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1 | 690-693 | e , component of E along the
normal to the area element times the magnitude of the area element The
unit of electric flux is N C–1 m2 The basic definition of electric flux given by Eq |
1 | 691-694 | , component of E along the
normal to the area element times the magnitude of the area element The
unit of electric flux is N C–1 m2 The basic definition of electric flux given by Eq (1 |
1 | 692-695 | The
unit of electric flux is N C–1 m2 The basic definition of electric flux given by Eq (1 11) can be used, in
principle, to calculate the total flux through any given surface |
1 | 693-696 | The basic definition of electric flux given by Eq (1 11) can be used, in
principle, to calculate the total flux through any given surface All we
have to do is to divide the surface into small area elements, calculate the
flux at each element and add them up |
1 | 694-697 | (1 11) can be used, in
principle, to calculate the total flux through any given surface All we
have to do is to divide the surface into small area elements, calculate the
flux at each element and add them up Thus, the total flux f through a
surface S is
f ~ S E |
1 | 695-698 | 11) can be used, in
principle, to calculate the total flux through any given surface All we
have to do is to divide the surface into small area elements, calculate the
flux at each element and add them up Thus, the total flux f through a
surface S is
f ~ S E DS
(1 |
1 | 696-699 | All we
have to do is to divide the surface into small area elements, calculate the
flux at each element and add them up Thus, the total flux f through a
surface S is
f ~ S E DS
(1 12)
The approximation sign is put because the electric field E is taken to
be constant over the small area element |
1 | 697-700 | Thus, the total flux f through a
surface S is
f ~ S E DS
(1 12)
The approximation sign is put because the electric field E is taken to
be constant over the small area element This is mathematically exact
only when you take the limit DS ® 0 and the sum in Eq |
1 | 698-701 | DS
(1 12)
The approximation sign is put because the electric field E is taken to
be constant over the small area element This is mathematically exact
only when you take the limit DS ® 0 and the sum in Eq (1 |
1 | 699-702 | 12)
The approximation sign is put because the electric field E is taken to
be constant over the small area element This is mathematically exact
only when you take the limit DS ® 0 and the sum in Eq (1 12) is written
as an integral |
1 | 700-703 | This is mathematically exact
only when you take the limit DS ® 0 and the sum in Eq (1 12) is written
as an integral 1 |
1 | 701-704 | (1 12) is written
as an integral 1 10 ELECTRIC DIPOLE
An electric dipole is a pair of equal and opposite point charges q and –q,
separated by a distance 2a |
1 | 702-705 | 12) is written
as an integral 1 10 ELECTRIC DIPOLE
An electric dipole is a pair of equal and opposite point charges q and –q,
separated by a distance 2a The line connecting the two charges defines
a direction in space |
1 | 703-706 | 1 10 ELECTRIC DIPOLE
An electric dipole is a pair of equal and opposite point charges q and –q,
separated by a distance 2a The line connecting the two charges defines
a direction in space By convention, the direction from –q to q is said to
be the direction of the dipole |
1 | 704-707 | 10 ELECTRIC DIPOLE
An electric dipole is a pair of equal and opposite point charges q and –q,
separated by a distance 2a The line connecting the two charges defines
a direction in space By convention, the direction from –q to q is said to
be the direction of the dipole The mid-point of locations of –q and q is
called the centre of the dipole |
1 | 705-708 | The line connecting the two charges defines
a direction in space By convention, the direction from –q to q is said to
be the direction of the dipole The mid-point of locations of –q and q is
called the centre of the dipole The total charge of the electric dipole is obviously zero |
