Capacitance (NCERT)

# Capacitance (NCERT)

``` Page 1

2.1  INTRODUCTION
In Chapters 6 and 8 (Class XI), the notion of potential energy was
introduced. When an external force does work in taking a body from a
point to another against a force like spring force or gravitational force,
that work gets stored as potential energy of the body. When the external
force is removed, the body moves, gaining kinetic energy and losing
an equal amount of potential energy. The sum of kinetic and
potential energies is thus conserved. Forces of this kind are called
conservative forces. Spring force and gravitational force are examples of
conservative forces.
Coulomb force between two (stationary) charges is also a conservative
force. This is not surprising, since both have inverse-square dependence
on distance and differ mainly in the proportionality constants – the
masses in the gravitational law are replaced by charges in Coulomb’s
law. Thus, like the potential energy of a mass in a gravitational
field, we can define electrostatic potential energy of a charge in an
electrostatic field.
Consider an electrostatic field E due to some charge configuration.
First, for simplicity, consider the field E due to a charge Q placed at the
origin. Now, imagine that we bring a test charge q from a point R to a
point P against the repulsive force on it due to the charge Q. With reference
Chapter Two
ELECTROSTATIC
POTENTIAL AND
CAPACITANCE
Page 2

2.1  INTRODUCTION
In Chapters 6 and 8 (Class XI), the notion of potential energy was
introduced. When an external force does work in taking a body from a
point to another against a force like spring force or gravitational force,
that work gets stored as potential energy of the body. When the external
force is removed, the body moves, gaining kinetic energy and losing
an equal amount of potential energy. The sum of kinetic and
potential energies is thus conserved. Forces of this kind are called
conservative forces. Spring force and gravitational force are examples of
conservative forces.
Coulomb force between two (stationary) charges is also a conservative
force. This is not surprising, since both have inverse-square dependence
on distance and differ mainly in the proportionality constants – the
masses in the gravitational law are replaced by charges in Coulomb’s
law. Thus, like the potential energy of a mass in a gravitational
field, we can define electrostatic potential energy of a charge in an
electrostatic field.
Consider an electrostatic field E due to some charge configuration.
First, for simplicity, consider the field E due to a charge Q placed at the
origin. Now, imagine that we bring a test charge q from a point R to a
point P against the repulsive force on it due to the charge Q. With reference
Chapter Two
ELECTROSTATIC
POTENTIAL AND
CAPACITANCE
Physics
52
to Fig. 2.1, this will happen if Q and q are both positive
or both negative. For definiteness, let us take Q, q > 0.
Two remarks may be made here. First, we assume
that the test charge q is so small that it does not disturb
the original configuration, namely the charge Q at the
origin (or else, we keep Q fixed at the origin by some
unspecified force). Second, in bringing the charge q from
R to P, we apply an external force F
ext
just enough to
counter the repulsive electric force F
E
(i.e, F
ext
= –F
E
).
This means there is no net force on or acceleration of
the charge q when it is brought from R to P, i.e., it is
brought with infinitesimally slow constant speed. In
this situation, work done by the external force is the negative of the work
done by the electric force, and gets fully stored in the form of potential
energy of the charge q. If the external force is removed on reaching P, the
electric force will take the charge away from Q – the stored energy (potential
energy) at P is used to provide kinetic energy to the charge q in such a
way that the sum of the kinetic and potential energies is conserved.
Thus, work done by external forces in moving a charge q from R to P is
W
RP
=
P
R
d
ext ?
F r i
=
P
R
d
E
-
?
F r i
(2.1)
This work done is against electrostatic repulsive force and gets stored
as potential energy.
At every point in electric field, a particle with charge q possesses a
certain electrostatic potential energy, this work done increases its potential
energy by an amount equal to potential energy difference between points
R and P.
Thus, potential energy difference
P R RP
U U U W ? = - = (2.2)
(Note here that this displacement is in an opposite sense to the electric
force and hence work done by electric field is negative, i.e., –W
RP
.)
Therefore, we can define electric potential energy difference between
two points as the work required to be done by an external force in moving
(without accelerating ) charge q from one point to another for electric field
of any arbitrary charge configuration.
