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ELECTRIC POTENTIAL AND CAPACITORSPotential energy is defined only for conservative forces. In the space occupied by conservative forces every
point is associated with certain energy which is called the energy of position or potential energy. Potential
energy generally are of three types :
potential energy etc.
Change in potential energy : Change in potential energy between any two points is defined in the terms
of the work done by the associated conservative force in displacing the particle between these two points
without any change in kinetic energy.
∫−=−1
12
r
rUU
We can define a unique value of potential energy only by assigning some arbitrary value to a fixed point
called the reference point. Whenever and wherever possible, we take the reference point at infinite and
assume potential energy to be zero there,
∫∞−=r
rdFUrr
.
In case of conservative force (field) potential energy is equal to negative of work done in shifting the
body from reference position to given position.
This is why in shifting a particle in a conservative field (say gravitational or electric), if the particle
moves opposite to the field, work done by the field will be negative and so change in potential energy will
be positive i.e. potential energy will i
be positive and change in potential energy will be negative
We can define electric potential energy difference between two points as the work required to be done by
an external force in moving (without acceleration) a test charge q from one point to another in an electric
field.
∫∫ −==P
R
Eext
P
R
RP rd.Frd.FWrrrr
Potential energy difference U =∆
here work done by the electric field is negative i.e.
i) Work done to move a charge q depends only on initial and final point and not on the path taken. This
is the fundamental characteristics of conservative force.
ii) The actual value of potential energy is
energy that is significant.
If we take electric potential energy at infinity as zero ie R is at infinity we get
PPPP UUUUW =−=−= ∞∞ 0
21
CHAPTER - 2
ELECTRIC POTENTIAL AND CAPACITORSPotential energy is defined only for conservative forces. In the space occupied by conservative forces every
point is associated with certain energy which is called the energy of position or potential energy. Potential
energy generally are of three types : Elastic potential energy, Electric potential energy and Gravitational
Change in potential energy between any two points is defined in the terms
of the work done by the associated conservative force in displacing the particle between these two points
without any change in kinetic energy.
−=2
1
.r
WrdFrr
……(i)
e value of potential energy only by assigning some arbitrary value to a fixed point
called the reference point. Whenever and wherever possible, we take the reference point at infinite and
assume potential energy to be zero there, i.e. if take ∞=1r and rr =2 then from equation (i)
−= W
In case of conservative force (field) potential energy is equal to negative of work done in shifting the
body from reference position to given position.
This is why in shifting a particle in a conservative field (say gravitational or electric), if the particle
moves opposite to the field, work done by the field will be negative and so change in potential energy will
potential energy will increase. When the particle moves in the direction of field, work will
be positive and change in potential energy will be negative i.e. potential energy will decrease.
nergy difference between two points as the work required to be done by
an external force in moving (without acceleration) a test charge q from one point to another in an electric
RPRP WUU =−=
done by the electric field is negative i.e. - WRP
Work done to move a charge q depends only on initial and final point and not on the path taken. This
is the fundamental characteristics of conservative force.
The actual value of potential energy is not physically significant. It is only the difference in potential
If we take electric potential energy at infinity as zero ie R is at infinity we get W
ELECTRIC POTENTIAL AND CAPACITORS Potential energy is defined only for conservative forces. In the space occupied by conservative forces every
point is associated with certain energy which is called the energy of position or potential energy. Potential
Elastic potential energy, Electric potential energy and Gravitational
Change in potential energy between any two points is defined in the terms
of the work done by the associated conservative force in displacing the particle between these two points
e value of potential energy only by assigning some arbitrary value to a fixed point
called the reference point. Whenever and wherever possible, we take the reference point at infinite and
from equation (i)
In case of conservative force (field) potential energy is equal to negative of work done in shifting the
This is why in shifting a particle in a conservative field (say gravitational or electric), if the particle
moves opposite to the field, work done by the field will be negative and so change in potential energy will
ncrease. When the particle moves in the direction of field, work will
potential energy will decrease.
nergy difference between two points as the work required to be done by
an external force in moving (without acceleration) a test charge q from one point to another in an electric
Work done to move a charge q depends only on initial and final point and not on the path taken. This
not physically significant. It is only the difference in potential
RPRP UUW −=
potential energy of a charge Q at a point on an electric field is the work done by the external force in
bringing the charge q from infinity to that point.
