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1 | 1690-1693 | 23) and Eq (2 25)],
U
rq q
rq q
q q
r
=
+
+
41
0
1
2
12
1
3
13
2
3
23
πε
(2 26)
Again, because of the conservative nature of the
electrostatic force (or equivalently, the path
independence of work done), the final expression for
U, Eq |
1 | 1691-1694 | (2 25)],
U
rq q
rq q
q q
r
=
+
+
41
0
1
2
12
1
3
13
2
3
23
πε
(2 26)
Again, because of the conservative nature of the
electrostatic force (or equivalently, the path
independence of work done), the final expression for
U, Eq (2 |
1 | 1692-1695 | 25)],
U
rq q
rq q
q q
r
=
+
+
41
0
1
2
12
1
3
13
2
3
23
πε
(2 26)
Again, because of the conservative nature of the
electrostatic force (or equivalently, the path
independence of work done), the final expression for
U, Eq (2 26), is independent of the manner in which
the configuration is assembled |
1 | 1693-1696 | 26)
Again, because of the conservative nature of the
electrostatic force (or equivalently, the path
independence of work done), the final expression for
U, Eq (2 26), is independent of the manner in which
the configuration is assembled The potential energy
FIGURE 2 |
1 | 1694-1697 | (2 26), is independent of the manner in which
the configuration is assembled The potential energy
FIGURE 2 13 Potential energy of a
system of charges q1 and q2 is
directly proportional to the product
of charges and inversely to the
distance between them |
1 | 1695-1698 | 26), is independent of the manner in which
the configuration is assembled The potential energy
FIGURE 2 13 Potential energy of a
system of charges q1 and q2 is
directly proportional to the product
of charges and inversely to the
distance between them FIGURE 2 |
1 | 1696-1699 | The potential energy
FIGURE 2 13 Potential energy of a
system of charges q1 and q2 is
directly proportional to the product
of charges and inversely to the
distance between them FIGURE 2 14 Potential energy of a
system of three charges is given by
Eq |
1 | 1697-1700 | 13 Potential energy of a
system of charges q1 and q2 is
directly proportional to the product
of charges and inversely to the
distance between them FIGURE 2 14 Potential energy of a
system of three charges is given by
Eq (2 |
1 | 1698-1701 | FIGURE 2 14 Potential energy of a
system of three charges is given by
Eq (2 26), with the notation given
in the figure |
1 | 1699-1702 | 14 Potential energy of a
system of three charges is given by
Eq (2 26), with the notation given
in the figure Rationalised 2023-24
Electrostatic Potential
and Capacitance
57
EXAMPLE 2 |
1 | 1700-1703 | (2 26), with the notation given
in the figure Rationalised 2023-24
Electrostatic Potential
and Capacitance
57
EXAMPLE 2 4
is characteristic of the present state of configuration, and not the way
the state is achieved |
1 | 1701-1704 | 26), with the notation given
in the figure Rationalised 2023-24
Electrostatic Potential
and Capacitance
57
EXAMPLE 2 4
is characteristic of the present state of configuration, and not the way
the state is achieved Example 2 |
1 | 1702-1705 | Rationalised 2023-24
Electrostatic Potential
and Capacitance
57
EXAMPLE 2 4
is characteristic of the present state of configuration, and not the way
the state is achieved Example 2 4 Four charges are arranged at the corners of a square
ABCD of side d, as shown in Fig |
1 | 1703-1706 | 4
is characteristic of the present state of configuration, and not the way
the state is achieved Example 2 4 Four charges are arranged at the corners of a square
ABCD of side d, as shown in Fig 2 |
1 | 1704-1707 | Example 2 4 Four charges are arranged at the corners of a square
ABCD of side d, as shown in Fig 2 15 |
1 | 1705-1708 | 4 Four charges are arranged at the corners of a square
ABCD of side d, as shown in Fig 2 15 (a) Find the work required to
put together this arrangement |
1 | 1706-1709 | 2 15 (a) Find the work required to
put together this arrangement (b) A charge q0 is brought to the centre
E of the square, the four charges being held fixed at its corners |
