NCERT Textbook - Moving Charges and Magnetism Class 12 Notes | EduRev

 Page 1


Physics
132
4.1  INTRODUCTION
Both Electricity and Magnetism have been known for more than 2000
years. However, it was only about 200 years ago, in 1820, that it was
realised that they were intimately related*. During a lecture demonstration
in the summer of 1820, the Danish physicist Hans Christian Oersted
noticed that a current in a straight wire caused a noticeable deflection in
a nearby magnetic compass needle. He investigated this phenomenon.
He found that the alignment of the needle is tangential to an imaginary
circle which has the straight wire as its centre and has its plane
perpendicular to the wire. This situation is depicted in Fig.4.1(a). It is
noticeable when the current is large and the needle sufficiently close to
the wire so that the earth’s magnetic field may be ignored. Reversing the
direction of the current reverses the orientation of the needle [Fig. 4.1(b)].
The deflection increases on increasing the current or bringing the needle
closer to the wire. Iron filings sprinkled around the wire arrange
themselves in concentric circles with the wire as the centre [Fig. 4.1(c)].
Oersted concluded that moving charges or currents produced a
magnetic field in the surrounding space.
Following this there was intense experimentation. In 1864, the laws
obeyed by electricity and magnetism were unified and formulated by
Chapter Four
MOVING CHARGES
AND MAGNETISM
* See the box in Chapter 1, Page 3.
2015-16(20/01/2015)
Page 2


Physics
132
4.1  INTRODUCTION
Both Electricity and Magnetism have been known for more than 2000
years. However, it was only about 200 years ago, in 1820, that it was
realised that they were intimately related*. During a lecture demonstration
in the summer of 1820, the Danish physicist Hans Christian Oersted
noticed that a current in a straight wire caused a noticeable deflection in
a nearby magnetic compass needle. He investigated this phenomenon.
He found that the alignment of the needle is tangential to an imaginary
circle which has the straight wire as its centre and has its plane
perpendicular to the wire. This situation is depicted in Fig.4.1(a). It is
noticeable when the current is large and the needle sufficiently close to
the wire so that the earth’s magnetic field may be ignored. Reversing the
direction of the current reverses the orientation of the needle [Fig. 4.1(b)].
The deflection increases on increasing the current or bringing the needle
closer to the wire. Iron filings sprinkled around the wire arrange
themselves in concentric circles with the wire as the centre [Fig. 4.1(c)].
Oersted concluded that moving charges or currents produced a
magnetic field in the surrounding space.
Following this there was intense experimentation. In 1864, the laws
obeyed by electricity and magnetism were unified and formulated by
Chapter Four
MOVING CHARGES
AND MAGNETISM
* See the box in Chapter 1, Page 3.
2015-16(20/01/2015)
Moving Charges and
Magnetism
133
James Maxwell who then realised that light was electromagnetic waves.
Radio waves were discovered by Hertz, and produced by  J.C.Bose and
G. Marconi by the end of the 19
th
 century. A  remarkable scientific and
technological progress has taken place in the 20
th
 century. This is due to
our increased understanding of electromagnetism and the invention of
devices  for production, amplification, transmission and detection of
electromagnetic waves.
In this chapter, we will see how magnetic field exerts
forces on moving charged particles, like electrons,
protons, and current-carrying wires. We shall also learn
how currents produce magnetic fields. We shall see how
particles can be accelerated to very high energies in a
cyclotron. We shall study how currents and voltages are
detected by a galvanometer.
In this and subsequent Chapter on magnetism,
we adopt the following convention:  A current or a
field (electric or magnetic) emerging out of the plane of the
paper is depicted by a dot (¤). A current or a field going
into the plane of the paper is depicted by a cross ( ? )*.
Figures. 4.1(a) and 4.1(b) correspond to these two
situations, respectively.
4.2  MAGNETIC FORCE
4.2.1  Sources and fields
Before we introduce the concept of a magnetic field B, we
shall recapitulate what we have learnt in Chapter 1 about
the electric field E. We have seen that the interaction
between two charges can be considered in two stages.
