Chap 1 Electric Charges & Fields with Annotations_35342811_2024_07_10_11_07.pdf
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Sep 01, 2024
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Chap 1: Electric Charges and Field
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Electric Charge
It is an intrinsic property of the elementary particles like
electrons, protons, etc., of which all the objects are
made up of.
It is a scalar quantity.
Its SI unit is Coulomb (C).
�=1.6×10
−19
coulomb
6
It is an intrinsic property of the elementary particles like
electrons, protons, etc., of which all the objects are
made up of.
It is a scalar quantity.
Its SI unit is Coulomb (C).
�=1.6×10
−19
coulomb
6
Electric Charge
??????Important
7
??????Important
7
Basic Properties of Charge
1.Additivity
2.Quantization
3.Conservation
8
1.Additivity
2.Quantization
3.Conservation
8
➢Addition of Charges
If a system contains three point charges q1,q2, and q3 , then the total
charge of the system will be algebraic additionof q1, q2 and q3 , i.e.,
charge will add up
q= q1+q2+q3
Basic Properties of Charge
9
If a system contains three point charges q1,q2, and q3 , then the total
charge of the system will be algebraic additionof q1, q2 and q3 , i.e.,
charge will add up
q= q1+q2+q3
Basic Properties of Charge
9
➢Quantization of charges
Electric charges is always quantized ,i.e., electric charge is always an
integral multiple of charge ‘e’.
�=��,??????ℎ����=0,±1,±2,……
Basic Properties of Charge
10
Electric charges is always quantized ,i.e., electric charge is always an
integral multiple of charge ‘e’.
�=��,??????ℎ����=0,±1,±2,……
Basic Properties of Charge
10
➢Conservation of charges
Law of conservation of charge
1.The total charge of an isolated system remains constant.
2. The electric charges can neither be created nor destroyed, they
can only be transferred from one body to another.
Basic Properties of Charge
11
Law of conservation of charge
1.The total charge of an isolated system remains constant.
2. The electric charges can neither be created nor destroyed, they
can only be transferred from one body to another.
Basic Properties of Charge
11
Coulomb’s Law
12
12
Forces between Charges .
Like Charges Repel .
Unlike Charges attract .
13
Like Charges Repel .
Unlike Charges attract .
13
French Scientist
Coulomb
How Much Force ? (Coulomb’s Law) .
??????
�=Permittivity of free Space
In Vacuum,
14
Coulomb
How Much Force ? (Coulomb’s Law) .
??????
�=Permittivity of free Space
In Vacuum,
14
➢Coulomb’s Law is valid for point Charges
When the size of charged bodies are much smaller than
the distance between them,then charged bodies are
called point Charges.
Ex. Stars in the sky
Not Small, but looks Small
Meaning of Point Charges .
Point Charge≠Small Charge
15
When the size of charged bodies are much smaller than
the distance between them,then charged bodies are
called point Charges.
Ex. Stars in the sky
Not Small, but looks Small
Meaning of Point Charges .
Point Charge≠Small Charge
15
Important Concepts .
➢Coulomb’s Law
follows inverse
Square Law
➢Coulomb’s Force is
a Central force
they act along the line
joining the centres of
two bodies
➢Two Charges exert
equal & opposite
forces on each other.
they form
Action-Reaction Pair
(follow Newton’s 3
rd
Law)
16
➢Coulomb’s Law
follows inverse
Square Law
➢Coulomb’s Force is
a Central force
they act along the line
joining the centres of
two bodies
➢Two Charges exert
equal & opposite
forces on each other.
they form
Action-Reaction Pair
(follow Newton’s 3
rd
Law)
16
Important Concepts .
➢Coulomb’s Force is a
Conservative Force
i.e., the work done
against these forces
does not depend on path
➢Coulomb’s Force
depends on the medium
➢Coulomb’s Force is a
long range Force
From Nuclear dimension
(r=10
-15
m)
To (r= 10
18
m)
17
➢Coulomb’s Force is a
Conservative Force
i.e., the work done
against these forces
does not depend on path
➢Coulomb’s Force
depends on the medium
➢Coulomb’s Force is a
long range Force
From Nuclear dimension
(r=10
-15
m)
To (r= 10
18
m)
17
Vector form of
Coulomb’s law
18
Coulomb’s law
18
Vector form of Coulomb’s Law .
�
21=force on Charge q2 due to q1
ෞ�
12=is a unit vector in the
direction from q1 to q2
19
�
21=force on Charge q2 due to q1
ෞ�
12=is a unit vector in the
direction from q1 to q2
19
Importance of Vector form .