1 | 1707-1710 | 15 (a) Find the work required to
put together this arrangement (b) A charge q0 is brought to the centre
E of the square, the four charges being held fixed at its corners How
much extra work is needed to do this |
1 | 1708-1711 | (a) Find the work required to
put together this arrangement (b) A charge q0 is brought to the centre
E of the square, the four charges being held fixed at its corners How
much extra work is needed to do this FIGURE 2 |
1 | 1709-1712 | (b) A charge q0 is brought to the centre
E of the square, the four charges being held fixed at its corners How
much extra work is needed to do this FIGURE 2 15
Solution
(a) Since the work done depends on the final arrangement of the
charges, and not on how they are put together, we calculate work
needed for one way of putting the charges at A, B, C and D |
1 | 1710-1713 | How
much extra work is needed to do this FIGURE 2 15
Solution
(a) Since the work done depends on the final arrangement of the
charges, and not on how they are put together, we calculate work
needed for one way of putting the charges at A, B, C and D Suppose,
first the charge +q is brought to A, and then the charges –q, +q, and
–q are brought to B, C and D, respectively |
1 | 1711-1714 | FIGURE 2 15
Solution
(a) Since the work done depends on the final arrangement of the
charges, and not on how they are put together, we calculate work
needed for one way of putting the charges at A, B, C and D Suppose,
first the charge +q is brought to A, and then the charges –q, +q, and
–q are brought to B, C and D, respectively The total work needed can
be calculated in steps:
(i)
Work needed to bring charge +q to A when no charge is present
elsewhere: this is zero |
1 | 1712-1715 | 15
Solution
(a) Since the work done depends on the final arrangement of the
charges, and not on how they are put together, we calculate work
needed for one way of putting the charges at A, B, C and D Suppose,
first the charge +q is brought to A, and then the charges –q, +q, and
–q are brought to B, C and D, respectively The total work needed can
be calculated in steps:
(i)
Work needed to bring charge +q to A when no charge is present
elsewhere: this is zero (ii) Work needed to bring –q to B when +q is at A |
1 | 1713-1716 | Suppose,
first the charge +q is brought to A, and then the charges –q, +q, and
–q are brought to B, C and D, respectively The total work needed can
be calculated in steps:
(i)
Work needed to bring charge +q to A when no charge is present
elsewhere: this is zero (ii) Work needed to bring –q to B when +q is at A This is given by
(charge at B) × (electrostatic potential at B due to charge +q at A)
= −
×
= −
q
q
d
q
d
4
4
0
2
0
π
π
ε
ε
(iii) Work needed to bring charge +q to C when +q is at A and –q is at
B |
1 | 1714-1717 | The total work needed can
be calculated in steps:
(i)
Work needed to bring charge +q to A when no charge is present
elsewhere: this is zero (ii) Work needed to bring –q to B when +q is at A This is given by
(charge at B) × (electrostatic potential at B due to charge +q at A)
= −
×
= −
q
q
d
q
d
4
4
0
2
0
π
π
ε
ε
(iii) Work needed to bring charge +q to C when +q is at A and –q is at
B This is given by (charge at C) × (potential at C due to charges
at A and B)
= +
+
+
−
q
q
d
q
d
4
2
4
0
0
π
π
ε
ε
=
−
−
q
d
2
0
4
1
1
2
πε
(iv) Work needed to bring –q to D when +q at A,–q at B, and +q at C |
1 | 1715-1718 | (ii) Work needed to bring –q to B when +q is at A This is given by
(charge at B) × (electrostatic potential at B due to charge +q at A)
= −
×
= −
q
q
d
q
d
4
4
0
2
0
π
π
ε
ε
(iii) Work needed to bring charge +q to C when +q is at A and –q is at
B This is given by (charge at C) × (potential at C due to charges
at A and B)
= +
+
+
−
q
q
d
q
d
4
2
4
0
0
π
π
ε
ε
=
−
−
q
d
2
0
4
1
1
2
πε
(iv) Work needed to bring –q to D when +q at A,–q at B, and +q at C This is given by (charge at D) × (potential at D due to charges at A,
B and C)
= −
+
+
−
+
q
q
d
q
d
q
d
4
4
2
4
0
0
0
π
π
π
ε
ε
ε
=
−
−
q
d
2
0
4
2
1
2
πε
Rationalised 2023-24
