Page 1
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, 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
Rationalised 2023-24
Page 2
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, 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
Rationalised 2023-24
Physics
108
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 took place in the 20
th
century. This was 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.
* 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.
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)
Rationalised 2023-24
Page 3
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, 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
Rationalised 2023-24
Physics
108
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 took place in the 20
th
century. This was 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.
* 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.
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)
Rationalised 2023-24
109
Moving Charges and
Magnetism
E = Q ˆ r / (4pe
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)] º 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.
Rationalised 2023-24
Page 4
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, 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
Rationalised 2023-24
Physics
108
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 took place in the 20
th
century. This was 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.
* 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.
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)
Rationalised 2023-24
109
Moving Charges and
Magnetism
E = Q ˆ r / (4pe
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)] º 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.
Rationalised 2023-24
Physics
110
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 q ˆ n , where q 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.
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
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 nq v
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 = [(nq v
d
)lA] × 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 l.
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.
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 q 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.
Rationalised 2023-24
Page 5
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, 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
Rationalised 2023-24
Physics
108
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 took place in the 20
th
century. This was 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.
* 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.
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)
Rationalised 2023-24
109
Moving Charges and
Magnetism
E = Q ˆ r / (4pe
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)] º 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.
Rationalised 2023-24
Physics
110
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 q ˆ n , where q 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.
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
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 nq v
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 = [(nq v
d
)lA] × 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 l.
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.
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 q 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.
Rationalised 2023-24
111
Moving Charges and
Magnetism
EXAMPLE 4.1
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?
FIGURE 4.3
Solution From Eq. (4.4), we find that there is an upward force F, of
magnitude IlB,. For mid-air suspension, this must be balanced by
the force due to gravity:
m g = I lB
m g
B
I l
=
0.2 9.8
0.65 T
2 1.5
×
= =
×
Note that it would have been sufficient to specify m/l, the mass per
unit length of the wire. The earth’s magnetic field is approximately
4 × 10
–5
T and we have ignored it.
Example 4.2 If the magnetic field is parallel to the positive y-axis
and the charged particle is moving along the positive x-axis (Fig. 4.4),
which way would the Lorentz force be for (a) an electron (negative
charge), (b) a proton (positive charge).
FIGURE 4.4
Solution The velocity v of particle is along the x-axis, while B, the
magnetic field is along the y-axis, so v × B is along the z-axis (screw
rule or right-hand thumb rule). So, (a) for electron it will be along –z
axis. (b) for a positive charge (proton) the force is along +z axis.
EXAMPLE 4.2
Charged particles moving in a magnetic field.
Interactive demonstration:
http://www.phys.hawaii.edu/~teb/optics/java/partmagn/index.html
Rationalised 2023-24
Read More