Thevectorformofcoulomb′slawgivesadditionalinformation:
1. Ƹ�
21=−Ƹ�
12therefore Ԧ�
21=−Ԧ�
12
twochargesexertequalandoppositeforcesoneachother.So
CoulombianforcesobeyNewton′sthirdlawofmotion.
2.Coulombianforcesactalong,Ԧ�
21��Ԧ�
12
i.e.,alongthelinejoiningthecentresoftwocharges,sothey
arecentralforces.
20
Thevectorformofcoulomb′slawgivesadditionalinformation:
1. Ƹ�
21=−Ƹ�
12therefore Ԧ�
21=−Ԧ�
12
twochargesexertequalandoppositeforcesoneachother.So
CoulombianforcesobeyNewton′sthirdlawofmotion.
2.Coulombianforcesactalong,Ԧ�
21��Ԧ�
12
i.e.,alongthelinejoiningthecentresoftwocharges,sothey
arecentralforces.
20
Limitations of Coulomb’s Law .
1. Electric charges must be at rest.
2. The electric charges must be point charges.
3. The separation between the charges must be greater
than the nuclear size (10
-15
m), because for distances
< 10
-15
m, the strong nuclear force dominates over the
electrostatic force.
21
1. Electric charges must be at rest.
2. The electric charges must be point charges.
3. The separation between the charges must be greater
than the nuclear size (10
-15
m), because for distances
< 10
-15
m, the strong nuclear force dominates over the
electrostatic force.
21
Permittivity .
➢Permittivity is a property of a medium which determines
the electric force between two charges situated in that
medium.
Ԧ�
21=��������ℎ��??????��
2������
1
=
1
4????????????0
�1�2
�
2
Ƹ�
12
??????
�=??????.??????���
−��
??????
�
??????
−�
�
−�
22
➢Permittivity is a property of a medium which determines
the electric force between two charges situated in that
medium.
Ԧ�
21=��������ℎ��??????��
2������
1
=
1
4????????????0
�1�2
�
2
Ƹ�
12
??????
�=??????.??????���
−��
??????
�
??????
−�
�
−�
22
Dielectric Constant or Relative Permittivity .
??????
����=
??????
??????
�
=
�
�??????�
�
���
��??????����=�
�????????????�=�.00054
��??????���=??????�
The dielectric constant or relative permittivity of amedium may be defined as the
ratio of the force between two charges placed some distance apart in free space to
the force between the same two charges when they are placed the same distance
apart in the given medium
�
���=
�
�??????�
�
23
??????
����=
??????
??????
�
=
�
�??????�
�
���
��??????����=�
�????????????�=�.00054
��??????���=??????�
The dielectric constant or relative permittivity of amedium may be defined as the
ratio of the force between two charges placed some distance apart in free space to
the force between the same two charges when they are placed the same distance
apart in the given medium
�
���=
�
�??????�
�
23
Q. Two identical charges, Q each, are kept at a distance r from each other. A third charge
q is placed on the line joining the above two charges such that all the three charges are in
equilibrium. What is the magnitude, sign and position of the charge q ?
24
q is placed on the line joining the above two charges such that all the three charges are in
equilibrium. What is the magnitude, sign and position of the charge q ?
24
Superposition Principle .
The Superposition Principlestates that when a number
of charges are interacting, the total force on a given
charge is the vector sum of the forcesexerted on it due to
all other charges.
Note: The force between two charges is not affected by
the presence of other charges.
25
The Superposition Principlestates that when a number
of charges are interacting, the total force on a given
charge is the vector sum of the forcesexerted on it due to
all other charges.
Note: The force between two charges is not affected by
the presence of other charges.
25
Q. Consider the charges q, q and -q placed at the vertices of an equilateral triangle, as shown
in Fig. What is the force on each charge ?
26
in Fig. What is the force on each charge ?
26
Electric field .
�=
�
�
�
The electric field at a point is defined as the electrostatic force per
unit test chargeacting on a vanishingly small positive test charge
placed at that point.
�=lim
�
0→0
�
�
�
Note: The electric field �is a vector quantity whose direction is
same as that of force �exerted on a positive charge test charge.
27
�=
�
�
�
The electric field at a point is defined as the electrostatic force per
unit test chargeacting on a vanishingly small positive test charge
placed at that point.
�=lim
�
0→0
�
�
�
Note: The electric field �is a vector quantity whose direction is
same as that of force �exerted on a positive charge test charge.
27
Electric field due to a point charge .
28
28
Electric Field Due To A System Of Point Charges .
The electric field at any point due to a group of
charges is equal to the vector sum of the electric
fields produced by each charge individually at
that point, when all other charges are assumed
to be absent.
