NCERT Textbook - Current Electricity Class 12 Notes | EduRev

Physics Class 12

Created by: Infinity Academy

Class 12 : NCERT Textbook - Current Electricity Class 12 Notes | EduRev

 Page 1


3.1  INTRODUCTION
In Chapter 1, all charges whether free or bound, were considered to be at
rest. Charges in motion constitute an electric current. Such currents occur
naturally in many situations. Lightning is one such phenomenon in
which charges flow from the clouds to the earth through the atmosphere,
sometimes with disastrous results. The flow of charges in lightning is not
steady, but in our everyday life we see many devices where charges flow
in a steady manner, like water flowing smoothly in a river. A torch and a
cell-driven clock are examples of such devices. In the present chapter, we
shall study some of the basic laws concerning steady electric currents.
3.2  ELECTRIC CURRENT
Imagine a small area held normal to the direction of flow of charges. Both
the positive and the negative charges may flow forward and backward
across the area. In a given time interval t, let q
+
 be the net amount (i.e.,
forward minus backward) of positive charge that flows in the forward
direction across the area. Similarly, let q
–
 be the net amount of negative
charge flowing across the area in the forward direction. The net amount
of charge flowing across the area in the forward direction in the time
interval t, then, is q = q
+
– q
–
. This is proportional to t for steady current
Chapter Three
CURRENT
ELECTRICITY
2015-16(20/01/2015)
Page 2


3.1  INTRODUCTION
In Chapter 1, all charges whether free or bound, were considered to be at
rest. Charges in motion constitute an electric current. Such currents occur
naturally in many situations. Lightning is one such phenomenon in
which charges flow from the clouds to the earth through the atmosphere,
sometimes with disastrous results. The flow of charges in lightning is not
steady, but in our everyday life we see many devices where charges flow
in a steady manner, like water flowing smoothly in a river. A torch and a
cell-driven clock are examples of such devices. In the present chapter, we
shall study some of the basic laws concerning steady electric currents.
3.2  ELECTRIC CURRENT
Imagine a small area held normal to the direction of flow of charges. Both
the positive and the negative charges may flow forward and backward
across the area. In a given time interval t, let q
+
 be the net amount (i.e.,
forward minus backward) of positive charge that flows in the forward
direction across the area. Similarly, let q
–
 be the net amount of negative
charge flowing across the area in the forward direction. The net amount
of charge flowing across the area in the forward direction in the time
interval t, then, is q = q
+
– q
–
. This is proportional to t for steady current
Chapter Three
CURRENT
ELECTRICITY
2015-16(20/01/2015)
Physics
94
and the quotient
q
I
t
= (3.1)
is defined to be the current across the area in the forward direction. (If it
turn out to be a negative number, it implies a current in the backward
direction.)
Currents are not always steady and hence more generally, we define
the current as follows. Let ?Q be the net charge flowing across a cross-
section of a conductor during the time interval ?t [i.e., between times t
and (t + ?t)]. Then, the current at time t across the cross-section of the
conductor is defined as the value of the ratio of ?Q to ?t in the limit of ?t
tending to zero,
( )
0
lim
t
Q
I t
t
? ?
?
=
?
(3.2)
In SI units, the unit of current is ampere. An ampere is defined
through magnetic effects of currents that we will study in the following
chapter. An ampere is typically the order of magnitude of currents in
domestic appliances. An average lightning carries currents of the order
of tens of thousands of amperes and at the other extreme, currents in
our nerves are in microamperes.
3.3 ELECTRIC CURRENTS IN CONDUCTORS
An electric charge will experience a force if an electric field is applied. If it is
free to move, it will thus move contributing to a current. In nature, free
charged particles do exist like in upper strata of atmosphere called the
ionosphere. However, in atoms and molecules, the negatively charged
electrons and the positively charged nuclei are bound to each other and
are thus not free to move. Bulk matter is made up of many molecules, a
gram of water, for example, contains approximately 10
22
 molecules. These
molecules are so closely packed that the electrons are no longer attached
to individual nuclei. In some materials, the electrons will still be bound,
i.e., they will not accelerate even if an electric field is applied. In other
materials, notably metals, some of the electrons are practically free to move
within the bulk material. These materials, generally called conductors,
develop electric currents in them when an electric field is applied.
