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1 | 5190-5193 | Now, let us recollect Experiment 6 3 in Section 6 2 In that experiment,
emf is induced in coil C1 wherever there was any change in current through
coil C2 |
1 | 5191-5194 | 3 in Section 6 2 In that experiment,
emf is induced in coil C1 wherever there was any change in current through
coil C2 Let F1 be the flux through coil C1 (say of N1 turns) when current in
coil C2 is I2 |
1 | 5192-5195 | 2 In that experiment,
emf is induced in coil C1 wherever there was any change in current through
coil C2 Let F1 be the flux through coil C1 (say of N1 turns) when current in
coil C2 is I2 Then, from Eq |
1 | 5193-5196 | In that experiment,
emf is induced in coil C1 wherever there was any change in current through
coil C2 Let F1 be the flux through coil C1 (say of N1 turns) when current in
coil C2 is I2 Then, from Eq (6 |
1 | 5194-5197 | Let F1 be the flux through coil C1 (say of N1 turns) when current in
coil C2 is I2 Then, from Eq (6 7), we have
N1F1 = MI2
For currents varrying with time,
(
)
(
)
1
1
2
d
d
d
d
N
MI
t
t
Φ
=
Since induced emf in coil C1 is given by
(
)
1
1
–d
d
N
t
Φ
ε1 =
We get,
d2
–
Id
M
t
ε1 =
Rationalised 2023-24
Physics
168
It shows that varying current in a coil can induce emf in a neighbouring
coil |
1 | 5195-5198 | Then, from Eq (6 7), we have
N1F1 = MI2
For currents varrying with time,
(
)
(
)
1
1
2
d
d
d
d
N
MI
t
t
Φ
=
Since induced emf in coil C1 is given by
(
)
1
1
–d
d
N
t
Φ
ε1 =
We get,
d2
–
Id
M
t
ε1 =
Rationalised 2023-24
Physics
168
It shows that varying current in a coil can induce emf in a neighbouring
coil The magnitude of the induced emf depends upon the rate of change
of current and mutual inductance of the two coils |
1 | 5196-5199 | (6 7), we have
N1F1 = MI2
For currents varrying with time,
(
)
(
)
1
1
2
d
d
d
d
N
MI
t
t
Φ
=
Since induced emf in coil C1 is given by
(
)
1
1
–d
d
N
t
Φ
ε1 =
We get,
d2
–
Id
M
t
ε1 =
Rationalised 2023-24
Physics
168
It shows that varying current in a coil can induce emf in a neighbouring
coil The magnitude of the induced emf depends upon the rate of change
of current and mutual inductance of the two coils 6 |
1 | 5197-5200 | 7), we have
N1F1 = MI2
For currents varrying with time,
(
)
(
)
1
1
2
d
d
d
d
N
MI
t
t
Φ
=
Since induced emf in coil C1 is given by
(
)
1
1
–d
d
N
t
Φ
ε1 =
We get,
d2
–
Id
M
t
ε1 =
Rationalised 2023-24
Physics
168
It shows that varying current in a coil can induce emf in a neighbouring
coil The magnitude of the induced emf depends upon the rate of change
of current and mutual inductance of the two coils 6 7 |
1 | 5198-5201 | The magnitude of the induced emf depends upon the rate of change
of current and mutual inductance of the two coils 6 7 2 Self-inductance
In the previous sub-section, we considered the flux in one solenoid due
to the current in the other |
1 | 5199-5202 | 6 7 2 Self-inductance
In the previous sub-section, we considered the flux in one solenoid due
to the current in the other It is also possible that emf is induced in a
single isolated coil due to change of flux through the coil by means of
varying the current through the same coil |
1 | 5200-5203 | 7 2 Self-inductance
In the previous sub-section, we considered the flux in one solenoid due
to the current in the other It is also possible that emf is induced in a
single isolated coil due to change of flux through the coil by means of
varying the current through the same coil This phenomenon is called
self-induction |
1 | 5201-5204 | 2 Self-inductance
In the previous sub-section, we considered the flux in one solenoid due
to the current in the other It is also possible that emf is induced in a
single isolated coil due to change of flux through the coil by means of
varying the current through the same coil This phenomenon is called
self-induction In this case, flux linkage through a coil of N turns is
proportional to the current through the coil and is expressed as
NB
I
Φ ∝
B
L
N
I
Φ
=
(6 |
1 | 5202-5205 | It is also possible that emf is induced in a
single isolated coil due to change of flux through the coil by means of
varying the current through the same coil This phenomenon is called
self-induction In this case, flux linkage through a coil of N turns is