(i) The right side of Eq. (2.2) depends only on the initial and final positions
of the charge. It means that the work done by an electrostatic field in
moving a charge from one point to another depends only on the initial
and the final points and is independent of the path taken to go from
one point to the other. This is the fundamental characteristic of a
conservative force. The concept of the potential energy would not be
meaningful if the work depended on the path. The path-independence
of work done by an electrostatic field can be proved using the
Coulomb’s law. We omit this proof here.
FIGURE 2.1 A test charge q (> 0) is
moved from the point R to the
point P against the repulsive
force on it by the charge Q (> 0)
placed at the origin.
Page 3

2.1  INTRODUCTION
In Chapters 6 and 8 (Class XI), the notion of potential energy was
introduced. When an external force does work in taking a body from a
point to another against a force like spring force or gravitational force,
that work gets stored as potential energy of the body. When the external
force is removed, the body moves, gaining kinetic energy and losing
an equal amount of potential energy. The sum of kinetic and
potential energies is thus conserved. Forces of this kind are called
conservative forces. Spring force and gravitational force are examples of
conservative forces.
Coulomb force between two (stationary) charges is also a conservative
force. This is not surprising, since both have inverse-square dependence
on distance and differ mainly in the proportionality constants – the
masses in the gravitational law are replaced by charges in Coulomb’s
law. Thus, like the potential energy of a mass in a gravitational
field, we can define electrostatic potential energy of a charge in an
electrostatic field.
Consider an electrostatic field E due to some charge configuration.
First, for simplicity, consider the field E due to a charge Q placed at the
origin. Now, imagine that we bring a test charge q from a point R to a
point P against the repulsive force on it due to the charge Q. With reference
Chapter Two
ELECTROSTATIC
POTENTIAL AND
CAPACITANCE
Physics
52
to Fig. 2.1, this will happen if Q and q are both positive
or both negative. For definiteness, let us take Q, q > 0.
Two remarks may be made here. First, we assume
that the test charge q is so small that it does not disturb
the original configuration, namely the charge Q at the
origin (or else, we keep Q fixed at the origin by some
unspecified force). Second, in bringing the charge q from
R to P, we apply an external force F
ext
just enough to
counter the repulsive electric force F
E
(i.e, F
ext
= –F
E
).
This means there is no net force on or acceleration of
the charge q when it is brought from R to P, i.e., it is
brought with infinitesimally slow constant speed. In
this situation, work done by the external force is the negative of the work
done by the electric force, and gets fully stored in the form of potential
energy of the charge q. If the external force is removed on reaching P, the
electric force will take the charge away from Q – the stored energy (potential
energy) at P is used to provide kinetic energy to the charge q in such a
way that the sum of the kinetic and potential energies is conserved.
Thus, work done by external forces in moving a charge q from R to P is
W
RP
=
P
R
d
ext ?
F r i
=
P
R
d
E
-
?
F r i
(2.1)
This work done is against electrostatic repulsive force and gets stored
as potential energy.
At every point in electric field, a particle with charge q possesses a
certain electrostatic potential energy, this work done increases its potential
energy by an amount equal to potential energy difference between points
R and P.
Thus, potential energy difference
P R RP
U U U W ? = - = (2.2)
(Note here that this displacement is in an opposite sense to the electric
force and hence work done by electric field is negative, i.e., –W
RP
.)
Therefore, we can define electric potential energy difference between
two points as the work required to be done by an external force in moving
(without accelerating ) charge q from one point to another for electric field
of any arbitrary charge configuration.
(i) The right side of Eq. (2.2) depends only on the initial and final positions
of the charge. It means that the work done by an electrostatic field in
moving a charge from one point to another depends only on the initial
and the final points and is independent of the path taken to go from
one point to the other. This is the fundamental characteristic of a
conservative force. The concept of the potential energy would not be
meaningful if the work depended on the path. The path-independence
of work done by an electrostatic field can be proved using the
Coulomb’s law. We omit this proof here.
FIGURE 2.1 A test charge q (> 0) is
moved from the point R to the
point P against the repulsive
force on it by the charge Q (> 0)
placed at the origin.