If we consider solid conductors, then of course the atoms are tightly
bound to each other so that the current is carried by the negatively
charged electrons. There are, however, other types of conductors like
electrolytic solutions where positive and negative charges both can move.
In our discussions, we will focus only on solid conductors so that the
current is carried by the negatively charged electrons in the background
of fixed positive ions.
Consider first the case when no electric field is present. The electrons
will be moving due to thermal motion during which they collide with the
fixed ions. An electron colliding with an ion emerges with the same speed
as before the collision. However, the direction of its velocity after the
collision is completely random. At a given time, there is no preferential
direction for the velocities of the electrons. Thus on the average, the
2015-16(20/01/2015)
Page 3


3.1  INTRODUCTION
In Chapter 1, all charges whether free or bound, were considered to be at
rest. Charges in motion constitute an electric current. Such currents occur
naturally in many situations. Lightning is one such phenomenon in
which charges flow from the clouds to the earth through the atmosphere,
sometimes with disastrous results. The flow of charges in lightning is not
steady, but in our everyday life we see many devices where charges flow
in a steady manner, like water flowing smoothly in a river. A torch and a
cell-driven clock are examples of such devices. In the present chapter, we
shall study some of the basic laws concerning steady electric currents.
3.2  ELECTRIC CURRENT
Imagine a small area held normal to the direction of flow of charges. Both
the positive and the negative charges may flow forward and backward
across the area. In a given time interval t, let q
+
 be the net amount (i.e.,
forward minus backward) of positive charge that flows in the forward
direction across the area. Similarly, let q
–
 be the net amount of negative
charge flowing across the area in the forward direction. The net amount
of charge flowing across the area in the forward direction in the time
interval t, then, is q = q
+
– q
–
. This is proportional to t for steady current
Chapter Three
CURRENT
ELECTRICITY
2015-16(20/01/2015)
Physics
94
and the quotient
q
I
t
= (3.1)
is defined to be the current across the area in the forward direction. (If it
turn out to be a negative number, it implies a current in the backward
direction.)
Currents are not always steady and hence more generally, we define
the current as follows. Let ?Q be the net charge flowing across a cross-
section of a conductor during the time interval ?t [i.e., between times t
and (t + ?t)]. Then, the current at time t across the cross-section of the
conductor is defined as the value of the ratio of ?Q to ?t in the limit of ?t
tending to zero,
( )
0
lim
t
Q
I t
t
? ?
?
=
?
(3.2)
In SI units, the unit of current is ampere. An ampere is defined
through magnetic effects of currents that we will study in the following
chapter. An ampere is typically the order of magnitude of currents in
domestic appliances. An average lightning carries currents of the order
of tens of thousands of amperes and at the other extreme, currents in
our nerves are in microamperes.
3.3 ELECTRIC CURRENTS IN CONDUCTORS
An electric charge will experience a force if an electric field is applied. If it is
free to move, it will thus move contributing to a current. In nature, free
charged particles do exist like in upper strata of atmosphere called the
ionosphere. However, in atoms and molecules, the negatively charged
electrons and the positively charged nuclei are bound to each other and
are thus not free to move. Bulk matter is made up of many molecules, a
gram of water, for example, contains approximately 10
22
 molecules. These
molecules are so closely packed that the electrons are no longer attached
to individual nuclei. In some materials, the electrons will still be bound,
i.e., they will not accelerate even if an electric field is applied. In other
materials, notably metals, some of the electrons are practically free to move
within the bulk material. These materials, generally called conductors,
develop electric currents in them when an electric field is applied.
If we consider solid conductors, then of course the atoms are tightly
bound to each other so that the current is carried by the negatively
charged electrons. There are, however, other types of conductors like
electrolytic solutions where positive and negative charges both can move.
In our discussions, we will focus only on solid conductors so that the
current is carried by the negatively charged electrons in the background
of fixed positive ions.
Consider first the case when no electric field is present. The electrons
will be moving due to thermal motion during which they collide with the
fixed ions. An electron colliding with an ion emerges with the same speed
as before the collision. However, the direction of its velocity after the
collision is completely random. At a given time, there is no preferential
direction for the velocities of the electrons. Thus on the average, the
2015-16(20/01/2015)
Current
Electricity
95
number of electrons travelling in any direction will be equal to the number
of electrons travelling in the opposite direction. So, there will be no net
electric current.