proportional to the current through the coil and is expressed as
NB
I
Φ ∝
B
L
N
I
Φ
=
(6 13)
where constant of proportionality L is called self-inductance of the coil |
1 | 5203-5206 | This phenomenon is called
self-induction In this case, flux linkage through a coil of N turns is
proportional to the current through the coil and is expressed as
NB
I
Φ ∝
B
L
N
I
Φ
=
(6 13)
where constant of proportionality L is called self-inductance of the coil It
is also called the coefficient of self-induction of the coil |
1 | 5204-5207 | In this case, flux linkage through a coil of N turns is
proportional to the current through the coil and is expressed as
NB
I
Φ ∝
B
L
N
I
Φ
=
(6 13)
where constant of proportionality L is called self-inductance of the coil It
is also called the coefficient of self-induction of the coil When the current
is varied, the flux linked with the coil also changes and an emf is induced
in the coil |
1 | 5205-5208 | 13)
where constant of proportionality L is called self-inductance of the coil It
is also called the coefficient of self-induction of the coil When the current
is varied, the flux linked with the coil also changes and an emf is induced
in the coil Using Eq |
1 | 5206-5209 | It
is also called the coefficient of self-induction of the coil When the current
is varied, the flux linked with the coil also changes and an emf is induced
in the coil Using Eq (6 |
1 | 5207-5210 | When the current
is varied, the flux linked with the coil also changes and an emf is induced
in the coil Using Eq (6 13), the induced emf is given by
(
B)
–d
d
N
t
Φ
ε =
d
–
d
I
L
t
ε =
(6 |
1 | 5208-5211 | Using Eq (6 13), the induced emf is given by
(
B)
–d
d
N
t
Φ
ε =
d
–
d
I
L
t
ε =
(6 14)
Thus, the self-induced emf always opposes any change (increase or
decrease) of current in the coil |
1 | 5209-5212 | (6 13), the induced emf is given by
(
B)
–d
d
N
t
Φ
ε =
d
–
d
I
L
t
ε =
(6 14)
Thus, the self-induced emf always opposes any change (increase or
decrease) of current in the coil It is possible to calculate the self-inductance for circuits with simple
geometries |
1 | 5210-5213 | 13), the induced emf is given by
(
B)
–d
d
N
t
Φ
ε =
d
–
d
I
L
t
ε =
(6 14)
Thus, the self-induced emf always opposes any change (increase or
decrease) of current in the coil It is possible to calculate the self-inductance for circuits with simple
geometries Let us calculate the self-inductance of a long solenoid of cross-
sectional area A and length l, having n turns per unit length |
1 | 5211-5214 | 14)
Thus, the self-induced emf always opposes any change (increase or
decrease) of current in the coil It is possible to calculate the self-inductance for circuits with simple
geometries Let us calculate the self-inductance of a long solenoid of cross-
sectional area A and length l, having n turns per unit length The magnetic
field due to a current I flowing in the solenoid is B = m0 n I (neglecting edge
effects, as before) |
1 | 5212-5215 | It is possible to calculate the self-inductance for circuits with simple
geometries Let us calculate the self-inductance of a long solenoid of cross-
sectional area A and length l, having n turns per unit length The magnetic
field due to a current I flowing in the solenoid is B = m0 n I (neglecting edge
effects, as before) The total flux linked with the solenoid is
(
)(
)( )
0
NB
nl
n I
A
Φ
µ
=
0n2AlI
where nl is the total number of turns |
1 | 5213-5216 | Let us calculate the self-inductance of a long solenoid of cross-
sectional area A and length l, having n turns per unit length The magnetic
field due to a current I flowing in the solenoid is B = m0 n I (neglecting edge
effects, as before) The total flux linked with the solenoid is
(
)(
)( )
0
NB
nl
n I
A
Φ
µ
=
0n2AlI
where nl is the total number of turns Thus, the self-inductance is,
L
I
ΝΦΒ
=
2
=µ0n Al
(6 |
1 | 5214-5217 | The magnetic
field due to a current I flowing in the solenoid is B = m0 n I (neglecting edge
effects, as before) The total flux linked with the solenoid is
(
)(
)( )
0
NB
nl
n I
A
Φ
µ
=
0n2AlI
where nl is the total number of turns Thus, the self-inductance is,
L
I
ΝΦΒ
=
2
=µ0n Al
(6 15)
If we fill the inside of the solenoid with a material of relative permeability
