that the magnetic dipole moment m
associated with a current loop was defined
to be m = NIA where N is the number of
turns in the loop, I the current and A the
area vector (Eq. 4.30).
The resemblance of magnetic field lines
for a bar magnet and a solenoid suggest that
a bar magnet may be thought of as a large
number of circulating currents in analogy
with a solenoid. Cutting a bar magnet in half
is like cutting a solenoid. We get two smaller
solenoids with weaker magnetic properties.
The field lines remain continuous, emerging
from one face of the solenoid and entering
into the other face. One can test this analogy
by moving a small compass needle in the
neighbourhood of a bar magnet and a
current-carrying finite solenoid and noting
that the deflections of the needle are similar
in both cases.
To make this analogy more firm we
calculate the axial field of a finite solenoid
depicted in Fig. 5.4 (a). We shall demonstrate
that at large distances this axial field
resembles that of a bar magnet.
Let the solenoid of Fig. 5.4(a) consists of
n turns per unit length. Let its length be 2l
and radius a. We can evaluate the axial field
at a point P, at a distance r from the centre O
of the solenoid. To do this, consider a circular element of thickness dx of
the solenoid at a distance x from its centre. It consists of n d x turns. Let
I be the current in the solenoid. In Section 4.6 of the previous chapter we
have calculated the magnetic field on the axis of a circular current loop.
From Eq. (4.13), the magnitude of the field at point P due to the circular
element is
FIGURE 5.4 Calculation of (a) The axial field of a
finite solenoid in order to demonstrate its similarity
to that of a bar magnet. (b) A magnetic needle
in a uniform magnetic field B. The
arrangement may be used to
determine either B or the magnetic
moment m of the needle.
Magnetism and
Matter
177
2
0
3
2 2
2
2[( ) ]
n dx I a
dB
r x a
m
=
- +
The magnitude of the total field is obtained by summing over all the
elements x = l to x = + l. Thus,
2
0
2
nIa
B
m
=
2 2 3 /2
[( ) ]
l
l
dx
r x a
-
- +
ò
This integration can be done by trigonometric substitutions. This
exercise, however, is not necessary for our purpose. Note that the range
of x is from  l to + l. Consider the far axial field of the solenoid, i.e.,
r >> a and r >> l. Then the denominator is approximated by
3
2 2 3
2
[( ) ] r x a r - + »
and 
2
0
3
2
l
l
n I a
B dx
r
m
-
=
ò
      =  
2
0
3
2
2
n I l a
r
m
 (5.1)
Note that the magnitude of the magnetic moment of the solenoid is,
m = n (2 l) I (pa
2
) 
area). Thus,
0
3
2
4
m
B
r
m
p
=
(5.2)
This is also the far axial magnetic field of a bar magnet which one may
obtain experimentally. Thus, a bar magnet and a solenoid produce similar
magnetic fields. The magnetic moment of a bar magnet is thus equal to
the magnetic moment of an equivalent solenoid that produces the same
magnetic field.
Some textbooks assign a magnetic charge (also called pole strength)
+q
m
to the north pole and q
m
 to the south pole of a bar magnet of length
2l, and magnetic moment q
m
(2l). The field strength due to q
m
 at a distance
r from it is given by m
0
q
m
/4pr
2
. The magnetic field due to the bar magnet
is then obtained, both for the axial and the equatorial case, in a manner
analogous to that of an electric dipole (Chapter 1). The method is simple
and appealing. However, magnetic monopoles do not exist, and we have
avoided this approach for that reason.
5.2.3  The dipole in a uniform magnetic field
The pattern of iron filings, i.e., the magnetic field lines gives us an
approximate idea of the magnetic field B. We may at times be required to
determine the magnitude of B accurately. This is done by placing a small
compass needle of known magnetic moment m and moment of inertia I
and allowing it to oscillate in the magnetic field. This arrangement is shown
in Fig. 5.4(b).
The torque on the needle is [see Eq. (4.29)],
t = m × B (5.3)
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