Electrostatic Potential:
Electrostatic potential difference between two points (P&R) is defined as the work done by the external
force in bringing a unit positive charge form one point (R) to the other point (P)
Electrostatic potential difference = V
Potential is a scalar quantity unit is J/C or volt.
1 J/C = 1 Volt = V.
Smaller units: 1) micro volt ( =10
Electrostatic potential at a point is defined as the work done by an external force in bring a unit positive
charge from infinity to that point.
Electrostatic potential difference V∆
1. Give the physical concept of electric potential ?
The concept of potential plays an important role and it may be made clear as follows
Consider a region free of any electric charge. Imagine a small charge, +q being placed in the region. It
can be made to move about any where in the region without doing any work of electrical origin because
the charge does not experience any electric force.
Let a charge +Q be placed in the region before +q is introduced. Now, work has to be done to move the
charge +q from one point to another because it experiences electrostatic force of repulsion
presence of the charge Q in space endows every point in the medium with a property by virtue of which
work is to be done to move a charge up to this point. This property is called electrostatic potential.
Electrostatic potential of a body represents the degree of electrification of the body. It determines the
direction of flow of charge between two charged bodies placed in contact with each other. The charge
always flows from a body at higher potential to another body at lower p
as soon as the potentials of the two bodies become equal
2.Define electrostatic potential at a point and electrostatic potential difference between two points?
Electrostatic potential at a point is defined as the work
is taken is taken from infinity up to the point.
Electrostatic potential different between 2 points is the work done per unit charge when a positive test
charge is taken between the points.
Potential due to a point charge
Potential at a point P due to a charge Q is defined as the work done in taking a unit +1C charge from
infinity up to that point. Work done in moving through a small distance
22
Q at a point on an electric field is the work done by the external force in
bringing the charge q from infinity to that point.
Electrostatic potential difference between two points (P&R) is defined as the work done by the external
force in bringing a unit positive charge form one point (R) to the other point (P)
q
UUVV RP
RP
−=−
Potential is a scalar quantity unit is J/C or volt. D. F of potential = ML2 T-3I -1.
=10-6
V 2) milli volt (m V) =10-3
V
Electrostatic potential at a point is defined as the work done by an external force in bring a unit positive
q
W
eargch
workV
∆∆
==
. Give the physical concept of electric potential ?
The concept of potential plays an important role and it may be made clear as follows
Consider a region free of any electric charge. Imagine a small charge, +q being placed in the region. It
can be made to move about any where in the region without doing any work of electrical origin because
the charge does not experience any electric force.
Let a charge +Q be placed in the region before +q is introduced. Now, work has to be done to move the
charge +q from one point to another because it experiences electrostatic force of repulsion
presence of the charge Q in space endows every point in the medium with a property by virtue of which
work is to be done to move a charge up to this point. This property is called electrostatic potential.
l of a body represents the degree of electrification of the body. It determines the
direction of flow of charge between two charged bodies placed in contact with each other. The charge
always flows from a body at higher potential to another body at lower potential. The flow of charge stops
as soon as the potentials of the two bodies become equal
.Define electrostatic potential at a point and electrostatic potential difference between two points?
Electrostatic potential at a point is defined as the work done per unit charge when a unit +ve test charge
is taken is taken from infinity up to the point.
Electrostatic potential different between 2 points is the work done per unit charge when a positive test
Potential at a point P due to a charge Q is defined as the work done in taking a unit +1C charge from
to that point. Work done in moving through a small distance r′∆ is
Q at a point on an electric field is the work done by the external force in
Electrostatic potential difference between two points (P&R) is defined as the work done by the external
.
Electrostatic potential at a point is defined as the work done by an external force in bring a unit positive
The concept of potential plays an important role and it may be made clear as follows :
Consider a region free of any electric charge. Imagine a small charge, +q being placed in the region. It
can be made to move about any where in the region without doing any work of electrical origin because
Let a charge +Q be placed in the region before +q is introduced. Now, work has to be done to move the
charge +q from one point to another because it experiences electrostatic force of repulsion. Thus the
presence of the charge Q in space endows every point in the medium with a property by virtue of which
work is to be done to move a charge up to this point. This property is called electrostatic potential.
l of a body represents the degree of electrification of the body. It determines the
direction of flow of charge between two charged bodies placed in contact with each other. The charge
otential. The flow of charge stops
.Define electrostatic potential at a point and electrostatic potential difference between two points?