Electrostatic Potential
and Capacitance
53
(ii) Equation (2.2) defines potential energy difference in terms
of the physically meaningful quantity work. Clearly,
potential energy so defined is undetermined to within an
additive constant.What this means is that the actual value
of potential energy is not physically significant; it is only
the difference of potential energy that is significant. We can
always add an arbitrary constant a to potential energy at
every point, since this will not change the potential energy
difference:
( ) ( )
P R P R
U U U U a a + - + = -
Put it differently, there is a freedom in choosing the point
where potential energy is zero. A convenient choice is to have
electrostatic potential energy zero at infinity. With this choice,
if we take the point R at infinity, we get from Eq. (2.2)
P P P
W U U U
8 8
= - = (2.3)
Since the point P is arbitrary, Eq. (2.3) provides us with a
definition of potential energy of a charge q at any point.
Potential energy of charge q at a point (in the presence of field
due to any charge configuration) is the work done by the
external force (equal and opposite to the electric force) in
bringing the charge q from infinity to that point.
2.2  ELECTROSTATIC POTENTIAL
Consider any general static charge configuration. We define
potential energy of a test charge q in terms of the work done
on the charge q. This work is obviously proportional to q, since
the force at any point is qE, where E is the electric field at that
point due to the given charge configuration. It is, therefore,
convenient to divide the work by the amount of charge q, so
that the resulting quantity is independent of q. In other words,
work done per unit test charge is characteristic of the electric
field associated with the charge configuration. This leads to
the idea of electrostatic potential V due to a given charge
configuration. From Eq. (2.1), we get:
Work done by external force in bringing a unit positive
charge from point R to P
= V
P
– V
R

P R
U U
q
? ? -
=
? ?
? ?
(2.4)
where V
P
and V
R
are the electrostatic potentials at P and R, respectively.
Note, as before, that it is not the actual value of potential but the potential
difference that is physically significant. If, as before, we choose the
potential to be zero at infinity, Eq. (2.4) implies:
Work done by an external force in bringing a unit positive charge
from infinity to a point = electrostatic potential (V ) at that point.
COUNT ALESSANDRO VOLTA (1745 –1827)
Count Alessandro Volta
(1745 – 1827) Italian
physicist, professor at
Pavia. Volta established
that the animal electri-
city observed by Luigi
Galvani, 1737–1798, in
experiments with frog
muscle tissue placed in
contact with dissimilar
metals, was not due to
any exceptional property
of animal tissues but
was also generated
whenever any wet body
was sandwiched between
dissimilar metals. This
led him to develop the
first voltaic pile, or
battery, consisting of a
large stack of moist disks
of cardboard (electro-
lyte) sandwiched
between disks of metal
(electrodes).
Page 4

2.1  INTRODUCTION
In Chapters 6 and 8 (Class XI), the notion of potential energy was
introduced. When an external force does work in taking a body from a
point to another against a force like spring force or gravitational force,
that work gets stored as potential energy of the body. When the external
force is removed, the body moves, gaining kinetic energy and losing
an equal amount of potential energy. The sum of kinetic and
potential energies is thus conserved. Forces of this kind are called
conservative forces. Spring force and gravitational force are examples of
conservative forces.
Coulomb force between two (stationary) charges is also a conservative
force. This is not surprising, since both have inverse-square dependence
on distance and differ mainly in the proportionality constants – the
masses in the gravitational law are replaced by charges in Coulomb’s
law. Thus, like the potential energy of a mass in a gravitational
field, we can define electrostatic potential energy of a charge in an
electrostatic field.
Consider an electrostatic field E due to some charge configuration.
First, for simplicity, consider the field E due to a charge Q placed at the
origin. Now, imagine that we bring a test charge q from a point R to a
point P against the repulsive force on it due to the charge Q. With reference
Chapter Two
ELECTROSTATIC
POTENTIAL AND
CAPACITANCE
Physics
52
to Fig. 2.1, this will happen if Q and q are both positive
or both negative. For definiteness, let us take Q, q > 0.
Two remarks may be made here. First, we assume
that the test charge q is so small that it does not disturb
the original configuration, namely the charge Q at the
origin (or else, we keep Q fixed at the origin by some
unspecified force). Second, in bringing the charge q from
R to P, we apply an external force F
ext
just enough to
counter the repulsive electric force F
E
(i.e, F
ext
= –F
E
).