The charge Q, the source of the field, produces an electric
field E, where
FIGURE 4.1 The magnetic field due to a straight long current-carrying
wire. The wire is perpendicular to the plane of the paper. A ring  of
compass needles surrounds the wire. The orientation of the needles is
shown when (a) the current emerges out of the plane of the paper,
(b) the current moves into the plane of the paper. (c) The arrangement of
iron filings around the wire. The darkened ends of the needle represent
north poles. The effect of the earth’s magnetic field is neglected.
Hans Christian Oersted
(1777–1851) Danish
physicist and chemist,
professor at Copenhagen.
He observed that a
compass needle suffers a
deflection when placed
near a wire carrying an
electric current. This
discovery gave the first
empirical evidence of a
connection between electric
and magnetic phenomena.
HANS CHRISTIAN OERSTED  (1777–1851)
* A dot appears like the tip of an arrow pointed at you, a cross is like the feathered
tail of an arrow moving away from you.
2015-16(20/01/2015)
Page 3


Physics
132
4.1  INTRODUCTION
Both Electricity and Magnetism have been known for more than 2000
years. However, it was only about 200 years ago, in 1820, that it was
realised that they were intimately related*. During a lecture demonstration
in the summer of 1820, the Danish physicist Hans Christian Oersted
noticed that a current in a straight wire caused a noticeable deflection in
a nearby magnetic compass needle. He investigated this phenomenon.
He found that the alignment of the needle is tangential to an imaginary
circle which has the straight wire as its centre and has its plane
perpendicular to the wire. This situation is depicted in Fig.4.1(a). It is
noticeable when the current is large and the needle sufficiently close to
the wire so that the earth’s magnetic field may be ignored. Reversing the
direction of the current reverses the orientation of the needle [Fig. 4.1(b)].
The deflection increases on increasing the current or bringing the needle
closer to the wire. Iron filings sprinkled around the wire arrange
themselves in concentric circles with the wire as the centre [Fig. 4.1(c)].
Oersted concluded that moving charges or currents produced a
magnetic field in the surrounding space.
Following this there was intense experimentation. In 1864, the laws
obeyed by electricity and magnetism were unified and formulated by
Chapter Four
MOVING CHARGES
AND MAGNETISM
* See the box in Chapter 1, Page 3.
2015-16(20/01/2015)
Moving Charges and
Magnetism
133
James Maxwell who then realised that light was electromagnetic waves.
Radio waves were discovered by Hertz, and produced by  J.C.Bose and
G. Marconi by the end of the 19
th
 century. A  remarkable scientific and
technological progress has taken place in the 20
th
 century. This is due to
our increased understanding of electromagnetism and the invention of
devices  for production, amplification, transmission and detection of
electromagnetic waves.
In this chapter, we will see how magnetic field exerts
forces on moving charged particles, like electrons,
protons, and current-carrying wires. We shall also learn
how currents produce magnetic fields. We shall see how
particles can be accelerated to very high energies in a
cyclotron. We shall study how currents and voltages are
detected by a galvanometer.
In this and subsequent Chapter on magnetism,
we adopt the following convention:  A current or a
field (electric or magnetic) emerging out of the plane of the
paper is depicted by a dot (¤). A current or a field going
into the plane of the paper is depicted by a cross ( ? )*.
Figures. 4.1(a) and 4.1(b) correspond to these two
situations, respectively.
4.2  MAGNETIC FORCE
4.2.1  Sources and fields
Before we introduce the concept of a magnetic field B, we
shall recapitulate what we have learnt in Chapter 1 about
the electric field E. We have seen that the interaction
between two charges can be considered in two stages.
The charge Q, the source of the field, produces an electric
field E, where
FIGURE 4.1 The magnetic field due to a straight long current-carrying
wire. The wire is perpendicular to the plane of the paper. A ring  of
compass needles surrounds the wire. The orientation of the needles is
shown when (a) the current emerges out of the plane of the paper,
(b) the current moves into the plane of the paper. (c) The arrangement of
iron filings around the wire. The darkened ends of the needle represent
north poles. The effect of the earth’s magnetic field is neglected.