done per unit charge when a unit +ve test charge
Electrostatic potential different between 2 points is the work done per unit charge when a positive test
Potential at a point P due to a charge Q is defined as the work done in taking a unit +1C charge from
rdFW ′=∆2
0r4
Q
′πε= rd ′
∴ Total work done′
−= ∫∞
dr
QW
r
2
04πε
∴ Potential r
1V,
r
Q
4
1V
0
∝πε
= .The graph connecting V and r is a rectangular hyperbola
Potential due to an electric dipole
Consider an electric dipole of dipole moment
distance r from the centre of the dipole and making an angle
Potential due to a point charge q at a distance r is given by
and –q. Hence total potential at P
From the figure arr 1
rrr+= (triangular law of vector addition)
arr1rrr
−=
θ−+= cosra2arr 2221
+θ−=
2
2
221
r
acos
r
a21rr since r > > a
θ−= cosr
a21rr 22
1 1 cosr
a21rr
−=
2/1
1
cosr
a21
r
1
r
1−
θ−= using binomial expansion
θ+= cosr
a1
r
1
r
1
1
similarly arr2
rrr+=
θ++= cosra2arr 2222
23
∞
′
=′r
r
Qrd
04πε =
r4
Q
0πε
.The graph connecting V and r is a rectangular hyperbola
Consider an electric dipole of dipole moment a2qPrr
= . We want to find the potential at a point P at a
distance r from the centre of the dipole and making an angle θ as shown in figure.
Potential due to a point charge q at a distance r is given by r/q4
1V
0πε= . Here there are two charges +q
−=
24
1
10 r
q
r
qV
πε
(triangular law of vector addition)
since r > > a 2
2
r
a can be neglected
2/1
cos
θ
using binomial expansion
.The graph connecting V and r is a rectangular hyperbola
∝2
r
1E
. We want to find the potential at a point P at a
as shown in figure.
. Here there are two charges +q
+θ+=
2
222
2a
rcos
r
a21rr then
−= θcos11
2
1
r
a
rr
θ−−
θ+=− cosr
a1cos
r
a1
r
1
r
1
r
1
21
θ=− cosa2r
1
r
1
r
12
21
∴2
0
2
0
cos
4
1cos2
4 r
P
r
aqV
θπε
θπε
==
2
0 r
cosP
4
1V
θπε
= r̂.PcosPr
=θ
∴r̂.P
r4
1V
20
r
πε= ( r > > a)
Along the axial line π=θ ,0 4
Vπε
=
Along the equatorial line 2
π=θ , cos
(i) potential due to a dipole depends on r and
fixed, the points corresponding to P on the cone so generated will have the same potential as at P.
ii) due to a dipole 2
r
1Vα , due to a single charge
Potential due to a system of charges
nVVVV +++= .......21
+++=
nP
n
PP r
q
r
q
r
qV ....
4
1
2
2
1
1
0πε
Potential due to charged shell out the shell is given by
the shell and R its radius. Electric field inside the shell is zero this implies potential is a constant inside
the shell and equals value at its surface
Equipotential Surfaces
If surface has all points at the same potential it is called equi
24
2/1
cos2
11
2
1−
+= θr
a
rr
20 r
P1
πε
, cos 0,0 == Vθ
potential due to a dipole depends on r and θ. If you rotate the position vector
fixed, the points corresponding to P on the cone so generated will have the same potential as at P.
, due to a single charge r
1V ∝
Potential due to a system of charges
Potential due to charged shell out the shell is given by r
q
4
1V
0πε= ( )Rr≥ where q is the total charge on
the shell and R its radius. Electric field inside the shell is zero this implies potential is a constant inside
the shell and equals value at its surface R
q
4
1V
0πε=
points at the same potential it is called equi potential surface.
. If you rotate the position vector rr
about Pr
keeping θ
fixed, the points corresponding to P on the cone so generated will have the same potential as at P.
where q is the total charge on
the shell and R its radius. Electric field inside the shell is zero this implies potential is a constant inside
25
q1 q2
r12
For a single charge r
q
4
1V
0πε= . If V is constant r is constant. Equipotential surfaces of a single point
charge are concentric spherical surfaces centered at the charge.
For any charge configuration, equipotential surface through a point is normal to the electric field at that
point. Work done in moving a charge in an equipotential surface is zero.
3.Explain why +ve charge flows from higher potential to lower potential while -ve charge flows from
lower potential to higher potential?