Physics
58
EXAMPLE 2 |
1 | 1716-1719 | This is given by
(charge at B) × (electrostatic potential at B due to charge +q at A)
= −
×
= −
q
q
d
q
d
4
4
0
2
0
π
π
ε
ε
(iii) Work needed to bring charge +q to C when +q is at A and –q is at
B This is given by (charge at C) × (potential at C due to charges
at A and B)
= +
+
+
−
q
q
d
q
d
4
2
4
0
0
π
π
ε
ε
=
−
−
q
d
2
0
4
1
1
2
πε
(iv) Work needed to bring –q to D when +q at A,–q at B, and +q at C This is given by (charge at D) × (potential at D due to charges at A,
B and C)
= −
+
+
−
+
q
q
d
q
d
q
d
4
4
2
4
0
0
0
π
π
π
ε
ε
ε
=
−
−
q
d
2
0
4
2
1
2
πε
Rationalised 2023-24
Physics
58
EXAMPLE 2 4
Add the work done in steps (i), (ii), (iii) and (iv) |
1 | 1717-1720 | This is given by (charge at C) × (potential at C due to charges
at A and B)
= +
+
+
−
q
q
d
q
d
4
2
4
0
0
π
π
ε
ε
=
−
−
q
d
2
0
4
1
1
2
πε
(iv) Work needed to bring –q to D when +q at A,–q at B, and +q at C This is given by (charge at D) × (potential at D due to charges at A,
B and C)
= −
+
+
−
+
q
q
d
q
d
q
d
4
4
2
4
0
0
0
π
π
π
ε
ε
ε
=
−
−
q
d
2
0
4
2
1
2
πε
Rationalised 2023-24
Physics
58
EXAMPLE 2 4
Add the work done in steps (i), (ii), (iii) and (iv) The total work
required is
=
−
+
+
−
+
−
q
d
2
0
4
0
1
1
1
2
2
1
2
πε
( )
( )
=
−
(−
)
q
d
2
0
4
4
2
πε
The work done depends only on the arrangement of the charges, and
not how they are assembled |
1 | 1718-1721 | This is given by (charge at D) × (potential at D due to charges at A,
B and C)
= −
+
+
−
+
q
q
d
q
d
q
d
4
4
2
4
0
0
0
π
π
π
ε
ε
ε
=
−
−
q
d
2
0
4
2
1
2
πε
Rationalised 2023-24
Physics
58
EXAMPLE 2 4
Add the work done in steps (i), (ii), (iii) and (iv) The total work
required is
=
−
+
+
−
+
−
q
d
2
0
4
0
1
1
1
2
2
1
2
πε
( )
( )
=
−
(−
)
q
d
2
0
4
4
2
πε
The work done depends only on the arrangement of the charges, and
not how they are assembled By definition, this is the total
electrostatic energy of the charges |
1 | 1719-1722 | 4
Add the work done in steps (i), (ii), (iii) and (iv) The total work
required is
=
−
+
+
−
+
−
q
d
2
0
4
0
1
1
1
2
2
1
2
πε
( )
( )
=
−
(−
)
q
d
2
0
4
4
2
πε
The work done depends only on the arrangement of the charges, and
not how they are assembled By definition, this is the total
electrostatic energy of the charges (Students may try calculating same work/energy by taking charges
in any other order they desire and convince themselves that the energy
will remain the same |
1 | 1720-1723 | The total work
required is
=
−
+
+
−
+
−
q
d
2
0
4
0
1
1
1
2
2
1
2
πε
( )
( )
=
−
(−
)
q
d
2
0
4
4
2
πε
The work done depends only on the arrangement of the charges, and
not how they are assembled By definition, this is the total
electrostatic energy of the charges (Students may try calculating same work/energy by taking charges
in any other order they desire and convince themselves that the energy
will remain the same )
(b) The extra work necessary to bring a charge q0 to the point E when
the four charges are at A, B, C and D is q0 × (electrostatic potential at
E due to the charges at A, B, C and D) |
1 | 1721-1724 | By definition, this is the total
electrostatic energy of the charges (Students may try calculating same work/energy by taking charges
in any other order they desire and convince themselves that the energy
will remain the same )
(b) The extra work necessary to bring a charge q0 to the point E when
the four charges are at A, B, C and D is q0 × (electrostatic potential at
E due to the charges at A, B, C and D) The electrostatic potential at
E is clearly zero since potential due to A and C is cancelled by that
due to B and D |
1 | 1722-1725 | (Students may try calculating same work/energy by taking charges
in any other order they desire and convince themselves that the energy
will remain the same )
(b) The extra work necessary to bring a charge q0 to the point E when
the four charges are at A, B, C and D is q0 × (electrostatic potential at
E due to the charges at A, B, C and D) The electrostatic potential at
E is clearly zero since potential due to A and C is cancelled by that