Hans Christian Oersted
(1777–1851) Danish
physicist and chemist,
professor at Copenhagen.
He observed that a
compass needle suffers a
deflection when placed
near a wire carrying an
electric current. This
discovery gave the first
empirical evidence of a
connection between electric
and magnetic phenomena.
HANS CHRISTIAN OERSTED  (1777–1851)
* A dot appears like the tip of an arrow pointed at you, a cross is like the feathered
tail of an arrow moving away from you.
2015-16(20/01/2015)
Physics
134
E = Q ˆ r / (4pe e e e e
0
)r
2
(4.1)
where ˆ r is unit vector along r,  and  the field E is a vector
field. A charge q interacts with this field and experiences
a force F given by
     F  =  q  E   = q Q ˆ r  / (4pe
0
) r
2
(4.2)
As pointed out in the Chapter 1, the field  E is not
just an artefact but has a physical role. It can  convey
energy and momentum and is not established
instantaneously but takes finite time to propagate. The
concept of a field was specially stressed by Faraday  and
was incorporated by Maxwell in his unification of
electricity and magnetism. In addition to depending on
each point in space, it can also vary with time, i.e., be a
function of time.  In our discussions in this chapter, we
will assume that the fields do not change with time.
The field at a particular point can be due to one or
more charges.  If there are more charges the fields add
vectorially. You have already learnt in Chapter 1 that this
is called the principle of superposition. Once the field is
known, the force on a test charge is given by Eq. (4.2).
Just as static charges produce an electric field, the
currents or moving charges produce (in addition) a
magnetic field, denoted by B (r), again a vector field. It
has several basic properties identical to the electric field.
It is defined at each point in space (and can in addition
depend on time). Experimentally, it is found to obey the
principle of superposition: the magnetic field of several
sources is the vector addition of magnetic field of each
individual source.
4.2.2  Magnetic Field,  Lorentz  Force
Let us suppose that there is  a point charge q (moving
with a velocity v and, located at r at a given time t) in
presence of both the electric field E (r) and the magnetic
field B (r).  The force on an electric charge q due to both of
them can be written as
F   = q [ E (r) +  v × B (r)] = F
electric
 +F
magnetic
(4.3)
This force was given first  by H.A. Lorentz based on the extensive
experiments of Ampere and others. It is called the Lorentz force. You
have already studied in detail the force due to the electric field.  If we
look at the interaction with the magnetic field, we find the following
features.
(i) It depends on q, v and B (charge of the particle, the velocity and the
magnetic field). Force on a negative charge is opposite to that on a
positive charge.
(ii) The magnetic force q [ v × B ] includes a vector product of velocity
and magnetic field. The vector product makes the force due to magnetic
HENDRIK ANTOON LORENTZ (1853 – 1928)
Hendrik Antoon Lorentz
(1853 – 1928) Dutch
theoretical physicist,
professor at Leiden. He
investigated the
relationship between
electricity, magnetism, and
mechanics. In order to
explain the observed effect
of magnetic fields on
emitters of light (Zeeman
effect), he postulated the
existence of electric charges
in the atom, for which he
was awarded the Nobel Prize
in 1902. He derived a set of
transformation equations
(known after him, as
Lorentz transformation
equations) by some tangled
mathematical arguments,
but he was not aware that
these equations hinge on a
new concept of space and
time.
2015-16(20/01/2015)
Page 4


Physics
132
4.1  INTRODUCTION
Both Electricity and Magnetism have been known for more than 2000
years. However, it was only about 200 years ago, in 1820, that it was
realised that they were intimately related*. During a lecture demonstration
in the summer of 1820, the Danish physicist Hans Christian Oersted
noticed that a current in a straight wire caused a noticeable deflection in
a nearby magnetic compass needle. He investigated this phenomenon.