Any system has a tendency to reduce the PE to q a minimum. The PE of a +vely charged particle will
decrease when it moves from higher potential to lower potential. The PE of a -ve charge will decrease
when it moves from lower potential to higher potential. This explains why a +ve charge flows from
higher potential to lower potential and –ve charge flows from lower potential to higher potential.
The mutual PE of a system of charges resides on the system of two charges and not with any one
single charge.
Relation between field and potential
A and B are two equi potential surfaces having potential V and V-δV respectively. Work done in taking a
unit positive charge from A to B is equal potential difference = ( ) VdVV −−
dV|E| −=δl
ld
dV|E|
−=
Thus electric field is equal to negative gradient of potential.
Notes
i) Electric field is in the direction in which the potential decreases the steepest.
ii) Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the
equi potential surface.
Potential energy of a system of charges
Consider two charges q1 and q2 separated by a distance r12. Potential at the position of q2 due to q1 is
12
1
0
1r
q
4
1V
πε= . From the definition of potential, work done in bringing q2 from infinity to the point distant
r12 is q2 times the potential at 12rr
due to q1,
∴ work done on 12
21
0
2r
4
1q
πε=
This work done is stored as potential energy of a system of two charges.
12
21
0 r
4
1U
πε=
If q1q2 > 0, PE is positive. It will be negative q
For a system of three charges.
Total PE
++
πε=
13
3
23
32
12
21
0 r
q
r
r
4
1U
The potential energy is characteristic of the present state of configuration, and not the way
achieved.
4.What is the relation between PE and force?
If the PE is +ve, the force between them will be repulsion. If the PE is
be attractive. If q1 and q2 are both +ve or both
If q1or q2 is +ve and the other –ve the PE will be
F=
Electric field is equal to negative gradient of potential
E=
5. Potential energy of a single charge.
If V is the potential at a point P, which is the work done
upto that point, then work done in bringing a charge q will be qV. This is stored as PE
( )rqVW = or PE = ( )rqV
6. Potential energy of a system of two charges in an external
q1 and q2 are two charges separated by a distance r
an external field. Their total potential energy isis given by
7. Potential energy of a dipole in an external field.
Consider a dipole of dipole moment
torque experienced at this position
angle dθ, is ∆W= pE sinθ dθ.
∴ Total work done in rotating the dipole from an angle
26
0, PE is positive. It will be negative q1q2 < 0
13
13
r
q
The potential energy is characteristic of the present state of configuration, and not the way
.What is the relation between PE and force?
If the PE is +ve, the force between them will be repulsion. If the PE is –ve the force between them will
are both +ve or both –ve then PE will be +ve and the force will be repulsion.
ve the PE will be –ve and the force will be attraction.
F= � ��
��
Electric field is equal to negative gradient of potential
E= � ��
��
Potential energy of a single charge.
If V is the potential at a point P, which is the work done in bringing a unit positive charge from infinity
upto that point, then work done in bringing a charge q will be qV. This is stored as PE
Potential energy of a system of two charges in an external field.
are two charges separated by a distance r12 residing in a potential V(r1) and V(r
an external field. Their total potential energy isis given by 2211 ()( rVqrVqPE +=
Potential energy of a dipole in an external field.
Consider a dipole of dipole moment a2qprr
= placed at an angle θ in a uniform electric field
torque experienced at this position Eprrr
×=τ . θ=τ sinPE . Work done ∆W in rotating through a small
Total work done in rotating the dipole from an angle θ0 to θ1 is
The potential energy is characteristic of the present state of configuration, and not the way state is
ve the force between them will
ve then PE will be +ve and the force will be repulsion.
ve and the force will be attraction.
in bringing a unit positive charge from infinity
upto that point, then work done in bringing a charge q will be qV. This is stored as PE
) and V(r2) respectively in
120
212
4)
r
πε+
in a uniform electric field Er
. The
W in rotating through a small
∫θ
θθ−θ=θθ=
1
0
cos(cospEdsinpEW 10
Let us take the zero potential energy as
pcospEcos2
cospEUr
−=θ−=
θ−
π=
Potential energy of the dipole is also given by
where V1 and V2 are the potential at the point where q and
Epa
qpEU
4.
24cos
0
2
πεθ −−=−−=
rr
Since a
q
24 0
2
πε is a constant that term is insignificance for potential energy.
8. Electrostatics of conductors.
i. Inside a conductor, electrostatic field is zero
If there is no current in the conductor electric field is zero. If there is an electric field there will be flow
of charges when the free charges distribute themselves , then electric field becomes zero.
ii. At the surface of a charged conductor, elec
point.