mr (for example soft iron, which has a high value of relative permeability),
then,
2
0
r
L
=µ µn Al
(6 |
1 | 5215-5218 | The total flux linked with the solenoid is
(
)(
)( )
0
NB
nl
n I
A
Φ
µ
=
0n2AlI
where nl is the total number of turns Thus, the self-inductance is,
L
I
ΝΦΒ
=
2
=µ0n Al
(6 15)
If we fill the inside of the solenoid with a material of relative permeability
mr (for example soft iron, which has a high value of relative permeability),
then,
2
0
r
L
=µ µn Al
(6 16)
The self-inductance of the coil depends on its geometry and on the
permeability of the medium |
1 | 5216-5219 | Thus, the self-inductance is,
L
I
ΝΦΒ
=
2
=µ0n Al
(6 15)
If we fill the inside of the solenoid with a material of relative permeability
mr (for example soft iron, which has a high value of relative permeability),
then,
2
0
r
L
=µ µn Al
(6 16)
The self-inductance of the coil depends on its geometry and on the
permeability of the medium The self-induced emf is also called the back emf as it opposes any
change in the current in a circuit |
1 | 5217-5220 | 15)
If we fill the inside of the solenoid with a material of relative permeability
mr (for example soft iron, which has a high value of relative permeability),
then,
2
0
r
L
=µ µn Al
(6 16)
The self-inductance of the coil depends on its geometry and on the
permeability of the medium The self-induced emf is also called the back emf as it opposes any
change in the current in a circuit Physically, the self-inductance plays
Rationalised 2023-24
Electromagnetic
Induction
169
the role of inertia |
1 | 5218-5221 | 16)
The self-inductance of the coil depends on its geometry and on the
permeability of the medium The self-induced emf is also called the back emf as it opposes any
change in the current in a circuit Physically, the self-inductance plays
Rationalised 2023-24
Electromagnetic
Induction
169
the role of inertia It is the electromagnetic analogue of mass in mechanics |
1 | 5219-5222 | The self-induced emf is also called the back emf as it opposes any
change in the current in a circuit Physically, the self-inductance plays
Rationalised 2023-24
Electromagnetic
Induction
169
the role of inertia It is the electromagnetic analogue of mass in mechanics So, work needs to be done against the back emf (e) in establishing the
current |
1 | 5220-5223 | Physically, the self-inductance plays
Rationalised 2023-24
Electromagnetic
Induction
169
the role of inertia It is the electromagnetic analogue of mass in mechanics So, work needs to be done against the back emf (e) in establishing the
current This work done is stored as magnetic potential energy |
1 | 5221-5224 | It is the electromagnetic analogue of mass in mechanics So, work needs to be done against the back emf (e) in establishing the
current This work done is stored as magnetic potential energy For the
current I at an instant in a circuit, the rate of work done is
d
d
W
I
t
ε
=
If we ignore the resistive losses and consider only inductive effect,
then using Eq |
1 | 5222-5225 | So, work needs to be done against the back emf (e) in establishing the
current This work done is stored as magnetic potential energy For the
current I at an instant in a circuit, the rate of work done is
d
d
W
I
t
ε
=
If we ignore the resistive losses and consider only inductive effect,
then using Eq (6 |
1 | 5223-5226 | This work done is stored as magnetic potential energy For the
current I at an instant in a circuit, the rate of work done is
d
d
W
I
t
ε
=
If we ignore the resistive losses and consider only inductive effect,
then using Eq (6 16),
d
d
d
d
W
I
L I
t
t
=
Total amount of work done in establishing the current I is
W
W
L I
I
I
=
=
∫
∫
d
d
0
Thus, the energy required to build up the current I is,
2
21
W
LI
=
(6 |
1 | 5224-5227 | For the
current I at an instant in a circuit, the rate of work done is
d
d
W
I
t
ε
=
If we ignore the resistive losses and consider only inductive effect,
then using Eq (6 16),
d
d
d
d
W
I
L I
t
t
=
Total amount of work done in establishing the current I is
W
W
L I
I
I
=
=
∫
∫
d
d
0
Thus, the energy required to build up the current I is,
2
21
W
LI
=
(6 17)
This expression reminds us of mv 2/2 for the (mechanical) kinetic energy
of a particle of mass m, and shows that L is analogous to m (i |