Let us now see what happens to such a
piece of conductor if an electric field is applied.
To focus our thoughts, imagine the conductor
in the shape of a cylinder of radius R (Fig. 3.1).
Suppose we now take two thin circular discs
of a dielectric of the same radius and put
positive charge +Q distributed over one disc
and similarly –Q at the other disc. We attach
the two discs on the two flat surfaces of the
cylinder. An electric field will be created and
is directed from the positive towards the
negative charge. The electrons will be accelerated due to this field towards
+Q. They will thus move to neutralise the charges. The electrons, as long
as they are moving, will constitute an electric current. Hence in the
situation considered, there will be a current for a very short while and no
current thereafter.
We can also imagine a mechanism where the ends of the cylinder are
supplied with fresh charges to make up for any charges neutralised by
electrons moving inside the conductor. In that case, there will be a steady
electric field in the body of the conductor. This will result in a continuous
current rather than a current for a short period of time. Mechanisms,
which maintain a steady electric field are cells or batteries that we shall
study  later in this chapter. In the next sections, we shall study the steady
current that results from a steady electric field in conductors.
3.4  OHM’S LAW
A basic law regarding flow of currents was discovered by G.S. Ohm in
1828, long before the physical mechanism responsible for flow of currents
was discovered. Imagine a conductor through which a current I is flowing
and let V be the potential difference between the ends of the conductor.
Then Ohm’s law states that
     V ? I
or, V = R I (3.3)
where the constant of proportionality R is called the resistance of the
conductor. The SI units of resistance is ohm, and is denoted by the symbol
O. The resistance R not only depends on the material of the conductor
but also on the dimensions of the conductor. The dependence of R on the
dimensions of the conductor can easily be determined as follows.
Consider a conductor satisfying Eq. (3.3) to be in the form of a slab of
length l and cross sectional area A [Fig. 3.2(a)]. Imagine placing two such
identical slabs side by side [Fig. 3.2(b)], so that the length of the
combination is 2l. The current flowing through the  combination is the
same as that flowing through either of the slabs. If V is the potential
difference across the ends of the first slab, then V is also the potential
difference across the ends of the second slab since the second slab is
FIGURE 3.1 Charges +Q and –Q put at the ends
of a metallic cylinder. The electrons will drift
because of the electric field created to
neutralise the charges. The current thus
will stop after a while unless the charges +Q
and –Q are continuously replenished.
FIGURE 3.2
Illustrating the
relation R = ?l/A for
a rectangular slab
of length l and area
of cross-section A.
2015-16(20/01/2015)
Page 4


3.1  INTRODUCTION
In Chapter 1, all charges whether free or bound, were considered to be at
rest. Charges in motion constitute an electric current. Such currents occur
naturally in many situations. Lightning is one such phenomenon in
which charges flow from the clouds to the earth through the atmosphere,
sometimes with disastrous results. The flow of charges in lightning is not
steady, but in our everyday life we see many devices where charges flow
in a steady manner, like water flowing smoothly in a river. A torch and a
cell-driven clock are examples of such devices. In the present chapter, we
shall study some of the basic laws concerning steady electric currents.
3.2  ELECTRIC CURRENT
Imagine a small area held normal to the direction of flow of charges. Both
the positive and the negative charges may flow forward and backward
across the area. In a given time interval t, let q
+
 be the net amount (i.e.,
forward minus backward) of positive charge that flows in the forward
direction across the area. Similarly, let q
–
 be the net amount of negative
charge flowing across the area in the forward direction. The net amount
of charge flowing across the area in the forward direction in the time
interval t, then, is q = q
+
– q
–
. This is proportional to t for steady current
Chapter Three
CURRENT
ELECTRICITY
2015-16(20/01/2015)
Physics
94
and the quotient
q
I
t
= (3.1)
is defined to be the current across the area in the forward direction. (If it
turn out to be a negative number, it implies a current in the backward
direction.)
Currents are not always steady and hence more generally, we define
the current as follows. Let ?Q be the net charge flowing across a cross-
section of a conductor during the time interval ?t [i.e., between times t
and (t + ?t)]. Then, the current at time t across the cross-section of the
conductor is defined as the value of the ratio of ?Q to ?t in the limit of ?t
tending to zero,
( )
0
lim
t
Q
I t
t
? ?
?
=
?