If Er
were not normal to the surface it would have some non
this component free charges will move. Hence field will be normal.
iii. The interior of a conductor can have no excess charge in the static situation
When a conductor is charged excess charge can reside only on the surface. This follows from Gauss law.
Since electric field inside is zero net charge is zero.
iv. Electrostatic potential is constant through out the volume of the conductor and has the same value
(as inside) on its surface
Electric field intensity inside the conductor is zero.
0dv,0dx
dv,0Eif,
dx
dvE ===
−= i.e potential , V= constant.
v. Electric field at the surface of a charged conductor
Where σ is the surface charge density and
direction. Flux contribution from the curved part of the Gaussian surface is zero.
27
)1
This work done is stored as PE.
Let us take the zero potential energy as θ0=2
π
E.prr
Potential energy of the dipole is also given by [ ]a
qVVqU
24 0
2
21 πε−−=
are the potential at the point where q and −q resides.
a
q
24 0
2
πε
is a constant that term is insignificance for potential energy.
Inside a conductor, electrostatic field is zero
If there is no current in the conductor electric field is zero. If there is an electric field there will be flow
of charges when the free charges distribute themselves , then electric field becomes zero.
At the surface of a charged conductor, electrostatic filed must be normal to the surface at every
were not normal to the surface it would have some non- zero component along the surface. Due to
this component free charges will move. Hence field will be normal.
The interior of a conductor can have no excess charge in the static situation
When a conductor is charged excess charge can reside only on the surface. This follows from Gauss law.
Since electric field inside is zero net charge is zero.
ectrostatic potential is constant through out the volume of the conductor and has the same value
Electric field intensity inside the conductor is zero.
i.e potential , V= constant.
Electric field at the surface of a charged conductor nE0
)r
εσ
=
is the surface charge density and n)
is a unit vector normal to the surface in the outward
Flux contribution from the curved part of the Gaussian surface is zero.
If there is no current in the conductor electric field is zero. If there is an electric field there will be flow
of charges when the free charges distribute themselves , then electric field becomes zero.
trostatic filed must be normal to the surface at every
zero component along the surface. Due to
.
When a conductor is charged excess charge can reside only on the surface. This follows from Gauss law.
ectrostatic potential is constant through out the volume of the conductor and has the same value
is a unit vector normal to the surface in the outward
Flux contribution from the curved part of the Gaussian surface is zero.
On the flat face 00
E,dsq
Edsεσ
=ε
=
vi. Electrostatic shielding
Since all charges resides only on the outer surface of the conductor electric field inside the conductor is
zero. Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded
from outside electrical influence. The
electrostatic shielding.
9. Dielectric and Polarisation:
Polar and non polar molecules:-
In certain molecules centes of positive charge and negative charge coincide. As a result net charge of
molecules becomes zero. Such molecules are said to have symmetric structure. its net dipole moment is
zero. This type of symmetrical molecules are known as non polar molecules. ex. O
There are molecules in which the centers of
a molecule will have net dipole moment. They are known as polar molecules. H
Conductors placed in an external electric field.
When a conductor is placed in an external electric field inside the conductor there will be induced field
(Ein) which is equal and opposite to the external field (E
Dielectric in an external electric field
Dielectrics are non- conducting substances having no free charge carriers. When it is placed in an external
electric field (E0) it induces a field (E
inside the dielectric is non zero.
Non- polar molecule in an external field
In an external electric field, the positive and negative charges of a non
the opposite directions. So that it develops a net dipole moment. The dielectric is said to be polarised by
the external field.
Polarisation( Pr) and Susceptibility(
When a polar or non- polar, dielectric develops a net dipole moment in the presence of an external field,
the dipole moment per unit volume is called polarisation
For linear isotropic dielectrics Pr
ψ=
the electric susceptibility of the dielectric medium.
Unit of Pr
is 23 m
C
m
Cm=
28
0εσ
=
Since all charges resides only on the outer surface of the conductor electric field inside the conductor is
zero. Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded
from outside electrical influence. The field inside the cavity is always zero. This is known as
In certain molecules centes of positive charge and negative charge coincide. As a result net charge of
molecules becomes zero. Such molecules are said to have symmetric structure. its net dipole moment is
zero. This type of symmetrical molecules are known as non polar molecules. ex. O
There are molecules in which the centers of positive charges and negative charges do not coincide. Such
a molecule will have net dipole moment. They are known as polar molecules. H
Conductors placed in an external electric field.
external electric field inside the conductor there will be induced field
) which is equal and opposite to the external field (E0) so that total field inside the conductor is zero.