He found that the alignment of the needle is tangential to an imaginary
circle which has the straight wire as its centre and has its plane
perpendicular to the wire. This situation is depicted in Fig.4.1(a). It is
noticeable when the current is large and the needle sufficiently close to
the wire so that the earth’s magnetic field may be ignored. Reversing the
direction of the current reverses the orientation of the needle [Fig. 4.1(b)].
The deflection increases on increasing the current or bringing the needle
closer to the wire. Iron filings sprinkled around the wire arrange
themselves in concentric circles with the wire as the centre [Fig. 4.1(c)].
Oersted concluded that moving charges or currents produced a
magnetic field in the surrounding space.
Following this there was intense experimentation. In 1864, the laws
obeyed by electricity and magnetism were unified and formulated by
Chapter Four
MOVING CHARGES
AND MAGNETISM
* See the box in Chapter 1, Page 3.
2015-16(20/01/2015)
Moving Charges and
Magnetism
133
James Maxwell who then realised that light was electromagnetic waves.
Radio waves were discovered by Hertz, and produced by  J.C.Bose and
G. Marconi by the end of the 19
th
 century. A  remarkable scientific and
technological progress has taken place in the 20
th
 century. This is due to
our increased understanding of electromagnetism and the invention of
devices  for production, amplification, transmission and detection of
electromagnetic waves.
In this chapter, we will see how magnetic field exerts
forces on moving charged particles, like electrons,
protons, and current-carrying wires. We shall also learn
how currents produce magnetic fields. We shall see how
particles can be accelerated to very high energies in a
cyclotron. We shall study how currents and voltages are
detected by a galvanometer.
In this and subsequent Chapter on magnetism,
we adopt the following convention:  A current or a
field (electric or magnetic) emerging out of the plane of the
paper is depicted by a dot (¤). A current or a field going
into the plane of the paper is depicted by a cross ( ? )*.
Figures. 4.1(a) and 4.1(b) correspond to these two
situations, respectively.
4.2  MAGNETIC FORCE
4.2.1  Sources and fields
Before we introduce the concept of a magnetic field B, we
shall recapitulate what we have learnt in Chapter 1 about
the electric field E. We have seen that the interaction
between two charges can be considered in two stages.
The charge Q, the source of the field, produces an electric
field E, where
FIGURE 4.1 The magnetic field due to a straight long current-carrying
wire. The wire is perpendicular to the plane of the paper. A ring  of
compass needles surrounds the wire. The orientation of the needles is
shown when (a) the current emerges out of the plane of the paper,
(b) the current moves into the plane of the paper. (c) The arrangement of
iron filings around the wire. The darkened ends of the needle represent
north poles. The effect of the earth’s magnetic field is neglected.
Hans Christian Oersted
(1777–1851) Danish
physicist and chemist,
professor at Copenhagen.
He observed that a
compass needle suffers a
deflection when placed
near a wire carrying an
electric current. This
discovery gave the first
empirical evidence of a
connection between electric
and magnetic phenomena.
HANS CHRISTIAN OERSTED  (1777–1851)
* A dot appears like the tip of an arrow pointed at you, a cross is like the feathered
tail of an arrow moving away from you.
2015-16(20/01/2015)
Physics
134
E = Q ˆ r / (4pe e e e e
0
)r
2
(4.1)
where ˆ r is unit vector along r,  and  the field E is a vector
field. A charge q interacts with this field and experiences
a force F given by
     F  =  q  E   = q Q ˆ r  / (4pe
0
) r
2
(4.2)
As pointed out in the Chapter 1, the field  E is not
just an artefact but has a physical role. It can  convey
energy and momentum and is not established
instantaneously but takes finite time to propagate. The
concept of a field was specially stressed by Faraday  and
was incorporated by Maxwell in his unification of
electricity and magnetism. In addition to depending on
each point in space, it can also vary with time, i.e., be a
function of time.  In our discussions in this chapter, we
will assume that the fields do not change with time.
The field at a particular point can be due to one or
more charges.  If there are more charges the fields add
vectorially. You have already learnt in Chapter 1 that this
is called the principle of superposition. Once the field is
known, the force on a test charge is given by Eq. (4.2).