1 | 706-709 | By convention, the direction from –q to q is said to
be the direction of the dipole The mid-point of locations of –q and q is
called the centre of the dipole The total charge of the electric dipole is obviously zero This does not
mean that the field of the electric dipole is zero |
1 | 707-710 | The mid-point of locations of –q and q is
called the centre of the dipole The total charge of the electric dipole is obviously zero This does not
mean that the field of the electric dipole is zero Since the charge q and
–q are separated by some distance, the electric fields due to them, when
added, do not exactly cancel out |
1 | 708-711 | The total charge of the electric dipole is obviously zero This does not
mean that the field of the electric dipole is zero Since the charge q and
–q are separated by some distance, the electric fields due to them, when
added, do not exactly cancel out However, at distances much larger than
the separation of the two charges forming a dipole (r >> 2a), the fields
due to q and –q nearly cancel out |
1 | 709-712 | This does not
mean that the field of the electric dipole is zero Since the charge q and
–q are separated by some distance, the electric fields due to them, when
added, do not exactly cancel out However, at distances much larger than
the separation of the two charges forming a dipole (r >> 2a), the fields
due to q and –q nearly cancel out The electric field due to a dipole
therefore falls off, at large distance, faster than like 1/r 2 (the dependence
on r of the field due to a single charge q) |
1 | 710-713 | Since the charge q and
–q are separated by some distance, the electric fields due to them, when
added, do not exactly cancel out However, at distances much larger than
the separation of the two charges forming a dipole (r >> 2a), the fields
due to q and –q nearly cancel out The electric field due to a dipole
therefore falls off, at large distance, faster than like 1/r 2 (the dependence
on r of the field due to a single charge q) These qualitative ideas are
borne out by the explicit calculation as follows:
1 |
1 | 711-714 | However, at distances much larger than
the separation of the two charges forming a dipole (r >> 2a), the fields
due to q and –q nearly cancel out The electric field due to a dipole
therefore falls off, at large distance, faster than like 1/r 2 (the dependence
on r of the field due to a single charge q) These qualitative ideas are
borne out by the explicit calculation as follows:
1 10 |
1 | 712-715 | The electric field due to a dipole
therefore falls off, at large distance, faster than like 1/r 2 (the dependence
on r of the field due to a single charge q) These qualitative ideas are
borne out by the explicit calculation as follows:
1 10 1 The field of an electric dipole
The electric field of the pair of charges (–q and q) at any point in space
can be found out from Coulomb’s law and the superposition principle |
1 | 713-716 | These qualitative ideas are
borne out by the explicit calculation as follows:
1 10 1 The field of an electric dipole
The electric field of the pair of charges (–q and q) at any point in space
can be found out from Coulomb’s law and the superposition principle The results are simple for the following two cases: (i) when the point is on
the dipole axis, and (ii) when it is in the equatorial plane of the dipole,
i |
1 | 714-717 | 10 1 The field of an electric dipole
The electric field of the pair of charges (–q and q) at any point in space
can be found out from Coulomb’s law and the superposition principle The results are simple for the following two cases: (i) when the point is on
the dipole axis, and (ii) when it is in the equatorial plane of the dipole,
i e |
1 | 715-718 | 1 The field of an electric dipole
The electric field of the pair of charges (–q and q) at any point in space
can be found out from Coulomb’s law and the superposition principle The results are simple for the following two cases: (i) when the point is on