due to B and D Hence, no work is required to bring any charge to
point E |
1 | 1723-1726 | )
(b) The extra work necessary to bring a charge q0 to the point E when
the four charges are at A, B, C and D is q0 × (electrostatic potential at
E due to the charges at A, B, C and D) The electrostatic potential at
E is clearly zero since potential due to A and C is cancelled by that
due to B and D Hence, no work is required to bring any charge to
point E 2 |
1 | 1724-1727 | The electrostatic potential at
E is clearly zero since potential due to A and C is cancelled by that
due to B and D Hence, no work is required to bring any charge to
point E 2 8 POTENTIAL ENERGY IN AN EXTERNAL FIELD
2 |
1 | 1725-1728 | Hence, no work is required to bring any charge to
point E 2 8 POTENTIAL ENERGY IN AN EXTERNAL FIELD
2 8 |
1 | 1726-1729 | 2 8 POTENTIAL ENERGY IN AN EXTERNAL FIELD
2 8 1 Potential energy of a single charge
In Section 2 |
1 | 1727-1730 | 8 POTENTIAL ENERGY IN AN EXTERNAL FIELD
2 8 1 Potential energy of a single charge
In Section 2 7, the source of the electric field was specified – the charges
and their locations - and the potential energy of the system of those charges
was determined |
1 | 1728-1731 | 8 1 Potential energy of a single charge
In Section 2 7, the source of the electric field was specified – the charges
and their locations - and the potential energy of the system of those charges
was determined In this section, we ask a related but a distinct question |
1 | 1729-1732 | 1 Potential energy of a single charge
In Section 2 7, the source of the electric field was specified – the charges
and their locations - and the potential energy of the system of those charges
was determined In this section, we ask a related but a distinct question What is the potential energy of a charge q in a given field |
1 | 1730-1733 | 7, the source of the electric field was specified – the charges
and their locations - and the potential energy of the system of those charges
was determined In this section, we ask a related but a distinct question What is the potential energy of a charge q in a given field This question
was, in fact, the starting point that led us to the notion of the electrostatic
potential (Sections 2 |
1 | 1731-1734 | In this section, we ask a related but a distinct question What is the potential energy of a charge q in a given field This question
was, in fact, the starting point that led us to the notion of the electrostatic
potential (Sections 2 1 and 2 |
1 | 1732-1735 | What is the potential energy of a charge q in a given field This question
was, in fact, the starting point that led us to the notion of the electrostatic
potential (Sections 2 1 and 2 2) |
1 | 1733-1736 | This question
was, in fact, the starting point that led us to the notion of the electrostatic
potential (Sections 2 1 and 2 2) But here we address this question again
to clarify in what way it is different from the discussion in Section 2 |
1 | 1734-1737 | 1 and 2 2) But here we address this question again
to clarify in what way it is different from the discussion in Section 2 7 |
1 | 1735-1738 | 2) But here we address this question again
to clarify in what way it is different from the discussion in Section 2 7 The main difference is that we are now concerned with the potential
energy of a charge (or charges) in an external field |
1 | 1736-1739 | But here we address this question again
to clarify in what way it is different from the discussion in Section 2 7 The main difference is that we are now concerned with the potential
energy of a charge (or charges) in an external field The external field E is
not produced by the given charge(s) whose potential energy we wish to
calculate |
1 | 1737-1740 | 7 The main difference is that we are now concerned with the potential
energy of a charge (or charges) in an external field The external field E is
not produced by the given charge(s) whose potential energy we wish to
calculate E is produced by sources external to the given charge(s) |