Dielectric in an external electric field
nducting substances having no free charge carriers. When it is placed in an external
) it induces a field (Ein) in the opposite direction. Magnitude of E
polar molecule in an external field
In an external electric field, the positive and negative charges of a non- polar molecule are displaced in
So that it develops a net dipole moment. The dielectric is said to be polarised by
) and Susceptibility( ψψψψe)
polar, dielectric develops a net dipole moment in the presence of an external field,
the dipole moment per unit volume is called polarisation Pr
.
Ee
rψ . ψe is a constant characteristics of the dielectric and is known as
the electric susceptibility of the dielectric medium.
Since all charges resides only on the outer surface of the conductor electric field inside the conductor is
zero. Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded
field inside the cavity is always zero. This is known as
In certain molecules centes of positive charge and negative charge coincide. As a result net charge of the
molecules becomes zero. Such molecules are said to have symmetric structure. its net dipole moment is
zero. This type of symmetrical molecules are known as non polar molecules. ex. O2, H2, N2, CO2 etc.
positive charges and negative charges do not coincide. Such
a molecule will have net dipole moment. They are known as polar molecules. H2O, HCl, CO etc.
external electric field inside the conductor there will be induced field
) so that total field inside the conductor is zero.
nducting substances having no free charge carriers. When it is placed in an external
) in the opposite direction. Magnitude of Ein < E0 so that net field
polar molecule are displaced in
So that it develops a net dipole moment. The dielectric is said to be polarised by
polar, dielectric develops a net dipole moment in the presence of an external field,
is a constant characteristics of the dielectric and is known as
[In linear isotropic dielectrics induced dipole moment is in the direction of the field and proportional to
field strength] σP is the induced surface charge density.
Polarisation tv
qP =
At
qtP =
This polarization is also defined as the induced surface charge per unit area or surface polarization charge
density.
10. Capacity of Capacitance
If charge(Q) on the capacitor is increased its potential (V) increase
ie Q αV or Q= CV or V
QC = where C is a constant known as capacity or capacitance of the conductor.
It is the ability to store charge
faradvolt
Coulomb
V
QC ===
A conductor has a capacity of one farad, if a charge of one coulomb raises its potential by one volt.
11.Principle of a capacitor
Consider A charged metal plate A as in figure (i) a charge Q and potential V. Its capacity is given by
When an uncharged identical metal plate B is brought near A polarization of charge takes place near B as
shown in figure because of this potential of A decreases
charge flows to earth hence potential of A is further lowered, increasing the capacity. This is the principle
of capacitor. A capacitor consists of two conductors, one of them is charged and the other usually
connected. This arrangement is used to increase the capacity of a conductor.
(A single conductor can be used as a capacitor, by assuming other end at infinity)
29
[In linear isotropic dielectrics induced dipole moment is in the direction of the field and proportional to
is the induced surface charge density.
At
qt, t is thickness of the slab
A/qP =
This polarization is also defined as the induced surface charge per unit area or surface polarization charge
If charge(Q) on the capacitor is increased its potential (V) increase
where C is a constant known as capacity or capacitance of the conductor.
A conductor has a capacity of one farad, if a charge of one coulomb raises its potential by one volt.
------
------
as in figure (i) a charge Q and potential V. Its capacity is given by
When an uncharged identical metal plate B is brought near A polarization of charge takes place near B as
shown in figure because of this potential of A decreases by increasing the capacity. If we earth B, +ve
charge flows to earth hence potential of A is further lowered, increasing the capacity. This is the principle
of capacitor. A capacitor consists of two conductors, one of them is charged and the other usually
connected. This arrangement is used to increase the capacity of a conductor.
(A single conductor can be used as a capacitor, by assuming other end at infinity)
[In linear isotropic dielectrics induced dipole moment is in the direction of the field and proportional to
This polarization is also defined as the induced surface charge per unit area or surface polarization charge
where C is a constant known as capacity or capacitance of the conductor.
A conductor has a capacity of one farad, if a charge of one coulomb raises its potential by one volt. -
as in figure (i) a charge Q and potential V. Its capacity is given by V
Q.