(3.2)
In SI units, the unit of current is ampere. An ampere is defined
through magnetic effects of currents that we will study in the following
chapter. An ampere is typically the order of magnitude of currents in
domestic appliances. An average lightning carries currents of the order
of tens of thousands of amperes and at the other extreme, currents in
our nerves are in microamperes.
3.3 ELECTRIC CURRENTS IN CONDUCTORS
An electric charge will experience a force if an electric field is applied. If it is
free to move, it will thus move contributing to a current. In nature, free
charged particles do exist like in upper strata of atmosphere called the
ionosphere. However, in atoms and molecules, the negatively charged
electrons and the positively charged nuclei are bound to each other and
are thus not free to move. Bulk matter is made up of many molecules, a
gram of water, for example, contains approximately 10
22
 molecules. These
molecules are so closely packed that the electrons are no longer attached
to individual nuclei. In some materials, the electrons will still be bound,
i.e., they will not accelerate even if an electric field is applied. In other
materials, notably metals, some of the electrons are practically free to move
within the bulk material. These materials, generally called conductors,
develop electric currents in them when an electric field is applied.
If we consider solid conductors, then of course the atoms are tightly
bound to each other so that the current is carried by the negatively
charged electrons. There are, however, other types of conductors like
electrolytic solutions where positive and negative charges both can move.
In our discussions, we will focus only on solid conductors so that the
current is carried by the negatively charged electrons in the background
of fixed positive ions.
Consider first the case when no electric field is present. The electrons
will be moving due to thermal motion during which they collide with the
fixed ions. An electron colliding with an ion emerges with the same speed
as before the collision. However, the direction of its velocity after the
collision is completely random. At a given time, there is no preferential
direction for the velocities of the electrons. Thus on the average, the
2015-16(20/01/2015)
Current
Electricity
95
number of electrons travelling in any direction will be equal to the number
of electrons travelling in the opposite direction. So, there will be no net
electric current.
Let us now see what happens to such a
piece of conductor if an electric field is applied.
To focus our thoughts, imagine the conductor
in the shape of a cylinder of radius R (Fig. 3.1).
Suppose we now take two thin circular discs
of a dielectric of the same radius and put
positive charge +Q distributed over one disc
and similarly –Q at the other disc. We attach
the two discs on the two flat surfaces of the
cylinder. An electric field will be created and
is directed from the positive towards the
negative charge. The electrons will be accelerated due to this field towards
+Q. They will thus move to neutralise the charges. The electrons, as long
as they are moving, will constitute an electric current. Hence in the
situation considered, there will be a current for a very short while and no
current thereafter.
We can also imagine a mechanism where the ends of the cylinder are
supplied with fresh charges to make up for any charges neutralised by
electrons moving inside the conductor. In that case, there will be a steady
electric field in the body of the conductor. This will result in a continuous
current rather than a current for a short period of time. Mechanisms,
which maintain a steady electric field are cells or batteries that we shall
study  later in this chapter. In the next sections, we shall study the steady
current that results from a steady electric field in conductors.
3.4  OHM’S LAW
A basic law regarding flow of currents was discovered by G.S. Ohm in
1828, long before the physical mechanism responsible for flow of currents
was discovered. Imagine a conductor through which a current I is flowing
and let V be the potential difference between the ends of the conductor.
Then Ohm’s law states that
     V ? I
or, V = R I (3.3)
where the constant of proportionality R is called the resistance of the
conductor. The SI units of resistance is ohm, and is denoted by the symbol
O. The resistance R not only depends on the material of the conductor
but also on the dimensions of the conductor. The dependence of R on the
dimensions of the conductor can easily be determined as follows.
Consider a conductor satisfying Eq. (3.3) to be in the form of a slab of
length l and cross sectional area A [Fig. 3.2(a)]. Imagine placing two such
identical slabs side by side [Fig. 3.2(b)], so that the length of the
combination is 2l. The current flowing through the  combination is the
same as that flowing through either of the slabs. If V is the potential
difference across the ends of the first slab, then V is also the potential
difference across the ends of the second slab since the second slab is
FIGURE 3.1 Charges +Q and –Q put at the ends
of a metallic cylinder. The electrons will drift
because of the electric field created to
neutralise the charges. The current thus
will stop after a while unless the charges +Q
and –Q are continuously replenished.
FIGURE 3.2
Illustrating the
relation R = ?l/A for
a rectangular slab
of length l and area
of cross-section A.