the dipole axis, and (ii) when it is in the equatorial plane of the dipole,
i e , on a plane perpendicular to the dipole axis through its centre |
1 | 716-719 | The results are simple for the following two cases: (i) when the point is on
the dipole axis, and (ii) when it is in the equatorial plane of the dipole,
i e , on a plane perpendicular to the dipole axis through its centre The
electric field at any general point P is obtained by adding the electric
fields E–q due to the charge –q and E+q due to the charge q, by the
parallelogram law of vectors |
1 | 717-720 | e , on a plane perpendicular to the dipole axis through its centre The
electric field at any general point P is obtained by adding the electric
fields E–q due to the charge –q and E+q due to the charge q, by the
parallelogram law of vectors (i) For points on the axis
Let the point P be at distance r from the centre of the dipole on the side of
the charge q, as shown in Fig |
1 | 718-721 | , on a plane perpendicular to the dipole axis through its centre The
electric field at any general point P is obtained by adding the electric
fields E–q due to the charge –q and E+q due to the charge q, by the
parallelogram law of vectors (i) For points on the axis
Let the point P be at distance r from the centre of the dipole on the side of
the charge q, as shown in Fig 1 |
1 | 719-722 | The
electric field at any general point P is obtained by adding the electric
fields E–q due to the charge –q and E+q due to the charge q, by the
parallelogram law of vectors (i) For points on the axis
Let the point P be at distance r from the centre of the dipole on the side of
the charge q, as shown in Fig 1 17(a) |
1 | 720-723 | (i) For points on the axis
Let the point P be at distance r from the centre of the dipole on the side of
the charge q, as shown in Fig 1 17(a) Then
E
p
−
= −
+
q
rq
a
4
0
2
πε (
)
�
[1 |
1 | 721-724 | 1 17(a) Then
E
p
−
= −
+
q
rq
a
4
0
2
πε (
)
�
[1 13(a)]
where ˆp is the unit vector along the dipole axis (from –q to q) |
1 | 722-725 | 17(a) Then
E
p
−
= −
+
q
rq
a
4
0
2
πε (
)
�
[1 13(a)]
where ˆp is the unit vector along the dipole axis (from –q to q) Also
E
p
+
=
−
q
rq
a
4
0
2
π ε (
)
�
[1 |
1 | 723-726 | Then
E
p
−
= −
+
q
rq
a
4
0
2
πε (
)
�
[1 13(a)]
where ˆp is the unit vector along the dipole axis (from –q to q) Also
E
p
+
=
−
q
rq
a
4
0
2
π ε (
)
�
[1 13(b)]
Rationalised 2023-24
24
Physics
The total field at P is
E
E
E
p
=
+
=
−
−
+
+
−
q
q
q
r
a
r
a
4
1
1
0
2
2
π ε
(
)
(
)
�
=
−
q
a r
r
a
o
4
4
2
2 2
π ε
(
)
�p
(1 |
1 | 724-727 | 13(a)]
where ˆp is the unit vector along the dipole axis (from –q to q) Also
E
p
+
=
−
q
rq
a
4
0
2
π ε (
)
�
[1 13(b)]
Rationalised 2023-24
24
Physics
The total field at P is
E
E
E
p
=
+
=
−
−
+
+
−
q
q
q
r
a
r
a
4
1
1
0
2
2
π ε
(
)
(
)
�
=
−
q
a r
r
a
o
4
4
2
2 2
π ε
(
)
�p
(1 14)
For r >> a
E
p
=
44
0
q a3
πεr
ˆ (r >> a)
(1 |
1 | 725-728 | Also
E
p
+
=
−
q
rq
a
4
0
2
π ε (
)
�
[1 13(b)]
Rationalised 2023-24
24
Physics
The total field at P is
E
E
E
p
=
+
=
−
−
+
+
−
q
q
q
r
a
r
a
4
1
1
0
2
2
π ε
(
)
(
)
�
=
−
q
a r
r
a
o
4
4
2
2 2
π ε
(
)
�p
(1 14)
For r >> a
E
p
=
44
0
q a3
πεr
ˆ (r >> a)
(1 15)
(ii) For points on the equatorial plane
The magnitudes of the electric fields due to the two
charges +q and –q are given by
E
q
r
a
+q
=
+
4
1
0
2
2
πε
[1 |
1 | 726-729 | 13(b)]
Rationalised 2023-24
24
Physics
The total field at P is
E
E
E
p
=
+
=
−
−
+
+
−
q
q
q
r
a
r
a
4
1
1
0
2
2
π ε
(
)
(
)
�
=
−
q
a r
r
a
o
4
4
2
2 2
π ε
(
)
�p
(1 14)
For r >> a
E
p
=
44
0
q a3
πεr
ˆ (r >> a)
(1 15)
(ii) For points on the equatorial plane
The magnitudes of the electric fields due to the two
charges +q and –q are given by
E
q
r
a
+q
=
+
4
1
0
2
2
πε
[1 16(a)]
E
q
r
a
–q
=
+
4
1
0
2
2
πε
[1 |
1 | 727-730 | 14)
For r >> a
E
p
=
44
0
q a3
πεr
ˆ (r >> a)
(1 15)
(ii) For points on the equatorial plane
The magnitudes of the electric fields due to the two
charges +q and –q are given by
E
q
r
a
+q
=
+
4
1
0
2
2