When an uncharged identical metal plate B is brought near A polarization of charge takes place near B as
by increasing the capacity. If we earth B, +ve
charge flows to earth hence potential of A is further lowered, increasing the capacity. This is the principle
of capacitor. A capacitor consists of two conductors, one of them is charged and the other usually earth
(A single conductor can be used as a capacitor, by assuming other end at infinity)
30
Dielectric Strength : The maximum electric field that a dielectric medium can withstand without break
down (of its insulating property) is called the dielectric strength dielectric strength of air is about 16 Vm103 −× .
12.Parallel plate capacitor
+ + + + + + + + +
A
Region III
Ed
Plate II
Plate I
A parallel plate capacitor consists of two large parallel conducting plates of area A separated by a
distance d (d2 << A). Assume that medium between the plates is vacuum.
Electric field in the outer region I is 022 00
=εσ
−εσ
Electric field in the outer region II is 022 00
=εσ
−εσ
Electric field in the inner region III is 000 22 εσ
=εσ
+εσ
A
QE
00 ε=
εσ
= [since σ =A
Q]
For uniform electric field d
VE = , where V is the potential difference between the plates
−=
ld
dvE
∴ V= Ed=A
Qd
0ε ∴
d
A
V
QC 0ε==
d
AC 0ε=
Capacitance depends only on geometry of the system,
Find the area of plates required to produce 1F capacitance having a separation 1cm is
39
12
2
0
m101085.8
101CdA =
×
×=
ε=
−
−
which is a plate about 30 km in length and breadth. This shows that 1F is
too big a unit in practical purposes.
13.Effect of dielectric on capacitance
When there is vacuum between the plates dEV,E 00
0
0 =εσ
=
When a dielectric slab is completely inserted between the plates it gets polarized pσ and pσ− are induced
surface charge densities ( )
0
p
0
p dEdV,E
ε
σ−σ==
ε
σ−σ=
( )pσ−σ is proportional to σ , ….∴K
p
σ=σ−σ where K >1
∴KA
Qd
K
dV
00 ε=
εσ
=
31
d
KA
V
QC 0ε==
d
KAC 0ε=
k0ε is called the permittivity of the medium and denoted by ε . 0ε=ε K for vacuum K=1,
0ε=ε is called permittivity of the vacuum 0
Kεε
= is called dielectric constant. It is dimensionless
d
KAC,
d
AC 00
0
ε=
ε=
∴0C
CK = .
Hence dielectric constant is the factor by which the capacitance increases from its vacuum value, when
the dielectric is inserted fully between the plates of a capacitor.
Capacity of a capacitor with thin dielectric
A thin dielectric of thickness t, dielectric constant k is inserted as shown in figure.
Capacity is given by
r
0
ttd
AC
ε+−
ε=
t
d
vacuum vacuum
C will be greater then d/AC 00 ε=
If slab is metal ∞=εr , td
AC 0
−
ε=′
C will be greater than C0 = d/A0ε
If slab is metal td
AC, 0
r −
ε=′∞=ε
14. Combination of capacitor
(i) Capacitors in series: Consider two capacitors C1 and C2 connected in series a potential of V voltage
is applied across them. In series connection charge on capacitors are same. Applied voltage V gets
divided as V1 and V2.
2211
21
21 , CVCVQC
Q
C
QVVV ==+=+=
,C
Q
C
Q
C
Q
21
+=21 C
1
C
1
C
1+=
where C is known as effective capacity of the combination 21
21
CC
CCC
+=
for n number of capacitor
n21 V...VVV +++= 21 C
Q
C
Q
C
Q+=
in series connection effective capacity decreases
(2) capacitors in parallel
In parallel connection voltage is same
Total charge stored in the system Q
CV = VCVC 21 + 21 CCC +=
for n capacitors in parallel 1CC +=
15.Energy stored in a capacitor
Consider a capacitor of capacity C. When fully charged potential difference across the capacitor is C and
charge is Q. When the charge in the capacitor is Q
Work done in adding a further charge dQ
∴ total work done in charging from 0 to Q is
This is stored as electrostatic PE
16.Energy density in a parallel plate capacitor
Consider a parallel plate capacitor with capacity
32
nC
Q....++
n21 C
1...