2015-16(20/01/2015)
Physics
96
identical to the first and the same current I flows through
both. The potential difference across the ends of the
combination is clearly sum of the potential difference
across the two individual slabs and hence equals 2V. The
current through the combination is I and the resistance
of the combination R
C
 is [from Eq. (3.3)],
2
2
C
V
R R
I
= =
(3.4)
since V/I = R, the resistance of either of the slabs. Thus,
doubling the length of a conductor doubles the
resistance. In general, then resistance is proportional to
length,
R l ? (3.5)
Next, imagine dividing the slab into two by cutting it
lengthwise so that the slab can be considered as a
combination of two identical slabs of length l , but each
having a cross sectional  area of A/2 [Fig. 3.2(c)].
For a given voltage V across the slab, if I is the current
through the entire slab, then clearly the current flowing
through each of the two half-slabs is I/2. Since the
potential difference across the ends of the half-slabs is V,
i.e., the same as across the full slab, the resistance of each
of the half-slabs R
1
 is
1
2 2 .
( /2)
V V
R R
I I
= = =
(3.6)
Thus, halving the area of the cross-section of a conductor doubles
the resistance. In general, then the resistance R is inversely proportional
to the cross-sectional area,
1
R
A
?
(3.7)
Combining Eqs. (3.5) and (3.7), we have
l
R
A
?
(3.8)
and hence for a given conductor
l
R
A
? =
(3.9)
where the constant of proportionality ? depends on the material of the
conductor but not on its dimensions. ? is called resistivity.
Using the last equation, Ohm’s law reads
I l
V I R
A
?
= × =
(3.10)
Current per unit area (taken normal to the current), I/A, is called
current density and is denoted by j. The SI units of the current density
are A/m
2
. Further, if E is the magnitude of uniform electric field in the
conductor whose length is l, then the potential difference V across its
ends is El. Using these, the last equation reads
GEORG SIMON OHM (1787–1854)
Georg Simon Ohm  (1787–
1854) German physicist,
professor at Munich. Ohm
was led to his law by an
analogy between the
conduction of heat: the
electric field is analogous to
the temperature gradient,
and the electric current is
analogous to the heat flow.
2015-16(20/01/2015)
Page 5


3.1  INTRODUCTION
In Chapter 1, all charges whether free or bound, were considered to be at
rest. Charges in motion constitute an electric current. Such currents occur
naturally in many situations. Lightning is one such phenomenon in
which charges flow from the clouds to the earth through the atmosphere,
sometimes with disastrous results. The flow of charges in lightning is not
steady, but in our everyday life we see many devices where charges flow
in a steady manner, like water flowing smoothly in a river. A torch and a
cell-driven clock are examples of such devices. In the present chapter, we
shall study some of the basic laws concerning steady electric currents.
3.2  ELECTRIC CURRENT
Imagine a small area held normal to the direction of flow of charges. Both
the positive and the negative charges may flow forward and backward
across the area. In a given time interval t, let q
+
 be the net amount (i.e.,
forward minus backward) of positive charge that flows in the forward
direction across the area. Similarly, let q
–
 be the net amount of negative
charge flowing across the area in the forward direction. The net amount
of charge flowing across the area in the forward direction in the time
interval t, then, is q = q
+
– q
–
. This is proportional to t for steady current
Chapter Three
CURRENT
ELECTRICITY
2015-16(20/01/2015)
Physics
94
and the quotient
q
I
t
= (3.1)
is defined to be the current across the area in the forward direction. (If it
turn out to be a negative number, it implies a current in the backward
direction.)
Currents are not always steady and hence more generally, we define
the current as follows. Let ?Q be the net charge flowing across a cross-
section of a conductor during the time interval ?t [i.e., between times t
and (t + ?t)]. Then, the current at time t across the cross-section of the
conductor is defined as the value of the ratio of ?Q to ?t in the limit of ?t
tending to zero,
( )
0
lim
t
Q
I t
t
? ?
?
=
?
(3.2)
In SI units, the unit of current is ampere. An ampere is defined
through magnetic effects of currents that we will study in the following
chapter. An ampere is typically the order of magnitude of currents in
domestic appliances. An average lightning carries currents of the order
of tens of thousands of amperes and at the other extreme, currents in
our nerves are in microamperes.