πε
[1 16(a)]
E
q
r
a
–q
=
+
4
1
0
2
2
πε
[1 16(b)]
and are equal |
1 | 728-731 | 15)
(ii) For points on the equatorial plane
The magnitudes of the electric fields due to the two
charges +q and –q are given by
E
q
r
a
+q
=
+
4
1
0
2
2
πε
[1 16(a)]
E
q
r
a
–q
=
+
4
1
0
2
2
πε
[1 16(b)]
and are equal The directions of E+q and E–q are as shown in
Fig |
1 | 729-732 | 16(a)]
E
q
r
a
–q
=
+
4
1
0
2
2
πε
[1 16(b)]
and are equal The directions of E+q and E–q are as shown in
Fig 1 |
1 | 730-733 | 16(b)]
and are equal The directions of E+q and E–q are as shown in
Fig 1 17(b) |
1 | 731-734 | The directions of E+q and E–q are as shown in
Fig 1 17(b) Clearly, the components normal to the dipole
axis cancel away |
1 | 732-735 | 1 17(b) Clearly, the components normal to the dipole
axis cancel away The components along the dipole axis
add up |
1 | 733-736 | 17(b) Clearly, the components normal to the dipole
axis cancel away The components along the dipole axis
add up The total electric field is opposite to ˆp |
1 | 734-737 | Clearly, the components normal to the dipole
axis cancel away The components along the dipole axis
add up The total electric field is opposite to ˆp We have
E = – (E +q + E –q ) cosq ˆp
= −
2+
4
2
2 3
2
rq a
a
o
π ε (
)
�
/
p
(1 |
1 | 735-738 | The components along the dipole axis
add up The total electric field is opposite to ˆp We have
E = – (E +q + E –q ) cosq ˆp
= −
2+
4
2
2 3
2
rq a
a
o
π ε (
)
�
/
p
(1 17)
At large distances (r >> a), this reduces to
E
p
= −
>>
42
q a3
r
r
a
π εo
ˆ
(
)
(1 |
1 | 736-739 | The total electric field is opposite to ˆp We have
E = – (E +q + E –q ) cosq ˆp
= −
2+
4
2
2 3
2
rq a
a
o
π ε (
)
�
/
p
(1 17)
At large distances (r >> a), this reduces to
E
p
= −
>>
42
q a3
r
r
a
π εo
ˆ
(
)
(1 18)
From Eqs |
1 | 737-740 | We have
E = – (E +q + E –q ) cosq ˆp
= −
2+
4
2
2 3
2
rq a
a
o
π ε (
)
�
/
p
(1 17)
At large distances (r >> a), this reduces to
E
p
= −
>>
42
q a3
r
r
a
π εo
ˆ
(
)
(1 18)
From Eqs (1 |
1 | 738-741 | 17)
At large distances (r >> a), this reduces to
E
p
= −
>>
42
q a3
r
r
a
π εo
ˆ
(
)
(1 18)
From Eqs (1 15) and (1 |
1 | 739-742 | 18)
From Eqs (1 15) and (1 18), it is clear that the dipole field at large
distances does not involve q and a separately; it depends on the product
qa |
1 | 740-743 | (1 15) and (1 18), it is clear that the dipole field at large
distances does not involve q and a separately; it depends on the product
qa This suggests the definition of dipole moment |
1 | 741-744 | 15) and (1 18), it is clear that the dipole field at large
distances does not involve q and a separately; it depends on the product
qa This suggests the definition of dipole moment The dipole moment
vector p of an electric dipole is defined by
p = q × 2a ˆp
(1 |
1 | 742-745 | 18), it is clear that the dipole field at large
distances does not involve q and a separately; it depends on the product
qa This suggests the definition of dipole moment The dipole moment
vector p of an electric dipole is defined by
p = q × 2a ˆp
(1 19)
that is, it is a vector whose magnitude is charge q times the separation
2a (between the pair of charges q, –q) and the direction is along the line
from –q to q |
1 | 743-746 | This suggests the definition of dipole moment The dipole moment
vector p of an electric dipole is defined by
p = q × 2a ˆp
(1 19)
that is, it is a vector whose magnitude is charge q times the separation
2a (between the pair of charges q, –q) and the direction is along the line
from –q to q In terms of p, the electric field of a dipole at large distances
takes simple forms:
At a point on the dipole axis
E
p
=
42
πεor3
(r >> a)
(1 |
1 | 744-747 | The dipole moment
vector p of an electric dipole is defined by
p = q × 2a ˆp
(1 19)
that is, it is a vector whose magnitude is charge q times the separation
2a (between the pair of charges q, –q) and the direction is along the line
from –q to q In terms of p, the electric field of a dipole at large distances
takes simple forms:
At a point on the dipole axis