1 | 1738-1741 | The main difference is that we are now concerned with the potential
energy of a charge (or charges) in an external field The external field E is
not produced by the given charge(s) whose potential energy we wish to
calculate E is produced by sources external to the given charge(s) The
external sources may be known, but often they are unknown or
unspecified; what is specified is the electric field E or the electrostatic
potential V due to the external sources |
1 | 1739-1742 | The external field E is
not produced by the given charge(s) whose potential energy we wish to
calculate E is produced by sources external to the given charge(s) The
external sources may be known, but often they are unknown or
unspecified; what is specified is the electric field E or the electrostatic
potential V due to the external sources We assume that the charge q
does not significantly affect the sources producing the external field |
1 | 1740-1743 | E is produced by sources external to the given charge(s) The
external sources may be known, but often they are unknown or
unspecified; what is specified is the electric field E or the electrostatic
potential V due to the external sources We assume that the charge q
does not significantly affect the sources producing the external field This
is true if q is very small, or the external sources are held fixed by other
unspecified forces |
1 | 1741-1744 | The
external sources may be known, but often they are unknown or
unspecified; what is specified is the electric field E or the electrostatic
potential V due to the external sources We assume that the charge q
does not significantly affect the sources producing the external field This
is true if q is very small, or the external sources are held fixed by other
unspecified forces Even if q is finite, its influence on the external sources
may still be ignored in the situation when very strong sources far away
at infinity produce a finite field E in the region of interest |
1 | 1742-1745 | We assume that the charge q
does not significantly affect the sources producing the external field This
is true if q is very small, or the external sources are held fixed by other
unspecified forces Even if q is finite, its influence on the external sources
may still be ignored in the situation when very strong sources far away
at infinity produce a finite field E in the region of interest Note again that
we are interested in determining the potential energy of a given charge q
(and later, a system of charges) in the external field; we are not interested
in the potential energy of the sources producing the external electric field |
1 | 1743-1746 | This
is true if q is very small, or the external sources are held fixed by other
unspecified forces Even if q is finite, its influence on the external sources
may still be ignored in the situation when very strong sources far away
at infinity produce a finite field E in the region of interest Note again that
we are interested in determining the potential energy of a given charge q
(and later, a system of charges) in the external field; we are not interested
in the potential energy of the sources producing the external electric field The external electric field E and the corresponding external potential
V may vary from point to point |
1 | 1744-1747 | Even if q is finite, its influence on the external sources
may still be ignored in the situation when very strong sources far away
at infinity produce a finite field E in the region of interest Note again that
we are interested in determining the potential energy of a given charge q
(and later, a system of charges) in the external field; we are not interested
in the potential energy of the sources producing the external electric field The external electric field E and the corresponding external potential
V may vary from point to point By definition, V at a point P is the work
done in bringing a unit positive charge from infinity to the point P |
1 | 1745-1748 | Note again that