Just as static charges produce an electric field, the
currents or moving charges produce (in addition) a
magnetic field, denoted by B (r), again a vector field. It
has several basic properties identical to the electric field.
It is defined at each point in space (and can in addition
depend on time). Experimentally, it is found to obey the
principle of superposition: the magnetic field of several
sources is the vector addition of magnetic field of each
individual source.
4.2.2  Magnetic Field,  Lorentz  Force
Let us suppose that there is  a point charge q (moving
with a velocity v and, located at r at a given time t) in
presence of both the electric field E (r) and the magnetic
field B (r).  The force on an electric charge q due to both of
them can be written as
F   = q [ E (r) +  v × B (r)] = F
electric
 +F
magnetic
(4.3)
This force was given first  by H.A. Lorentz based on the extensive
experiments of Ampere and others. It is called the Lorentz force. You
have already studied in detail the force due to the electric field.  If we
look at the interaction with the magnetic field, we find the following
features.
(i) It depends on q, v and B (charge of the particle, the velocity and the
magnetic field). Force on a negative charge is opposite to that on a
positive charge.
(ii) The magnetic force q [ v × B ] includes a vector product of velocity
and magnetic field. The vector product makes the force due to magnetic
HENDRIK ANTOON LORENTZ (1853 – 1928)
Hendrik Antoon Lorentz
(1853 – 1928) Dutch
theoretical physicist,
professor at Leiden. He
investigated the
relationship between
electricity, magnetism, and
mechanics. In order to
explain the observed effect
of magnetic fields on
emitters of light (Zeeman
effect), he postulated the
existence of electric charges
in the atom, for which he
was awarded the Nobel Prize
in 1902. He derived a set of
transformation equations
(known after him, as
Lorentz transformation
equations) by some tangled
mathematical arguments,
but he was not aware that
these equations hinge on a
new concept of space and
time.
2015-16(20/01/2015)
Moving Charges and
Magnetism
135
field vanish (become zero) if  velocity and magnetic field are parallel
or anti-parallel. The force acts in a (sideways) direction perpendicular
to both the velocity and the magnetic field.
Its direction  is given by the screw rule or
right hand rule  for vector (or cross) product
as illustrated in Fig. 4.2.
(iii) The magnetic force is zero if  charge is not
moving (as then |v|= 0).  Only a moving
charge feels the magnetic force.
The expression for the magnetic force helps
us to define the unit of the magnetic field, if
one takes q, F and v, all to be unity in the force
equation F = q [ v × B] =q v B sin ? ˆ n ,  where ?
is the angle between v and B [see  Fig. 4.2 (a)].
The magnitude of magnetic field B is 1 SI unit,
when the force acting on a unit charge (1 C),
moving perpendicular to B with a speed 1m/s,
is one newton.
Dimensionally, we have [B] = [F/qv] and the unit
of B are Newton second / (coulomb metre).  This
unit is called tesla (T) named after Nikola Tesla
(1856 – 1943). Tesla is a rather large unit. A smaller  unit (non-SI) called
gauss (=10
–4
 tesla) is also often used. The earth’s magnetic field is about
3.6 × 10
–5
 T. Table  4.1 lists magnetic fields over a wide range in the
universe.
FIGURE 4.2 The direction of the magnetic
force acting on a charged particle. (a) The
force on a positively charged particle with
velocity v and making an angle ? with the
magnetic field B is given by the right-hand
rule. (b) A moving charged particle q is
deflected in an opposite sense to –q in the
presence of magnetic field.