This means there is no net force on or acceleration of
the charge q when it is brought from R to P, i.e., it is
brought with infinitesimally slow constant speed. In
this situation, work done by the external force is the negative of the work
done by the electric force, and gets fully stored in the form of potential
energy of the charge q. If the external force is removed on reaching P, the
electric force will take the charge away from Q – the stored energy (potential
energy) at P is used to provide kinetic energy to the charge q in such a
way that the sum of the kinetic and potential energies is conserved.
Thus, work done by external forces in moving a charge q from R to P is
W
RP
=
P
R
d
ext ?
F r i
=
P
R
d
E
-
?
F r i
(2.1)
This work done is against electrostatic repulsive force and gets stored
as potential energy.
At every point in electric field, a particle with charge q possesses a
certain electrostatic potential energy, this work done increases its potential
energy by an amount equal to potential energy difference between points
R and P.
Thus, potential energy difference
P R RP
U U U W ? = - = (2.2)
(Note here that this displacement is in an opposite sense to the electric
force and hence work done by electric field is negative, i.e., –W
RP
.)
Therefore, we can define electric potential energy difference between
two points as the work required to be done by an external force in moving
(without accelerating ) charge q from one point to another for electric field
of any arbitrary charge configuration.
(i) The right side of Eq. (2.2) depends only on the initial and final positions
of the charge. It means that the work done by an electrostatic field in
moving a charge from one point to another depends only on the initial
and the final points and is independent of the path taken to go from
one point to the other. This is the fundamental characteristic of a
conservative force. The concept of the potential energy would not be
meaningful if the work depended on the path. The path-independence
of work done by an electrostatic field can be proved using the
Coulomb’s law. We omit this proof here.
FIGURE 2.1 A test charge q (> 0) is
moved from the point R to the
point P against the repulsive
force on it by the charge Q (> 0)
placed at the origin.
Electrostatic Potential
and Capacitance
53
(ii) Equation (2.2) defines potential energy difference in terms
of the physically meaningful quantity work. Clearly,
potential energy so defined is undetermined to within an
additive constant.What this means is that the actual value
of potential energy is not physically significant; it is only
the difference of potential energy that is significant. We can
always add an arbitrary constant a to potential energy at
every point, since this will not change the potential energy
difference:
( ) ( )
P R P R
U U U U a a + - + = -
Put it differently, there is a freedom in choosing the point
where potential energy is zero. A convenient choice is to have
electrostatic potential energy zero at infinity. With this choice,
if we take the point R at infinity, we get from Eq. (2.2)
P P P
W U U U
8 8
= - = (2.3)
Since the point P is arbitrary, Eq. (2.3) provides us with a
definition of potential energy of a charge q at any point.
Potential energy of charge q at a point (in the presence of field
due to any charge configuration) is the work done by the
external force (equal and opposite to the electric force) in
bringing the charge q from infinity to that point.
2.2  ELECTROSTATIC POTENTIAL
Consider any general static charge configuration. We define
potential energy of a test charge q in terms of the work done
on the charge q. This work is obviously proportional to q, since
the force at any point is qE, where E is the electric field at that
point due to the given charge configuration. It is, therefore,
convenient to divide the work by the amount of charge q, so
that the resulting quantity is independent of q. In other words,
work done per unit test charge is characteristic of the electric
field associated with the charge configuration. This leads to
the idea of electrostatic potential V due to a given charge
configuration. From Eq. (2.1), we get:
Work done by external force in bringing a unit positive
charge from point R to P
= V
P
– V
R

P R
U U
q
? ? -
=
? ?
? ?
(2.4)
where V
P
and V
R
are the electrostatic potentials at P and R, respectively.
Note, as before, that it is not the actual value of potential but the potential
difference that is physically significant. If, as before, we choose the
potential to be zero at infinity, Eq. (2.4) implies:
Work done by an external force in bringing a unit positive charge
from infinity to a point = electrostatic potential (V ) at that point.
COUNT ALESSANDRO VOLTA (1745 –1827)
Count Alessandro Volta
(1745 – 1827) Italian
physicist, professor at
Pavia. Volta established
that the animal electri-
city observed by Luigi
Galvani, 1737–1798, in
experiments with frog
muscle tissue placed in
contact with dissimilar
metals, was not due to
any exceptional property
of animal tissues but
was also generated
whenever any wet body
was sandwiched between
dissimilar metals. This
led him to develop the
first voltaic pile, or
battery, consisting of a
large stack of moist disks
of cardboard (electro-
lyte) sandwiched
between disks of metal
(electrodes).