we are interested in determining the potential energy of a given charge q
(and later, a system of charges) in the external field; we are not interested
in the potential energy of the sources producing the external electric field The external electric field E and the corresponding external potential
V may vary from point to point By definition, V at a point P is the work
done in bringing a unit positive charge from infinity to the point P Rationalised 2023-24
Electrostatic Potential
and Capacitance
59
EXAMPLE 2 |
1 | 1746-1749 | The external electric field E and the corresponding external potential
V may vary from point to point By definition, V at a point P is the work
done in bringing a unit positive charge from infinity to the point P Rationalised 2023-24
Electrostatic Potential
and Capacitance
59
EXAMPLE 2 5
(We continue to take potential at infinity to be zero |
1 | 1747-1750 | By definition, V at a point P is the work
done in bringing a unit positive charge from infinity to the point P Rationalised 2023-24
Electrostatic Potential
and Capacitance
59
EXAMPLE 2 5
(We continue to take potential at infinity to be zero ) Thus, work done in
bringing a charge q from infinity to the point P in the external field is qV |
1 | 1748-1751 | Rationalised 2023-24
Electrostatic Potential
and Capacitance
59
EXAMPLE 2 5
(We continue to take potential at infinity to be zero ) Thus, work done in
bringing a charge q from infinity to the point P in the external field is qV This work is stored in the form of potential energy of q |
1 | 1749-1752 | 5
(We continue to take potential at infinity to be zero ) Thus, work done in
bringing a charge q from infinity to the point P in the external field is qV This work is stored in the form of potential energy of q If the point P has
position vector r relative to some origin, we can write:
Potential energy of q at r in an external field
= qV(r)
(2 |
1 | 1750-1753 | ) Thus, work done in
bringing a charge q from infinity to the point P in the external field is qV This work is stored in the form of potential energy of q If the point P has
position vector r relative to some origin, we can write:
Potential energy of q at r in an external field
= qV(r)
(2 27)
where V(r) is the external potential at the point r |
1 | 1751-1754 | This work is stored in the form of potential energy of q If the point P has
position vector r relative to some origin, we can write:
Potential energy of q at r in an external field
= qV(r)
(2 27)
where V(r) is the external potential at the point r Thus, if an electron with charge q = e = 1 |
1 | 1752-1755 | If the point P has
position vector r relative to some origin, we can write:
Potential energy of q at r in an external field
= qV(r)
(2 27)
where V(r) is the external potential at the point r Thus, if an electron with charge q = e = 1 6×10–19 C is accelerated by
a potential difference of DV = 1 volt, it would gain energy of qDV = 1 |
1 | 1753-1756 | 27)
where V(r) is the external potential at the point r Thus, if an electron with charge q = e = 1 6×10–19 C is accelerated by
a potential difference of DV = 1 volt, it would gain energy of qDV = 1 6 ×
10–19J |
1 | 1754-1757 | Thus, if an electron with charge q = e = 1 6×10–19 C is accelerated by
a potential difference of DV = 1 volt, it would gain energy of qDV = 1 6 ×
10–19J This unit of energy is defined as 1 electron volt or 1eV, i |
1 | 1755-1758 | 6×10–19 C is accelerated by
a potential difference of DV = 1 volt, it would gain energy of qDV = 1 6 ×
10–19J This unit of energy is defined as 1 electron volt or 1eV, i e |
1 | 1756-1759 | 6 ×
10–19J This unit of energy is defined as 1 electron volt or 1eV, i e ,
1 eV=1 |
1 | 1757-1760 | This unit of energy is defined as 1 electron volt or 1eV, i e ,
1 eV=1 6 × 10–19J |
1 | 1758-1761 | e ,
1 eV=1 6 × 10–19J The units based on eV are most commonly used in
atomic, nuclear and particle physics, (1 keV = 103eV = 1 |
1 | 1759-1762 | ,
1 eV=1 6 × 10–19J The units based on eV are most commonly used in