3.3 ELECTRIC CURRENTS IN CONDUCTORS
An electric charge will experience a force if an electric field is applied. If it is
free to move, it will thus move contributing to a current. In nature, free
charged particles do exist like in upper strata of atmosphere called the
ionosphere. However, in atoms and molecules, the negatively charged
electrons and the positively charged nuclei are bound to each other and
are thus not free to move. Bulk matter is made up of many molecules, a
gram of water, for example, contains approximately 10
22
 molecules. These
molecules are so closely packed that the electrons are no longer attached
to individual nuclei. In some materials, the electrons will still be bound,
i.e., they will not accelerate even if an electric field is applied. In other
materials, notably metals, some of the electrons are practically free to move
within the bulk material. These materials, generally called conductors,
develop electric currents in them when an electric field is applied.
If we consider solid conductors, then of course the atoms are tightly
bound to each other so that the current is carried by the negatively
charged electrons. There are, however, other types of conductors like
electrolytic solutions where positive and negative charges both can move.
In our discussions, we will focus only on solid conductors so that the
current is carried by the negatively charged electrons in the background
of fixed positive ions.
Consider first the case when no electric field is present. The electrons
will be moving due to thermal motion during which they collide with the
fixed ions. An electron colliding with an ion emerges with the same speed
as before the collision. However, the direction of its velocity after the
collision is completely random. At a given time, there is no preferential
direction for the velocities of the electrons. Thus on the average, the
2015-16(20/01/2015)
Current
Electricity
95
number of electrons travelling in any direction will be equal to the number
of electrons travelling in the opposite direction. So, there will be no net
electric current.
Let us now see what happens to such a
piece of conductor if an electric field is applied.
To focus our thoughts, imagine the conductor
in the shape of a cylinder of radius R (Fig. 3.1).
Suppose we now take two thin circular discs
of a dielectric of the same radius and put
positive charge +Q distributed over one disc
and similarly –Q at the other disc. We attach
the two discs on the two flat surfaces of the
cylinder. An electric field will be created and
is directed from the positive towards the
negative charge. The electrons will be accelerated due to this field towards
+Q. They will thus move to neutralise the charges. The electrons, as long
as they are moving, will constitute an electric current. Hence in the
situation considered, there will be a current for a very short while and no
current thereafter.
We can also imagine a mechanism where the ends of the cylinder are
supplied with fresh charges to make up for any charges neutralised by
electrons moving inside the conductor. In that case, there will be a steady
electric field in the body of the conductor. This will result in a continuous
current rather than a current for a short period of time. Mechanisms,
which maintain a steady electric field are cells or batteries that we shall
study  later in this chapter. In the next sections, we shall study the steady
current that results from a steady electric field in conductors.
3.4  OHM’S LAW
A basic law regarding flow of currents was discovered by G.S. Ohm in
1828, long before the physical mechanism responsible for flow of currents
was discovered. Imagine a conductor through which a current I is flowing
and let V be the potential difference between the ends of the conductor.
Then Ohm’s law states that
     V ? I
or, V = R I (3.3)
where the constant of proportionality R is called the resistance of the
conductor. The SI units of resistance is ohm, and is denoted by the symbol
O. The resistance R not only depends on the material of the conductor
but also on the dimensions of the conductor. The dependence of R on the
dimensions of the conductor can easily be determined as follows.
Consider a conductor satisfying Eq. (3.3) to be in the form of a slab of
length l and cross sectional area A [Fig. 3.2(a)]. Imagine placing two such
identical slabs side by side [Fig. 3.2(b)], so that the length of the
combination is 2l. The current flowing through the  combination is the
same as that flowing through either of the slabs. If V is the potential
difference across the ends of the first slab, then V is also the potential
difference across the ends of the second slab since the second slab is
FIGURE 3.1 Charges +Q and –Q put at the ends
of a metallic cylinder. The electrons will drift
because of the electric field created to
neutralise the charges. The current thus
will stop after a while unless the charges +Q
and –Q are continuously replenished.
FIGURE 3.2
Illustrating the
relation R = ?l/A for
a rectangular slab
of length l and area
of cross-section A.