C
1
C
1
C
1+++=
in series connection effective capacity decreases
In parallel connection voltage is same VCQ,VCQ 2211 ==
21 QQ +=
n2 C....C ++
Consider a capacitor of capacity C. When fully charged potential difference across the capacitor is C and
When the charge in the capacitor is Q′ and potential is V′, V′=C
Q′,
charge dQ′ is dW=Vq=V′dQ′ = QdC
Q ′′
total work done in charging from 0 to Q is
∫ =′=Q
0
21
C2
QQd
C
QW
QV
2
1CV
2
1
C2
QW 2
2
=== [since Q=CV]
density in a parallel plate capacitor
Consider a parallel plate capacitor with capacity d
AC 0ε=
Consider a capacitor of capacity C. When fully charged potential difference across the capacitor is C and
[since Q=CV]
33
Energy density ( )
Add
A2
A
CAd2
Q
volume
Energy
0
22
εσ
=== =2
02
1Eε
If there is a dielectric between the plates energy density is given by 2r0 E
2
1U εε= .
17.Potential due to a hollow sphere or solid conducting charged sphere
We have � = ���
�� .
For a hollow sphere of radius “R” carrying a charge “Q”
Electric field intensity outside Eo =
��
�
��
Potential outside is Vo = � �
��[
��
�
��] =
��
�
�
Electric field intensity on the surface Es =
��
�
��
Potential outside is Vs = � �
��[
��
�
��] =
��
�
�
Electric field intensity inside is E i = zero
�� = ����
�� . That is 0 = �
���
��
Then ��� = 0 . So �� = �������� ��� �� ��� �� �ℎ�� �� �ℎ � "#$�� .
Then Vi= Vs =
��
�
�
Potential outside is Vs = � �
��[
��
�
��] =
��
�
�
18.What is Van De Graff Generator?
Van de Graff generator is a machine that can build up high voltages of the order of few million volts. The
resulting large electric fields are used to accelerate charge particles (electrons, protons, ions) to high
energies needed for experiments to problem the small scale structure of matter.
19.What is the principle of van De Graff generator?
Consider two concentric sphere shown in figure insulated from each other.
Potential on the surface of sphere of radius R is ( )
+πε
=R
q
R
Q
4
1V
0
R
Potential on the surface of a sphere of radius r is ( )
+πε
=R
Q
r
q
4
1V
0
r
∴Potential difference ( ) ( ) =−4
VV Rr
ie inner sphere is greater potential than outer sphere. This means that if we connect smaller and larger
sphere by a wire, the charge q will flow to the outer sphere.
What ever charge is given to smaller sphere it
charges on the outer sphere until we reach breakdown field of air 93
Van de Graff generator
20.Working of Van de Graff generator.
A Schematic diagram of Van de Graff
supported at height several metres above the ground on an insulating column. A long narrow endless belt
insulating material line rubber or silk is wound around two pulleys
centre of the shell. This belt is kept continuously moving by a motor driving the lower pulley. It
continuously carriers positive charge, sprayed on to it by a brush at ground level, to the top. These it
transistors its positive charge to another conducting brush connected to the larger shell positive charge
spreads out uniformly on the outer surface. In this way voltage differences as much as 6 or 8 million volts
with respect to ground can be built up.
34
−πε R
1
r
1
4
q
0
is +ve.
ie inner sphere is greater potential than outer sphere. This means that if we connect smaller and larger
sphere by a wire, the charge q will flow to the outer sphere.
What ever charge is given to smaller sphere it goes to outer sphere. In this way we can keep piling up
charges on the outer sphere until we reach breakdown field of air 93×106V/m)
.Working of Van de Graff generator.
Graff generator is shown in figure. A large spherical conducting shell is
supported at height several metres above the ground on an insulating column. A long narrow endless belt
insulating material line rubber or silk is wound around two pulleys – one at ground
centre of the shell. This belt is kept continuously moving by a motor driving the lower pulley. It
continuously carriers positive charge, sprayed on to it by a brush at ground level, to the top. These it
rge to another conducting brush connected to the larger shell positive charge
spreads out uniformly on the outer surface. In this way voltage differences as much as 6 or 8 million volts
with respect to ground can be built up.
ie inner sphere is greater potential than outer sphere. This means that if we connect smaller and larger
goes to outer sphere. In this way we can keep piling up
V/m). This is the principle of
generator is shown in figure. A large spherical conducting shell is
supported at height several metres above the ground on an insulating column. A long narrow endless belt
one at ground level and the other at
centre of the shell. This belt is kept continuously moving by a motor driving the lower pulley. It
continuously carriers positive charge, sprayed on to it by a brush at ground level, to the top. These it
rge to another conducting brush connected to the larger shell positive charge
spreads out uniformly on the outer surface. In this way voltage differences as much as 6 or 8 million volts