atomic, nuclear and particle physics, (1 keV = 103eV = 1 6 × 10–16J, 1 MeV
= 106eV = 1 |
1 | 1760-1763 | 6 × 10–19J The units based on eV are most commonly used in
atomic, nuclear and particle physics, (1 keV = 103eV = 1 6 × 10–16J, 1 MeV
= 106eV = 1 6 × 10–13J, 1 GeV = 109eV = 1 |
1 | 1761-1764 | The units based on eV are most commonly used in
atomic, nuclear and particle physics, (1 keV = 103eV = 1 6 × 10–16J, 1 MeV
= 106eV = 1 6 × 10–13J, 1 GeV = 109eV = 1 6 × 10–10J and 1 TeV = 1012eV
= 1 |
1 | 1762-1765 | 6 × 10–16J, 1 MeV
= 106eV = 1 6 × 10–13J, 1 GeV = 109eV = 1 6 × 10–10J and 1 TeV = 1012eV
= 1 6 × 10–7J) |
1 | 1763-1766 | 6 × 10–13J, 1 GeV = 109eV = 1 6 × 10–10J and 1 TeV = 1012eV
= 1 6 × 10–7J) [This has already been defined on Page 117, XI Physics
Part I, Table 6 |
1 | 1764-1767 | 6 × 10–10J and 1 TeV = 1012eV
= 1 6 × 10–7J) [This has already been defined on Page 117, XI Physics
Part I, Table 6 1 |
1 | 1765-1768 | 6 × 10–7J) [This has already been defined on Page 117, XI Physics
Part I, Table 6 1 ]
2 |
1 | 1766-1769 | [This has already been defined on Page 117, XI Physics
Part I, Table 6 1 ]
2 8 |
1 | 1767-1770 | 1 ]
2 8 2
Potential energy of a system of two charges in an
external field
Next, we ask: what is the potential energy of a system of two charges q1
and q2 located at r1and r2, respectively, in an external field |
1 | 1768-1771 | ]
2 8 2
Potential energy of a system of two charges in an
external field
Next, we ask: what is the potential energy of a system of two charges q1
and q2 located at r1and r2, respectively, in an external field First, we
calculate the work done in bringing the charge q1 from infinity to r1 |
1 | 1769-1772 | 8 2
Potential energy of a system of two charges in an
external field
Next, we ask: what is the potential energy of a system of two charges q1
and q2 located at r1and r2, respectively, in an external field First, we
calculate the work done in bringing the charge q1 from infinity to r1 Work done in this step is q1 V(r1), using Eq |
1 | 1770-1773 | 2
Potential energy of a system of two charges in an
external field
Next, we ask: what is the potential energy of a system of two charges q1
and q2 located at r1and r2, respectively, in an external field First, we
calculate the work done in bringing the charge q1 from infinity to r1 Work done in this step is q1 V(r1), using Eq (2 |
1 | 1771-1774 | First, we
calculate the work done in bringing the charge q1 from infinity to r1 Work done in this step is q1 V(r1), using Eq (2 27) |
1 | 1772-1775 | Work done in this step is q1 V(r1), using Eq (2 27) Next, we consider the
work done in bringing q2 to r2 |
1 | 1773-1776 | (2 27) Next, we consider the
work done in bringing q2 to r2 In this step, work is done not only against
the external field E but also against the field due to q1 |
1 | 1774-1777 | 27) Next, we consider the
work done in bringing q2 to r2 In this step, work is done not only against
the external field E but also against the field due to q1 Work done on q2 against the external field
= q2 V (r2)
Work done on q2 against the field due to q1
1
2
12
4
o
q q
εr
=
π
where r12 is the distance between q1 and q2 |
1 | 1775-1778 | Next, we consider the
work done in bringing q2 to r2 In this step, work is done not only against
the external field E but also against the field due to q1 Work done on q2 against the external field
= q2 V (r2)
Work done on q2 against the field due to q1
1
2
12
4
o
q q
εr
=
π
where r12 is the distance between q1 and q2 We have made use of Eqs |
1 | 1776-1779 | In this step, work is done not only against
the external field E but also against the field due to q1 Work done on q2 against the external field
= q2 V (r2)
Work done on q2 against the field due to q1
1
2
12
4
o
q q
εr
=
π
where r12 is the distance between q1 and q2 We have made use of Eqs (2 |
1 | 1777-1780 | Work done on q2 against the external field
= q2 V (r2)
Work done on q2 against the field due to q1
1
2
12
4
o
q q
εr
=
π
where r12 is the distance between q1 and q2 We have made use of Eqs (2 27) and (2 |