Hence, the electric field at point P due to the
system of N charges is
�=�
1+�
2+⋯+�
??????
29
The electric field at any point due to a group of
charges is equal to the vector sum of the electric
fields produced by each charge individually at
that point, when all other charges are assumed
to be absent.
Hence, the electric field at point P due to the
system of N charges is
�=�
1+�
2+⋯+�
??????
29
Q. Two point charges of +16????????????���−9????????????are placed 8 cm apart in air.
Determine the position of the point at which the resultant field is zero.
30
Determine the position of the point at which the resultant field is zero.
30
Electric lines of force .
An electric line of force may be defined as the
curve along which a small positive charge would
tend to move when free to do soin an electric
field and the tangent to which at any point gives
the direction of the electric field at that point.
NOTE:The lines of force do not really exist,
they are imaginary curves. Yet the concept
of lines of force is very useful.
Michael Faraday gave simple explanations
for many of his discoveries (in electricity
and magnetism) in terms of such lines of
force.
31
An electric line of force may be defined as the
curve along which a small positive charge would
tend to move when free to do soin an electric
field and the tangent to which at any point gives
the direction of the electric field at that point.
NOTE:The lines of force do not really exist,
they are imaginary curves. Yet the concept
of lines of force is very useful.
Michael Faraday gave simple explanations
for many of his discoveries (in electricity
and magnetism) in terms of such lines of
force.
31
Properties of electric lines of force .
1.The lines of force are continuous smooth curves
without any breaks.
2.The line of force start at positive charge and end at
negative charges they cannot form closed loops. If
there is a single charge, then the lines of force will
start or end at infinity.
3.The tangent to a line of force at any point gives the
direction of the electric field at that point.
32
1.The lines of force are continuous smooth curves
without any breaks.
2.The line of force start at positive charge and end at
negative charges they cannot form closed loops. If
there is a single charge, then the lines of force will
start or end at infinity.
3.The tangent to a line of force at any point gives the
direction of the electric field at that point.
32
Properties of electric lines of force .
4. No two lines of force can cross each other.
Reason. If they intersect, then there
will be two tangents at the point of
intersection (Fig.) and hence two
directions of the electric field at the
same point, which is not possible.
33
4. No two lines of force can cross each other.
Reason. If they intersect, then there
will be two tangents at the point of
intersection (Fig.) and hence two
directions of the electric field at the
same point, which is not possible.
33
5. The lines of force are always normal to the surface of a conductor on
which the charges are in equilibrium.
Reason.If the lines of force are not normal to the conductor, the
component of the field �parallel to the surface would cause the electrons
to move and would set up a current on the surface. But no current flows in
the equilibrium condition.
Properties of electric lines of force .
34
which the charges are in equilibrium.
Reason.If the lines of force are not normal to the conductor, the
component of the field �parallel to the surface would cause the electrons
to move and would set up a current on the surface. But no current flows in
the equilibrium condition.
Properties of electric lines of force .
34
6. The lines of force have a tendency to contract
lengthwise.This explains attraction between two
unlike charges.
7. The lines of force have a tendency to expand
laterallyso as to exert a lateral pressure on
neighbouring lines of force. This explains repulsion
between two similar charges.
Properties of electric lines of force .
35
lengthwise.This explains attraction between two
unlike charges.
7. The lines of force have a tendency to expand
laterallyso as to exert a lateral pressure on
neighbouring lines of force. This explains repulsion
between two similar charges.
Properties of electric lines of force .
35
8. The relative closeness of the lines of force gives a measure
of the strength of the electric field in any region. The lines of
force are
(i) close together in a strong field.
(ii) far apart in a weak field.
(iii) parallel and equally spaced in a uniform field.
9. The lines of force do not pass through a conductor because
the electric field inside a charged conductor is zero.
Properties of electric lines of force .
36
of the strength of the electric field in any region. The lines of
force are
(i) close together in a strong field.
(ii) far apart in a weak field.
(iii) parallel and equally spaced in a uniform field.
9. The lines of force do not pass through a conductor because
the electric field inside a charged conductor is zero.
Properties of electric lines of force .
36
Properties of electric lines of force .
37
37
Electric field lines for different charged conductors .
(i)Field lines of a positive point charge.
radially outwards
They extend to infinity.
The field is spherically symmetric
(ii) Field lines of a negative point charge.
radially inwards
They start from infinity.
spherically symmetric
38
(i)Field lines of a positive point charge.
radially outwards
They extend to infinity.