TABLE 4.1 ORDER OF MAGNITUDES OF MAGNETIC FIELDS IN A VARIETY OF PHYSICAL SITUATIONS
Physical situation Magnitude of B (in tesla)
Surface of a neutron star 10
8
Typical large field in a laboratory   1
Near a small bar magnet 10
–2
On the earth’s surface 10
–5
Human nerve fibre 10
–10
Interstellar space 10
–12
4.2.3  Magnetic force on a current-carrying conductor
We can extend the analysis for force due to magnetic field on a single
moving charge to a straight rod carrying current. Consider a rod of a
uniform cross-sectional area A and length l. We shall assume one kind
of mobile carriers as in a conductor (here electrons). Let the number
density of these mobile charge carriers in it be n. Then the total number
of mobile charge carriers in it is nlA. For a steady current I in this
conducting rod, we may assume that each mobile carrier has an average
2015-16(20/01/2015)
Page 5


Physics
132
4.1  INTRODUCTION
Both Electricity and Magnetism have been known for more than 2000
years. However, it was only about 200 years ago, in 1820, that it was
realised that they were intimately related*. During a lecture demonstration
in the summer of 1820, the Danish physicist Hans Christian Oersted
noticed that a current in a straight wire caused a noticeable deflection in
a nearby magnetic compass needle. He investigated this phenomenon.
He found that the alignment of the needle is tangential to an imaginary
circle which has the straight wire as its centre and has its plane
perpendicular to the wire. This situation is depicted in Fig.4.1(a). It is
noticeable when the current is large and the needle sufficiently close to
the wire so that the earth’s magnetic field may be ignored. Reversing the
direction of the current reverses the orientation of the needle [Fig. 4.1(b)].
The deflection increases on increasing the current or bringing the needle
closer to the wire. Iron filings sprinkled around the wire arrange
themselves in concentric circles with the wire as the centre [Fig. 4.1(c)].
Oersted concluded that moving charges or currents produced a
magnetic field in the surrounding space.
Following this there was intense experimentation. In 1864, the laws
obeyed by electricity and magnetism were unified and formulated by
Chapter Four
MOVING CHARGES
AND MAGNETISM
* See the box in Chapter 1, Page 3.
2015-16(20/01/2015)
Moving Charges and
Magnetism
133
James Maxwell who then realised that light was electromagnetic waves.
Radio waves were discovered by Hertz, and produced by  J.C.Bose and
G. Marconi by the end of the 19
th
 century. A  remarkable scientific and
technological progress has taken place in the 20
th
 century. This is due to
our increased understanding of electromagnetism and the invention of
devices  for production, amplification, transmission and detection of
electromagnetic waves.
In this chapter, we will see how magnetic field exerts
forces on moving charged particles, like electrons,
protons, and current-carrying wires. We shall also learn
how currents produce magnetic fields. We shall see how
particles can be accelerated to very high energies in a
cyclotron. We shall study how currents and voltages are
detected by a galvanometer.
In this and subsequent Chapter on magnetism,
we adopt the following convention:  A current or a
field (electric or magnetic) emerging out of the plane of the
paper is depicted by a dot (¤). A current or a field going
into the plane of the paper is depicted by a cross ( ? )*.
Figures. 4.1(a) and 4.1(b) correspond to these two
situations, respectively.
4.2  MAGNETIC FORCE
4.2.1  Sources and fields
Before we introduce the concept of a magnetic field B, we
shall recapitulate what we have learnt in Chapter 1 about
the electric field E. We have seen that the interaction
between two charges can be considered in two stages.
The charge Q, the source of the field, produces an electric
field E, where
FIGURE 4.1 The magnetic field due to a straight long current-carrying
wire. The wire is perpendicular to the plane of the paper. A ring  of
compass needles surrounds the wire. The orientation of the needles is
shown when (a) the current emerges out of the plane of the paper,
(b) the current moves into the plane of the paper. (c) The arrangement of
iron filings around the wire. The darkened ends of the needle represent
north poles. The effect of the earth’s magnetic field is neglected.
Hans Christian Oersted
(1777–1851) Danish
physicist and chemist,
professor at Copenhagen.
He observed that a
compass needle suffers a
deflection when placed
near a wire carrying an
electric current. This
discovery gave the first
empirical evidence of a
connection between electric
and magnetic phenomena.
HANS CHRISTIAN OERSTED  (1777–1851)
* A dot appears like the tip of an arrow pointed at you, a cross is like the feathered
tail of an arrow moving away from you.