Physics
54
In other words, the electrostatic potential (V )
at any point in a region with electrostatic field is
the work done in bringing a unit positive
charge (without acceleration) from infinity to
that point.
The qualifying remarks made earlier regarding
potential energy also apply to the definition of
potential. To obtain the work done per unit test
charge, we should take an infinitesimal test charge
dq, obtain the work done dW in bringing it from
infinity to the point and determine the ratio
dW/dq. Also, the external force at every point of
the path is to be equal and opposite to the
electrostatic force on the test charge at that point.
2.3  POTENTIAL DUE TO A POINT CHARGE
Consider a point charge Q at the origin (Fig. 2.3). For definiteness, take Q
to be positive. We wish to determine the potential at any point P with
position vector r from the origin. For that we must
calculate the work done in bringing a unit positive
test charge from infinity to the point P. For Q > 0,
the work done against the repulsive force on the
test charge is positive. Since work done is
independent of the path, we choose a convenient
path – along the radial direction from infinity to
the point P.
At some intermediate point P' on the path, the
electrostatic force on a unit positive charge is
2
0
1
ˆ
4 '
Q
r e
×
'
p
r
(2.5)
where ˆ' r is the unit vector along OP'. Work done
against this force from  r' to r' + ?r' is
2
0
4 '
Q
W r
r e
? = - ? '
p
(2.6)
The negative sign appears because for ?r' < 0, ?W is positive . Total
work done (W) by the external force is obtained by integrating Eq. (2.6)
from r' = 8 to r' = r,
2
0 0 0
4 4 4 '
r
r
Q Q Q
W dr
r r r e e e 8
8
= - = = '
p p p '
? (2.7)
This, by definition is the potential at P due to the charge Q
0
( )
4
Q
V r
r e
=
p
(2.8)
FIGURE 2.2 Work done on a test charge q
by the electrostatic field due to any given
charge configuration is independent
of the path, and depends only on
its initial and final positions.
FIGURE 2.3 Work done in bringing a unit
positive test charge from infinity to the
point P, against the repulsive force of
charge Q (Q > 0), is the potential at P due to
the charge Q.
Page 5

2.1  INTRODUCTION
In Chapters 6 and 8 (Class XI), the notion of potential energy was
introduced. When an external force does work in taking a body from a
point to another against a force like spring force or gravitational force,
that work gets stored as potential energy of the body. When the external
force is removed, the body moves, gaining kinetic energy and losing
an equal amount of potential energy. The sum of kinetic and
potential energies is thus conserved. Forces of this kind are called
conservative forces. Spring force and gravitational force are examples of
conservative forces.
Coulomb force between two (stationary) charges is also a conservative
force. This is not surprising, since both have inverse-square dependence
on distance and differ mainly in the proportionality constants – the
masses in the gravitational law are replaced by charges in Coulomb’s
law. Thus, like the potential energy of a mass in a gravitational
field, we can define electrostatic potential energy of a charge in an
electrostatic field.
Consider an electrostatic field E due to some charge configuration.
First, for simplicity, consider the field E due to a charge Q placed at the
origin. Now, imagine that we bring a test charge q from a point R to a
point P against the repulsive force on it due to the charge Q. With reference
Chapter Two
ELECTROSTATIC
POTENTIAL AND
CAPACITANCE
Physics
52
to Fig. 2.1, this will happen if Q and q are both positive
or both negative. For definiteness, let us take Q, q > 0.
Two remarks may be made here. First, we assume
that the test charge q is so small that it does not disturb
the original configuration, namely the charge Q at the
origin (or else, we keep Q fixed at the origin by some
unspecified force). Second, in bringing the charge q from
R to P, we apply an external force F
ext
just enough to
counter the repulsive electric force F
E
(i.e, F
ext
= –F
E
).
This means there is no net force on or acceleration of
the charge q when it is brought from R to P, i.e., it is
brought with infinitesimally slow constant speed. In
this situation, work done by the external force is the negative of the work
done by the electric force, and gets fully stored in the form of potential
energy of the charge q. If the external force is removed on reaching P, the
electric force will take the charge away from Q – the stored energy (potential
energy) at P is used to provide kinetic energy to the charge q in such a
way that the sum of the kinetic and potential energies is conserved.