E
p
=
42
πεor3
(r >> a)
(1 20)
At a point on the equatorial plane
E
= − 4p
πεor3
(r >> a)
(1 |
1 | 745-748 | 19)
that is, it is a vector whose magnitude is charge q times the separation
2a (between the pair of charges q, –q) and the direction is along the line
from –q to q In terms of p, the electric field of a dipole at large distances
takes simple forms:
At a point on the dipole axis
E
p
=
42
πεor3
(r >> a)
(1 20)
At a point on the equatorial plane
E
= − 4p
πεor3
(r >> a)
(1 21)
FIGURE 1 |
1 | 746-749 | In terms of p, the electric field of a dipole at large distances
takes simple forms:
At a point on the dipole axis
E
p
=
42
πεor3
(r >> a)
(1 20)
At a point on the equatorial plane
E
= − 4p
πεor3
(r >> a)
(1 21)
FIGURE 1 17 Electric field of a dipole
at (a) a point on the axis, (b) a point
on the equatorial plane of the dipole |
1 | 747-750 | 20)
At a point on the equatorial plane
E
= − 4p
πεor3
(r >> a)
(1 21)
FIGURE 1 17 Electric field of a dipole
at (a) a point on the axis, (b) a point
on the equatorial plane of the dipole p is the dipole moment vector of
magnitude p = q × 2a and
directed from –q to q |
1 | 748-751 | 21)
FIGURE 1 17 Electric field of a dipole
at (a) a point on the axis, (b) a point
on the equatorial plane of the dipole p is the dipole moment vector of
magnitude p = q × 2a and
directed from –q to q Rationalised 2023-24
Electric Charges
and Fields
25
EXAMPLE 1 |
1 | 749-752 | 17 Electric field of a dipole
at (a) a point on the axis, (b) a point
on the equatorial plane of the dipole p is the dipole moment vector of
magnitude p = q × 2a and
directed from –q to q Rationalised 2023-24
Electric Charges
and Fields
25
EXAMPLE 1 9
Notice the important point that the dipole field at large distances
falls off not as 1/r 2 but as1/r 3 |
1 | 750-753 | p is the dipole moment vector of
magnitude p = q × 2a and
directed from –q to q Rationalised 2023-24
Electric Charges
and Fields
25
EXAMPLE 1 9
Notice the important point that the dipole field at large distances
falls off not as 1/r 2 but as1/r 3 Further, the magnitude and the direction
of the dipole field depends not only on the distance r but also on the
angle between the position vector r and the dipole moment p |
1 | 751-754 | Rationalised 2023-24
Electric Charges
and Fields
25
EXAMPLE 1 9
Notice the important point that the dipole field at large distances
falls off not as 1/r 2 but as1/r 3 Further, the magnitude and the direction
of the dipole field depends not only on the distance r but also on the
angle between the position vector r and the dipole moment p We can think of the limit when the dipole size 2a approaches zero,
the charge q approaches infinity in such a way that the product
p = q × 2a is finite |
1 | 752-755 | 9
Notice the important point that the dipole field at large distances
falls off not as 1/r 2 but as1/r 3 Further, the magnitude and the direction
of the dipole field depends not only on the distance r but also on the
angle between the position vector r and the dipole moment p We can think of the limit when the dipole size 2a approaches zero,
the charge q approaches infinity in such a way that the product
p = q × 2a is finite Such a dipole is referred to as a point dipole |
1 | 753-756 | Further, the magnitude and the direction
of the dipole field depends not only on the distance r but also on the
angle between the position vector r and the dipole moment p We can think of the limit when the dipole size 2a approaches zero,
the charge q approaches infinity in such a way that the product
p = q × 2a is finite Such a dipole is referred to as a point dipole For a
point dipole, Eqs |
1 | 754-757 | We can think of the limit when the dipole size 2a approaches zero,
the charge q approaches infinity in such a way that the product
p = q × 2a is finite Such a dipole is referred to as a point dipole For a
point dipole, Eqs (1 |
1 | 755-758 | Such a dipole is referred to as a point dipole For a
point dipole, Eqs (1 20) and (1 |
1 | 756-759 | For a