2015-16(20/01/2015)
Physics
134
E = Q ˆ r / (4pe e e e e
0
)r
2
(4.1)
where ˆ r is unit vector along r,  and  the field E is a vector
field. A charge q interacts with this field and experiences
a force F given by
     F  =  q  E   = q Q ˆ r  / (4pe
0
) r
2
(4.2)
As pointed out in the Chapter 1, the field  E is not
just an artefact but has a physical role. It can  convey
energy and momentum and is not established
instantaneously but takes finite time to propagate. The
concept of a field was specially stressed by Faraday  and
was incorporated by Maxwell in his unification of
electricity and magnetism. In addition to depending on
each point in space, it can also vary with time, i.e., be a
function of time.  In our discussions in this chapter, we
will assume that the fields do not change with time.
The field at a particular point can be due to one or
more charges.  If there are more charges the fields add
vectorially. You have already learnt in Chapter 1 that this
is called the principle of superposition. Once the field is
known, the force on a test charge is given by Eq. (4.2).
Just as static charges produce an electric field, the
currents or moving charges produce (in addition) a
magnetic field, denoted by B (r), again a vector field. It
has several basic properties identical to the electric field.
It is defined at each point in space (and can in addition
depend on time). Experimentally, it is found to obey the
principle of superposition: the magnetic field of several
sources is the vector addition of magnetic field of each
individual source.
4.2.2  Magnetic Field,  Lorentz  Force
Let us suppose that there is  a point charge q (moving
with a velocity v and, located at r at a given time t) in
presence of both the electric field E (r) and the magnetic
field B (r).  The force on an electric charge q due to both of
them can be written as
F   = q [ E (r) +  v × B (r)] = F
electric
 +F
magnetic
(4.3)
This force was given first  by H.A. Lorentz based on the extensive
experiments of Ampere and others. It is called the Lorentz force. You
have already studied in detail the force due to the electric field.  If we
look at the interaction with the magnetic field, we find the following
features.
(i) It depends on q, v and B (charge of the particle, the velocity and the
magnetic field). Force on a negative charge is opposite to that on a
positive charge.
(ii) The magnetic force q [ v × B ] includes a vector product of velocity
and magnetic field. The vector product makes the force due to magnetic
HENDRIK ANTOON LORENTZ (1853 – 1928)
Hendrik Antoon Lorentz
(1853 – 1928) Dutch
theoretical physicist,
professor at Leiden. He
investigated the
relationship between
electricity, magnetism, and
mechanics. In order to
explain the observed effect
of magnetic fields on
emitters of light (Zeeman
effect), he postulated the
existence of electric charges
in the atom, for which he
was awarded the Nobel Prize
in 1902. He derived a set of
transformation equations
(known after him, as
Lorentz transformation
equations) by some tangled
mathematical arguments,
but he was not aware that
these equations hinge on a
new concept of space and
time.
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Moving Charges and
Magnetism
135
field vanish (become zero) if  velocity and magnetic field are parallel
or anti-parallel. The force acts in a (sideways) direction perpendicular
to both the velocity and the magnetic field.
Its direction  is given by the screw rule or
right hand rule  for vector (or cross) product
as illustrated in Fig. 4.2.
(iii) The magnetic force is zero if  charge is not
moving (as then |v|= 0).  Only a moving
charge feels the magnetic force.
The expression for the magnetic force helps
us to define the unit of the magnetic field, if
one takes q, F and v, all to be unity in the force
equation F = q [ v × B] =q v B sin ? ˆ n ,  where ?
is the angle between v and B [see  Fig. 4.2 (a)].
The magnitude of magnetic field B is 1 SI unit,
when the force acting on a unit charge (1 C),
moving perpendicular to B with a speed 1m/s,
is one newton.
Dimensionally, we have [B] = [F/qv] and the unit
of B are Newton second / (coulomb metre).  This
unit is called tesla (T) named after Nikola Tesla
(1856 – 1943). Tesla is a rather large unit. A smaller  unit (non-SI) called
gauss (=10
–4
 tesla) is also often used. The earth’s magnetic field is about
3.6 × 10
–5
 T. Table  4.1 lists magnetic fields over a wide range in the
universe.