2015-16(20/01/2015)
Physics
96
identical to the first and the same current I flows through
both. The potential difference across the ends of the
combination is clearly sum of the potential difference
across the two individual slabs and hence equals 2V. The
current through the combination is I and the resistance
of the combination R
C
 is [from Eq. (3.3)],
2
2
C
V
R R
I
= =
(3.4)
since V/I = R, the resistance of either of the slabs. Thus,
doubling the length of a conductor doubles the
resistance. In general, then resistance is proportional to
length,
R l ? (3.5)
Next, imagine dividing the slab into two by cutting it
lengthwise so that the slab can be considered as a
combination of two identical slabs of length l , but each
having a cross sectional  area of A/2 [Fig. 3.2(c)].
For a given voltage V across the slab, if I is the current
through the entire slab, then clearly the current flowing
through each of the two half-slabs is I/2. Since the
potential difference across the ends of the half-slabs is V,
i.e., the same as across the full slab, the resistance of each
of the half-slabs R
1
 is
1
2 2 .
( /2)
V V
R R
I I
= = =
(3.6)
Thus, halving the area of the cross-section of a conductor doubles
the resistance. In general, then the resistance R is inversely proportional
to the cross-sectional area,
1
R
A
?
(3.7)
Combining Eqs. (3.5) and (3.7), we have
l
R
A
?
(3.8)
and hence for a given conductor
l
R
A
? =
(3.9)
where the constant of proportionality ? depends on the material of the
conductor but not on its dimensions. ? is called resistivity.
Using the last equation, Ohm’s law reads
I l
V I R
A
?
= × =
(3.10)
Current per unit area (taken normal to the current), I/A, is called
current density and is denoted by j. The SI units of the current density
are A/m
2
. Further, if E is the magnitude of uniform electric field in the
conductor whose length is l, then the potential difference V across its
ends is El. Using these, the last equation reads
GEORG SIMON OHM (1787–1854)
Georg Simon Ohm  (1787–
1854) German physicist,
professor at Munich. Ohm
was led to his law by an
analogy between the
conduction of heat: the
electric field is analogous to
the temperature gradient,
and the electric current is
analogous to the heat flow.
2015-16(20/01/2015)
Current
Electricity
97
        E l = j ? l
or,    E = j ? (3.11)
The above relation for magnitudes E and j can indeed be cast in a
vector  form. The current density, (which we have defined as the current
through unit area normal to the current) is also directed along E, and is
also a vector j (= = = = = j E/E). Thus, the last equation can be written as,
        E = j? (3.12)
or,   j = s E (3.13)
where s =1/? is called the conductivity. Ohm’s law is often stated in an
equivalent form, Eq. (3.13) in addition to Eq.(3.3). In the next section, we
will try to understand the origin of the Ohm’s law as arising from the
characteristics of the drift of electrons.
3.5 DRIFT OF ELECTRONS AND THE ORIGIN OF
RESISTIVITY
As remarked before, an electron will suffer collisions with the heavy fixed
ions, but after collision, it will emerge with the same speed but in random
directions. If we consider all the electrons, their average velocity will be
zero since their directions are random. Thus, if there are N electrons and
the velocity of the i
th
 electron (i = 1, 2, 3, ... N ) at a given time is v
i
, then
1
1
0
N
i
i
N
=
=
?
v
(3.14)
Consider now the situation when an electric field is
present. Electrons will be accelerated due to this
field by
–e
m
=
E
a
(3.15)
where –e is the charge and m is the mass of an electron.
Consider again the i
th
 electron at a given time t. This
electron would have had its last collision some time
before t, and let t
i
 be the time elapsed after its last
collision. If v
i
 was its velocity immediately after the last
collision, then its velocity V
i
 at time t is
–
i i i
e
t
m
=
E
V v
(3.16)
since starting with its last collision it was accelerated
(Fig. 3.3) with an acceleration given by Eq. (3.15) for a
time interval t
i
. The average velocity of the electrons at
time t is the average of all the V
i
’s. The average of v
i
’s is
zero [Eq. (3.14)] since immediately after any collision,
the direction of the velocity of an electron is completely
random. The collisions of the electrons do not occur at
regular intervals but at random times. Let us denote by
t, the average time between successive collisions. Then
at a given time, some of the electrons would have spent
FIGURE 3.3 A schematic picture of
an electron moving from a point A to
another point B through repeated
collisions, and straight line travel
between collisions (full lines). If an
electric field is applied as shown, the
electron ends up at point B' (dotted
lines). A slight drift in a direction
opposite the electric field is visible.
2015-16(20/01/2015)
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