1 | 1778-1781 | We have made use of Eqs (2 27) and (2 22) |
1 | 1779-1782 | (2 27) and (2 22) By the superposition principle for fields, we add up
the work done on q2 against the two fields (E and that due to q1):
Work done in bringing q2 to r2
1
2
2
2
12
(
)
4
o
q q
q V
εr
=
+
π
r
(2 |
1 | 1780-1783 | 27) and (2 22) By the superposition principle for fields, we add up
the work done on q2 against the two fields (E and that due to q1):
Work done in bringing q2 to r2
1
2
2
2
12
(
)
4
o
q q
q V
εr
=
+
π
r
(2 28)
Thus,
Potential energy of the system
= the total work done in assembling the configuration
1
2
1
1
2
2
0 12
(
)
(
)
4
q q
q V
q V
εr
=
+
+
π
r
r
(2 |
1 | 1781-1784 | 22) By the superposition principle for fields, we add up
the work done on q2 against the two fields (E and that due to q1):
Work done in bringing q2 to r2
1
2
2
2
12
(
)
4
o
q q
q V
εr
=
+
π
r
(2 28)
Thus,
Potential energy of the system
= the total work done in assembling the configuration
1
2
1
1
2
2
0 12
(
)
(
)
4
q q
q V
q V
εr
=
+
+
π
r
r
(2 29)
Example 2 |
1 | 1782-1785 | By the superposition principle for fields, we add up
the work done on q2 against the two fields (E and that due to q1):
Work done in bringing q2 to r2
1
2
2
2
12
(
)
4
o
q q
q V
εr
=
+
π
r
(2 28)
Thus,
Potential energy of the system
= the total work done in assembling the configuration
1
2
1
1
2
2
0 12
(
)
(
)
4
q q
q V
q V
εr
=
+
+
π
r
r
(2 29)
Example 2 5
(a) Determine the electrostatic potential energy of a system consisting
of two charges 7 mC and –2 mC (and with no external field) placed
at (–9 cm, 0, 0) and (9 cm, 0, 0) respectively |
1 | 1783-1786 | 28)
Thus,
Potential energy of the system
= the total work done in assembling the configuration
1
2
1
1
2
2
0 12
(
)
(
)
4
q q
q V
q V
εr
=
+
+
π
r
r
(2 29)
Example 2 5
(a) Determine the electrostatic potential energy of a system consisting
of two charges 7 mC and –2 mC (and with no external field) placed
at (–9 cm, 0, 0) and (9 cm, 0, 0) respectively (b) How much work is required to separate the two charges infinitely
away from each other |
1 | 1784-1787 | 29)
Example 2 5
(a) Determine the electrostatic potential energy of a system consisting
of two charges 7 mC and –2 mC (and with no external field) placed
at (–9 cm, 0, 0) and (9 cm, 0, 0) respectively (b) How much work is required to separate the two charges infinitely
away from each other Rationalised 2023-24
Physics
60
EXAMPLE 2 |
1 | 1785-1788 | 5
(a) Determine the electrostatic potential energy of a system consisting
of two charges 7 mC and –2 mC (and with no external field) placed
at (–9 cm, 0, 0) and (9 cm, 0, 0) respectively (b) How much work is required to separate the two charges infinitely
away from each other Rationalised 2023-24
Physics
60
EXAMPLE 2 5
(c) Suppose that the same system of charges is now placed in an
external electric field E = A (1/r 2); A = 9 × 105 NC–1 m2 |
1 | 1786-1789 | (b) How much work is required to separate the two charges infinitely
away from each other Rationalised 2023-24
Physics
60
EXAMPLE 2 5
(c) Suppose that the same system of charges is now placed in an
external electric field E = A (1/r 2); A = 9 × 105 NC–1 m2 What would
the electrostatic energy of the configuration be |
1 | 1787-1790 | Rationalised 2023-24
Physics
60
EXAMPLE 2 5
(c) Suppose that the same system of charges is now placed in an
external electric field E = A (1/r 2); A = 9 × 105 NC–1 m2 What would
the electrostatic energy of the configuration be Solution
(a)
12
9
1
2
0
1
7
( 2)
10
9
10
4
0 |
1 | 1788-1791 | 5
(c) Suppose that the same system of charges is now placed in an
external electric field E = A (1/r 2); A = 9 × 105 NC–1 m2 What would
the electrostatic energy of the configuration be Solution
(a)
12
9
1
2
0
1
7
( 2)
10
9
10
4
0 18
q q
U
r
ε
−
× −
×
=
=
×
×
π
= –0 |
1 | 1789-1792 | What would
the electrostatic energy of the configuration be Solution
(a)
12
9
1
2
0
1
7
( 2)
10
9
10
4
0 18
q q
U
r
ε
−
× −
×
=
=
×
×
π
= –0 7 J |
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