point dipole, Eqs (1 20) and (1 21) are exact, true for any r |
1 | 757-760 | (1 20) and (1 21) are exact, true for any r 1 |
1 | 758-761 | 20) and (1 21) are exact, true for any r 1 10 |
1 | 759-762 | 21) are exact, true for any r 1 10 2 Physical significance of dipoles
In most molecules, the centres of positive charges and of negative charges*
lie at the same place |
1 | 760-763 | 1 10 2 Physical significance of dipoles
In most molecules, the centres of positive charges and of negative charges*
lie at the same place Therefore, their dipole moment is zero |
1 | 761-764 | 10 2 Physical significance of dipoles
In most molecules, the centres of positive charges and of negative charges*
lie at the same place Therefore, their dipole moment is zero CO2 and
CH4 are of this type of molecules |
1 | 762-765 | 2 Physical significance of dipoles
In most molecules, the centres of positive charges and of negative charges*
lie at the same place Therefore, their dipole moment is zero CO2 and
CH4 are of this type of molecules However, they develop a dipole moment
when an electric field is applied |
1 | 763-766 | Therefore, their dipole moment is zero CO2 and
CH4 are of this type of molecules However, they develop a dipole moment
when an electric field is applied But in some molecules, the centres of
negative charges and of positive charges do not coincide |
1 | 764-767 | CO2 and
CH4 are of this type of molecules However, they develop a dipole moment
when an electric field is applied But in some molecules, the centres of
negative charges and of positive charges do not coincide Therefore they
have a permanent electric dipole moment, even in the absence of an electric
field |
1 | 765-768 | However, they develop a dipole moment
when an electric field is applied But in some molecules, the centres of
negative charges and of positive charges do not coincide Therefore they
have a permanent electric dipole moment, even in the absence of an electric
field Such molecules are called polar molecules |
1 | 766-769 | But in some molecules, the centres of
negative charges and of positive charges do not coincide Therefore they
have a permanent electric dipole moment, even in the absence of an electric
field Such molecules are called polar molecules Water molecules, H2O,
is an example of this type |
1 | 767-770 | Therefore they
have a permanent electric dipole moment, even in the absence of an electric
field Such molecules are called polar molecules Water molecules, H2O,
is an example of this type Various materials give rise to interesting
properties and important applications in the presence or absence of
electric field |
1 | 768-771 | Such molecules are called polar molecules Water molecules, H2O,
is an example of this type Various materials give rise to interesting
properties and important applications in the presence or absence of
electric field Example 1 |
1 | 769-772 | Water molecules, H2O,
is an example of this type Various materials give rise to interesting
properties and important applications in the presence or absence of
electric field Example 1 9 Two charges ±10 mC are placed 5 |
1 | 770-773 | Various materials give rise to interesting
properties and important applications in the presence or absence of
electric field Example 1 9 Two charges ±10 mC are placed 5 0 mm apart |
1 | 771-774 | Example 1 9 Two charges ±10 mC are placed 5 0 mm apart Determine
the electric field at (a) a point P on the axis of the dipole 15 cm away
from its centre O on the side of the positive charge, as shown in Fig |
1 | 772-775 | 9 Two charges ±10 mC are placed 5 0 mm apart Determine
the electric field at (a) a point P on the axis of the dipole 15 cm away
from its centre O on the side of the positive charge, as shown in Fig 1 |
1 | 773-776 | 0 mm apart Determine
the electric field at (a) a point P on the axis of the dipole 15 cm away
from its centre O on the side of the positive charge, as shown in Fig 1 18(a), and (b) a point Q, 15 cm away from O on a line passing through