Thus, work done by external forces in moving a charge q from R to P is
W
RP
=
P
R
d
ext ?
F r i
=
P
R
d
E
-
?
F r i
(2.1)
This work done is against electrostatic repulsive force and gets stored
as potential energy.
At every point in electric field, a particle with charge q possesses a
certain electrostatic potential energy, this work done increases its potential
energy by an amount equal to potential energy difference between points
R and P.
Thus, potential energy difference
P R RP
U U U W ? = - = (2.2)
(Note here that this displacement is in an opposite sense to the electric
force and hence work done by electric field is negative, i.e., –W
RP
.)
Therefore, we can define electric potential energy difference between
two points as the work required to be done by an external force in moving
(without accelerating ) charge q from one point to another for electric field
of any arbitrary charge configuration.
(i) The right side of Eq. (2.2) depends only on the initial and final positions
of the charge. It means that the work done by an electrostatic field in
moving a charge from one point to another depends only on the initial
and the final points and is independent of the path taken to go from
one point to the other. This is the fundamental characteristic of a
conservative force. The concept of the potential energy would not be
meaningful if the work depended on the path. The path-independence
of work done by an electrostatic field can be proved using the
Coulomb’s law. We omit this proof here.
FIGURE 2.1 A test charge q (> 0) is
moved from the point R to the
point P against the repulsive
force on it by the charge Q (> 0)
placed at the origin.
Electrostatic Potential
and Capacitance
53
(ii) Equation (2.2) defines potential energy difference in terms
of the physically meaningful quantity work. Clearly,
potential energy so defined is undetermined to within an
additive constant.What this means is that the actual value
of potential energy is not physically significant; it is only
the difference of potential energy that is significant. We can
always add an arbitrary constant a to potential energy at
every point, since this will not change the potential energy
difference:
( ) ( )
P R P R
U U U U a a + - + = -
Put it differently, there is a freedom in choosing the point
where potential energy is zero. A convenient choice is to have
electrostatic potential energy zero at infinity. With this choice,
if we take the point R at infinity, we get from Eq. (2.2)
P P P
W U U U
8 8
= - = (2.3)
Since the point P is arbitrary, Eq. (2.3) provides us with a
definition of potential energy of a charge q at any point.
Potential energy of charge q at a point (in the presence of field
due to any charge configuration) is the work done by the
external force (equal and opposite to the electric force) in
bringing the charge q from infinity to that point.
2.2  ELECTROSTATIC POTENTIAL
Consider any general static charge configuration. We define
potential energy of a test charge q in terms of the work done
on the charge q. This work is obviously proportional to q, since
the force at any point is qE, where E is the electric field at that
point due to the given charge configuration. It is, therefore,
convenient to divide the work by the amount of charge q, so
that the resulting quantity is independent of q. In other words,
work done per unit test charge is characteristic of the electric
field associated with the charge configuration. This leads to
the idea of electrostatic potential V due to a given charge
configuration. From Eq. (2.1), we get:
Work done by external force in bringing a unit positive
charge from point R to P
= V
P
– V
R

P R
U U
q
? ? -
=
? ?
? ?
(2.4)
where V
P
and V
R
are the electrostatic potentials at P and R, respectively.
Note, as before, that it is not the actual value of potential but the potential
difference that is physically significant. If, as before, we choose the
potential to be zero at infinity, Eq. (2.4) implies:
Work done by an external force in bringing a unit positive charge
from infinity to a point = electrostatic potential (V ) at that point.
COUNT ALESSANDRO VOLTA (1745 –1827)
Count Alessandro Volta
(1745 – 1827) Italian
physicist, professor at
Pavia. Volta established
that the animal electri-
city observed by Luigi
Galvani, 1737–1798, in
experiments with frog
muscle tissue placed in
contact with dissimilar
metals, was not due to
any exceptional property
of animal tissues but
was also generated
whenever any wet body
was sandwiched between
dissimilar metals. This
led him to develop the
first voltaic pile, or
battery, consisting of a
large stack of moist disks
of cardboard (electro-
lyte) sandwiched
between disks of metal
(electrodes).