The field is spherically symmetric
(ii) Field lines of a negative point charge.
radially inwards
They start from infinity.
spherically symmetric
38
Electric field lines for equal & opposite Charges (Dipole) .
The field is Cylindrically Symmetric about the dipole axis
39
The field is Cylindrically Symmetric about the dipole axis
39
Thefield�iszeroatthemiddlepointNof
thejoinoftwocharges.Thispointiscalled
neutralpointfromwhichnolineofforce
passes.
Thisfieldalsohascylindricalsymmetry.
Electric field lines for equal & positive Charges .
40
thejoinoftwocharges.Thispointiscalled
neutralpointfromwhichnolineofforce
passes.
Thisfieldalsohascylindricalsymmetry.
Electric field lines for equal & positive Charges .
40
Thus the lines of force are parallel and normal
to the surface of the conductor.
They are equispaced, indicating that electric
field �is uniformat all points near the plane
conductor.
Electric field lines for Charged Plane Conductor .
41
to the surface of the conductor.
They are equispaced, indicating that electric
field �is uniformat all points near the plane
conductor.
Electric field lines for Charged Plane Conductor .
41
Relation between electric field strength and density of lines of force .
Electricfieldstrengthisproportionalto
thedensityoflinesofforce
42
Electricfieldstrengthisproportionalto
thedensityoflinesofforce
42
43
44
Continuous Charge Distribution .
In practice, we deal with charges much greater in
magnitude than the charge on an electron, so we can
ignore the quantum nature of charges and imagine that
the charge is spread in a region in a continuous manner.
Such a charge distribution is known as a continuous
charge distribution.
Ԧ�=
�
0
4????????????
0
න
��
�
2
.Ƹ�
45
In practice, we deal with charges much greater in
magnitude than the charge on an electron, so we can
ignore the quantum nature of charges and imagine that
the charge is spread in a region in a continuous manner.
Such a charge distribution is known as a continuous
charge distribution.
Ԧ�=
�
0
4????????????
0
න
��
�
2
.Ƹ�
45
Continuous Charge Distribution (Volume Charge distribution) .
46
46
Continuous Charge Distribution (Surface Charge distribution) .
47
47
Continuous Charge Distribution (Linear Charge distribution) .
48
48
Electric Dipole .
➢A pair of equal and opposite charges separated by a small
distanceis called an electric dipole.
49
➢A pair of equal and opposite charges separated by a small
distanceis called an electric dipole.
49
Dipole Moment .
50
50
Examples of electric Dipoles .
Dipoles are common in nature.
In molecules like ????????????�,??????
2�,??????
2??????
5�??????���.
The centre of positive charges does not fall exactly over the centre
of negative charges.Such molecules are electric dipoles. They have
a permanent dipole moment.
51
Dipoles are common in nature.
In molecules like ????????????�,??????
2�,??????
2??????
5�??????���.
The centre of positive charges does not fall exactly over the centre
of negative charges.Such molecules are electric dipoles. They have
a permanent dipole moment.
51
Ideal or Point Dipole .
We can think of a dipole in which size
2�→0���??????ℎ��??????��→∞
in such a way that the dipole moment, p = q x 2a has a finite
value. Such a dipole of negligibly small size is called an ideal
or point dipole.
Dipoles associated with individual atoms or molecules may
be treated as ideal dipoles.
52
We can think of a dipole in which size
2�→0���??????ℎ��??????��→∞
in such a way that the dipole moment, p = q x 2a has a finite
value. Such a dipole of negligibly small size is called an ideal
or point dipole.
Dipoles associated with individual atoms or molecules may
be treated as ideal dipoles.
52
➢The electric field produced by an electric dipole is called dipole field
Dipole Field .
53
Dipole Field .
53
Variation of dipole field with distance .
The total charge of an electric dipole is zero.
But the electric field of an electric dipole is not zero.
Dipole Electric field at larger distances falls off as 1/�
3
.
Where as Electric field due to point charge falls off as
1/�
2
.
54
The total charge of an electric dipole is zero.
But the electric field of an electric dipole is not zero.
Dipole Electric field at larger distances falls off as 1/�
3
.
Where as Electric field due to point charge falls off as
1/�
2
.
54
Electric Field at an Axial Point of a Dipole .
For point P at distance r from centre of dipole on charge q, for r≫a, total field at
Point P is
�(??????�????????????�)=
�
�????????????
�
.
��
�
�
(??????�??????≪�)
�??????�????????????�=
�
�????????????
�
.
��
�
�
ෝ�(??????�??????≪�)
55
For point P at distance r from centre of dipole on charge q, for r≫a, total field at
Point P is
�(??????�????????????�)=
�
�????????????
�
.