O and normal to the axis of the dipole, as shown in Fig |
1 | 774-777 | Determine
the electric field at (a) a point P on the axis of the dipole 15 cm away
from its centre O on the side of the positive charge, as shown in Fig 1 18(a), and (b) a point Q, 15 cm away from O on a line passing through
O and normal to the axis of the dipole, as shown in Fig 1 |
1 | 775-778 | 1 18(a), and (b) a point Q, 15 cm away from O on a line passing through
O and normal to the axis of the dipole, as shown in Fig 1 18(b) |
1 | 776-779 | 18(a), and (b) a point Q, 15 cm away from O on a line passing through
O and normal to the axis of the dipole, as shown in Fig 1 18(b) FIGURE 1 |
1 | 777-780 | 1 18(b) FIGURE 1 18
*
Centre of a collection of positive point charges is defined much the same way
as the centre of mass: r
r
cm =
∑
∑
q
q
i i
i
i
i |
1 | 778-781 | 18(b) FIGURE 1 18
*
Centre of a collection of positive point charges is defined much the same way
as the centre of mass: r
r
cm =
∑
∑
q
q
i i
i
i
i Rationalised 2023-24
26
Physics
EXAMPLE 1 |
1 | 779-782 | FIGURE 1 18
*
Centre of a collection of positive point charges is defined much the same way
as the centre of mass: r
r
cm =
∑
∑
q
q
i i
i
i
i Rationalised 2023-24
26
Physics
EXAMPLE 1 9
Solution (a) Field at P due to charge +10 mC
=
5
12
2
1
2
10
C
4 (8 |
1 | 780-783 | 18
*
Centre of a collection of positive point charges is defined much the same way
as the centre of mass: r
r
cm =
∑
∑
q
q
i i
i
i
i Rationalised 2023-24
26
Physics
EXAMPLE 1 9
Solution (a) Field at P due to charge +10 mC
=
5
12
2
1
2
10
C
4 (8 854 10
C N
m
)
−
−
−
−
π
×
2
4
2
1
(15
0 |
1 | 781-784 | Rationalised 2023-24
26
Physics
EXAMPLE 1 9
Solution (a) Field at P due to charge +10 mC
=
5
12
2
1
2
10
C
4 (8 854 10
C N
m
)
−
−
−
−
π
×
2
4
2
1
(15
0 25)
10
m
−
×
−
×
= 4 |
1 | 782-785 | 9
Solution (a) Field at P due to charge +10 mC
=
5
12
2
1
2
10
C
4 (8 854 10
C N
m
)
−
−
−
−
π
×
2
4
2
1
(15
0 25)
10
m
−
×
−
×
= 4 13 × 106 N C–1 along BP
Field at P due to charge –10 mC
–5
12
2
1
2
10
C
4 (8 |
1 | 783-786 | 854 10
C N
m
)
−
−
−
−
π
×
2
4
2
1
(15
0 25)
10
m
−
×
−
×
= 4 13 × 106 N C–1 along BP
Field at P due to charge –10 mC
–5
12
2
1
2
10
C
4 (8 854 10
C N
m
)
−
−
−
=
π
×
2
4
2
1
(15
0 |
1 | 784-787 | 25)
10
m
−
×
−
×
= 4 13 × 106 N C–1 along BP
Field at P due to charge –10 mC
–5
12
2
1
2
10
C
4 (8 854 10
C N
m
)
−
−
−
=
π
×
2
4
2
1
(15
0 25)
10
m
−
×
+
×
= 3 |
1 | 785-788 | 13 × 106 N C–1 along BP
Field at P due to charge –10 mC
–5
12
2
1
2
10
C
4 (8 854 10
C N
m
)
−
−
−
=
π
×
2
4
2
1
(15
0 25)
10
m
−
×
+
×
= 3 86 × 106 N C–1 along PA
The resultant electric field at P due to the two charges at A and B is
= 2 |
1 | 786-789 | 854 10
C N
m
)
−
−
−
=
π
×
2
4
2
1
(15
0 25)
10
m
−
×
+
×
= 3 86 × 106 N C–1 along PA
The resultant electric field at P due to the two charges at A and B is
= 2 7 × 105 N C–1 along BP |
1 | 787-790 | 25)
10
m
−
×
+
×
= 3 86 × 106 N C–1 along PA
The resultant electric field at P due to the two charges at A and B is
= 2 7 × 105 N C–1 along BP In this example, the ratio OP/OB is quite large (= 60) |
1 | 788-791 | 86 × 106 N C–1 along PA
The resultant electric field at P due to the two charges at A and B is
= 2 7 × 105 N C–1 along BP In this example, the ratio OP/OB is quite large (= 60) Thus, we can
expect to get approximately the same result as above by directly using
the formula for electric field at a far-away point on the axis of a dipole |
1 | 789-792 | 7 × 105 N C–1 along BP In this example, the ratio OP/OB is quite large (= 60) Thus, we can
expect to get approximately the same result as above by directly using
the formula for electric field at a far-away point on the axis of a dipole For a dipole consisting of charges ± q, 2a distance apart, the electric
field at a distance r from the centre on the axis of the dipole has a
magnitude
E
p
r
=
2
4
0
3
πε
(r/a >> 1)
where p = 2a q is the magnitude of the dipole moment |
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