Physics
54
In other words, the electrostatic potential (V )
at any point in a region with electrostatic field is
the work done in bringing a unit positive
charge (without acceleration) from infinity to
that point.
The qualifying remarks made earlier regarding
potential energy also apply to the definition of
potential. To obtain the work done per unit test
charge, we should take an infinitesimal test charge
dq, obtain the work done dW in bringing it from
infinity to the point and determine the ratio
dW/dq. Also, the external force at every point of
the path is to be equal and opposite to the
electrostatic force on the test charge at that point.
2.3  POTENTIAL DUE TO A POINT CHARGE
Consider a point charge Q at the origin (Fig. 2.3). For definiteness, take Q
to be positive. We wish to determine the potential at any point P with
position vector r from the origin. For that we must
calculate the work done in bringing a unit positive
test charge from infinity to the point P. For Q > 0,
the work done against the repulsive force on the
test charge is positive. Since work done is
independent of the path, we choose a convenient
path – along the radial direction from infinity to
the point P.
At some intermediate point P' on the path, the
electrostatic force on a unit positive charge is
2
0
1
ˆ
4 '
Q
r e
×
'
p
r
(2.5)
where ˆ' r is the unit vector along OP'. Work done
against this force from  r' to r' + ?r' is
2
0
4 '
Q
W r
r e
? = - ? '
p
(2.6)
The negative sign appears because for ?r' < 0, ?W is positive . Total
work done (W) by the external force is obtained by integrating Eq. (2.6)
from r' = 8 to r' = r,
2
0 0 0
4 4 4 '
r
r
Q Q Q
W dr
r r r e e e 8
8
= - = = '
p p p '
? (2.7)
This, by definition is the potential at P due to the charge Q
0
( )
4
Q
V r
r e
=
p
(2.8)
FIGURE 2.2 Work done on a test charge q
by the electrostatic field due to any given
charge configuration is independent
of the path, and depends only on
its initial and final positions.
FIGURE 2.3 Work done in bringing a unit
positive test charge from infinity to the
point P, against the repulsive force of
charge Q (Q > 0), is the potential at P due to
the charge Q.
Electrostatic Potential
and Capacitance
55
EXAMPLE 2.1
Equation (2.8) is true for any
sign of the charge Q, though we
considered Q > 0 in its derivation.
For Q < 0, V < 0, i.e., work done (by
the external force) per unit positive
test charge in bringing it from
infinity to the point is negative. This
is equivalent to saying that work
done by the electrostatic force in
bringing the unit positive charge
form infinity to the point P is
positive. [This is as it should be,
since for Q < 0, the force on a unit
positive test charge is attractive, so
that the electrostatic force and the
displacement (from infinity to P) are
in the same direction.] Finally, we
note that Eq. (2.8) is consistent with
the choice that potential at infinity
be zero.
Figure (2.4) shows how the electrostatic potential ( ? 1/r) and the
electrostatic field ( ? 1/r
2
) varies with r.
Example 2.1
(a) Calculate the potential at a point P due to a charge of 4 × 10
–7
C
located 9 cm away.
(b) Hence obtain the work done in bringing a charge of 2 × 10
–9
C
from infinity to the point P. Does the answer depend on the path
along which the charge is brought?
Solution
(a)
= 4 × 10
4
V
(b) W = qV = 2 × 10
–9
C × 4 × 10
4
V
= 8 × 10
–5
J
No, work done will be path independent. Any arbitrary infinitesimal
path can be resolved into two perpendicular displacements: One along
r and another perpendicular to r. The work done corresponding to
the later will be zero.
2.4  POTENTIAL DUE TO AN ELECTRIC DIPOLE
As we learnt in the last chapter, an electric dipole consists of two charges
q and  –q separated by a (small) distance 2a. Its total charge is zero. It is
characterised by a dipole moment vector p whose magnitude is q × 2a
and which points in the direction from –q to q (Fig. 2.5). We also saw that
the electric field of a dipole at a point with position vector r depends not
just on the magnitude r, but also on the angle between r and p. Further,
FIGURE 2.4 Variation of potential V with r [in units of
(Q/4pe
0
) m
-1
] (blue curve) and field with r [in units
of (Q/4pe
0
) m
-2
] (black curve) for a point charge Q.
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