��
�
�
(??????�??????≪�)
�??????�????????????�=
�
�????????????
�
.
��
�
�
ෝ�(??????�??????≪�)
55
Derivation: Electric Field at an Axial Point of a Dipole .
56
56
Derivation: Electric Field at an Axial Point of a Dipole .
Note: Electric Field at axial position is
in the direction of dipole moment.
57
Note: Electric Field at axial position is
in the direction of dipole moment.
57
Electric Field at an Equatorial Point of a Dipole .
For point P on the equatorial plane due to charges +q and –q, electric field of
dipole at a large distance
�(���??????���????????????�)=
�
�????????????
�
�
�
�
����??????���????????????�=−
�
�????????????
�
�
�
�
ෝ�
58
For point P on the equatorial plane due to charges +q and –q, electric field of
dipole at a large distance
�(���??????���????????????�)=
�
�????????????
�
�
�
�
����??????���????????????�=−
�
�????????????
�
�
�
�
ෝ�
58
Derivation: Electric Field at an Equatorial Point of a Dipole .
59
59
Derivation: Electric Field at an Equatorial Point of a Dipole .
60
60
Comparison of electric fields of a short dipole
at axial and equatorial points.
�(??????�????????????�)=
�
�????????????
�
��
�
�
����??????=
�
�????????????
�
�
�
�
61
at axial and equatorial points.
�(??????�????????????�)=
�
�????????????
�
��
�
�
����??????=
�
�????????????
�
�
�
�
61
Torque on a Dipole in a Uniform Electric Field .
??????=��sin??????
Ԧ??????=Ԧ�×�
??????
??????????????????=��sin90°=��
62
??????=��sin??????
Ԧ??????=Ԧ�×�
??????
??????????????????=��sin90°=��
62
Derivation:Torqueon a Dipole in a Uniform Electric Field .
63
63
Torque on a Dipole .
64
64
Stable & unstable equilibrium of Dipole in a
Uniform Electric Field
65
Uniform Electric Field
65
Q. An electric dipole, when held at 30°with respect to a uniform
electric field of 10
4
�??????
−1
experiences a torque of 9×10
−26
��.
Calculate dipole moment of the dipole.
66
electric field of 10
4
�??????
−1
experiences a torque of 9×10
−26
��.
Calculate dipole moment of the dipole.
66
Area Vector .
67
67
Electric Flux .
68
68
Units of Electric Flux .
??????Electric flux is a scalar quantity.
??????SI unit of electric flux
=��
2
??????
−1
??????Equivalently, SI unit of electric flux = V m.
69
??????Electric flux is a scalar quantity.
??????SI unit of electric flux
=��
2
??????
−1
??????Equivalently, SI unit of electric flux = V m.
69
Q.If�=6Ƹ�+3Ƹ�+4�,calculatetheelectricfluxthrougha
surfaceofarea20unitsin�−�plane.
70
surfaceofarea20unitsin�−�plane.
70
Gauss's Theorem .
➢Gauss Theorem states that the total flux through a closed surface is 1/??????
0
times the net charge enclosed by the closed surface. Mathematically,
71
➢Gauss Theorem states that the total flux through a closed surface is 1/??????
0
times the net charge enclosed by the closed surface. Mathematically,
71
Imp. Points Gauss's Theorem .
72
72
Gaussian Surface .
Gaussian surface:Any hypothetical closed surface
enclosing a charge is called the Gaussian surface of
that charge.
Importance: By a clever choice of Gaussian surface,
we can easily find the electric fields produced by
certain symmetric charge configurations
which are otherwise quite difficult to evaluate by
the direct application of Coulomb's law and the
principle of superposition.
73
Gaussian surface:Any hypothetical closed surface
enclosing a charge is called the Gaussian surface of
that charge.
Importance: By a clever choice of Gaussian surface,
we can easily find the electric fields produced by
certain symmetric charge configurations
which are otherwise quite difficult to evaluate by
the direct application of Coulomb's law and the
principle of superposition.
73
74
Application of
Gauss’s Theorem
75
Gauss’s Theorem
75
76
Electric Field .
Electric Field .
Electric Field due to an Infinitely Long Charged Wire .
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77
Derivation: Electric Field due to an Infinitely Long Charged Wire .
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78
Electric Field due to an Infinitely Plane sheet .
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79
Electric Field due to an Infinitely Plane sheet .
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80
Electric field of two oppositely charged plane parallel plates .
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81
Field Due To A Uniformly Charged Thin Spherical Shell .
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82
Derivation: Field Due To A Uniformly Charged Thin Spherical Shell .
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83
Arvind Academy
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84
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