FIGURE 4.2 The direction of the magnetic
force acting on a charged particle. (a) The
force on a positively charged particle with
velocity v and making an angle ? with the
magnetic field B is given by the right-hand
rule. (b) A moving charged particle q is
deflected in an opposite sense to –q in the
presence of magnetic field.
TABLE 4.1 ORDER OF MAGNITUDES OF MAGNETIC FIELDS IN A VARIETY OF PHYSICAL SITUATIONS
Physical situation Magnitude of B (in tesla)
Surface of a neutron star 10
8
Typical large field in a laboratory   1
Near a small bar magnet 10
–2
On the earth’s surface 10
–5
Human nerve fibre 10
–10
Interstellar space 10
–12
4.2.3  Magnetic force on a current-carrying conductor
We can extend the analysis for force due to magnetic field on a single
moving charge to a straight rod carrying current. Consider a rod of a
uniform cross-sectional area A and length l. We shall assume one kind
of mobile carriers as in a conductor (here electrons). Let the number
density of these mobile charge carriers in it be n. Then the total number
of mobile charge carriers in it is nlA. For a steady current I in this
conducting rod, we may assume that each mobile carrier has an average
2015-16(20/01/2015)
Physics
136 EXAMPLE 4.1
drift velocity v
d
 (see Chapter 3). In the presence of an external magnetic
field B, the force on these carriers is:
F = (nlA)q v
d
 ×× × × × × B
where q is the value of the charge on  a carrier.  Now nqv
d
 is the current
density j and |(nq v
d
)|A is the current I (see Chapter 3 for the discussion
of current and current density). Thus,
F = [(nqv
d 
)l A] × B = [ jAl ] × × × × × B
   = Il × × × × × B (4.4)
where l is a vector of magnitude l, the length of the rod, and with a direction
identical to the current I. Note that the current I is not a vector. In the last
step leading to Eq. (4.4), we have transferred the vector sign from  j to I.
Equation (4.4) holds for a straight rod. In this equation, B is the external
magnetic field. It is not the field produced by the current-carrying rod. If
the wire has an arbitrary shape we can calculate the Lorentz force on it
by considering it as a collection of linear strips dl
j
 and summing
j
j
Id × =
?
F B l
This summation can be converted to an integral in most cases.
ON PERMITTIVITY AND PERMEABILITY
In the universal law of gravitation, we say that any two point masses exert a force on
each other which is proportional to the product of the masses m
1
, m
2
 and inversely
proportional to the square of the distance r between them. We write it as F = Gm
1
m
2
/r
2
where G is the universal constant of gravitation. Similarly in Coulomb’s law of electrostatics
we write the force between two point charges q
1
, q
2
, separated by a distance r as
F = kq
1
q
2
/r
2
 where k is a constant of proportionality. In SI units, k is taken as
1/4pe where e is the permittivity of the medium. Also in magnetism, we get another
constant, which in SI units, is taken as µ/4p where µ is the permeability of the medium.
Although G, e and µ arise as proportionality constants, there is a difference between
gravitational force and electromagnetic force. While the gravitational force does not depend
on the intervening medium, the electromagnetic force depends on the medium between
the two charges or magnets. Hence while G is a universal constant, e and µ depend on
the medium. They have different values for different media. The product eµ turns out to
be related to the speed v of electromagnetic radiation in the medium through eµ =1/ v
 2
.
Electric permittivity e is a physical quantity that describes how an electric field affects
and is affected by a medium. It is determined by the ability of a material to polarise in
response to an applied field, and thereby to cancel, partially, the field inside the material.
Similarly, magnetic permeability µ is the ability of a substance to acquire magnetisation in
magnetic fields. It is a measure of the extent to which magnetic field can penetrate matter.
Example 4.1 A straight wire of mass 200 g and length 1.5 m carries
a current of 2 A. It is suspended in mid-air by a uniform horizontal
magnetic field B (Fig. 4.3). What is the magnitude of the magnetic
field?
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