1 | 5225-5228 | (6 16),
d
d
d
d
W
I
L I
t
t
=
Total amount of work done in establishing the current I is
W
W
L I
I
I
=
=
∫
∫
d
d
0
Thus, the energy required to build up the current I is,
2
21
W
LI
=
(6 17)
This expression reminds us of mv 2/2 for the (mechanical) kinetic energy
of a particle of mass m, and shows that L is analogous to m (i e |
1 | 5226-5229 | 16),
d
d
d
d
W
I
L I
t
t
=
Total amount of work done in establishing the current I is
W
W
L I
I
I
=
=
∫
∫
d
d
0
Thus, the energy required to build up the current I is,
2
21
W
LI
=
(6 17)
This expression reminds us of mv 2/2 for the (mechanical) kinetic energy
of a particle of mass m, and shows that L is analogous to m (i e , L is
electrical inertia and opposes growth and decay of current in the circuit) |
1 | 5227-5230 | 17)
This expression reminds us of mv 2/2 for the (mechanical) kinetic energy
of a particle of mass m, and shows that L is analogous to m (i e , L is
electrical inertia and opposes growth and decay of current in the circuit) Consider the general case of currents flowing simultaneously in two
nearby coils |
1 | 5228-5231 | e , L is
electrical inertia and opposes growth and decay of current in the circuit) Consider the general case of currents flowing simultaneously in two
nearby coils The flux linked with one coil will be the sum of two fluxes
which exist independently |
1 | 5229-5232 | , L is
electrical inertia and opposes growth and decay of current in the circuit) Consider the general case of currents flowing simultaneously in two
nearby coils The flux linked with one coil will be the sum of two fluxes
which exist independently Equation (6 |
1 | 5230-5233 | Consider the general case of currents flowing simultaneously in two
nearby coils The flux linked with one coil will be the sum of two fluxes
which exist independently Equation (6 7) would be modified into
N1
1
11
1
12
2
M
I
M
I
Φ =
+
where M11 represents inductance due to the same coil |
1 | 5231-5234 | The flux linked with one coil will be the sum of two fluxes
which exist independently Equation (6 7) would be modified into
N1
1
11
1
12
2
M
I
M
I
Φ =
+
where M11 represents inductance due to the same coil Therefore, using Faraday’s law,
1
2
1
11
12
d
d
d
d
I
I
M
M
t
t
ε = −
−
M11 is the self-inductance and is written as L1 |
1 | 5232-5235 | Equation (6 7) would be modified into
N1
1
11
1
12
2
M
I
M
I
Φ =
+
where M11 represents inductance due to the same coil Therefore, using Faraday’s law,
1
2
1
11
12
d
d
d
d
I
I
M
M
t
t
ε = −
−
M11 is the self-inductance and is written as L1 Therefore,
1
2
1
1
12
d
d
d
d
I
I
L
M
t
t
ε = −
−
Example 6 |
1 | 5233-5236 | 7) would be modified into
N1
1
11
1
12
2
M
I
M
I
Φ =
+
where M11 represents inductance due to the same coil Therefore, using Faraday’s law,
1
2
1
11
12
d
d
d
d
I
I
M
M
t
t
ε = −
−
M11 is the self-inductance and is written as L1 Therefore,
1
2
1
1
12
d
d
d
d
I
I
L
M
t
t
ε = −
−
Example 6 9 (a) Obtain the expression for the magnetic energy stored
in a solenoid in terms of magnetic field B, area A and length l of the
solenoid |
1 | 5234-5237 | Therefore, using Faraday’s law,
1
2
1
11
12
d
d
d
d
I
I
M
M
t
t
ε = −
−
M11 is the self-inductance and is written as L1 Therefore,
1
2
1
1
12
d
d
d
d
I
I
L
M
t
t
ε = −
−
Example 6 9 (a) Obtain the expression for the magnetic energy stored
in a solenoid in terms of magnetic field B, area A and length l of the
solenoid (b) How does this magnetic energy compare with the
electrostatic energy stored in a capacitor |
1 | 5235-5238 | Therefore,
1
2
1
1
12
d
d
d
d
I
I
L
M
t
t
ε = −
−
Example 6 9 (a) Obtain the expression for the magnetic energy stored
in a solenoid in terms of magnetic field B, area A and length l of the
solenoid (b) How does this magnetic energy compare with the
electrostatic energy stored in a capacitor Solution
(a)
From Eq |
1 | 5236-5239 | 9 (a) Obtain the expression for the magnetic energy stored
in a solenoid in terms of magnetic field B, area A and length l of the
solenoid (b) How does this magnetic energy compare with the
electrostatic energy stored in a capacitor Solution
(a)
From Eq (6 |
1 | 5237-5240 | (b) How does this magnetic energy compare with the
electrostatic energy stored in a capacitor Solution
(a)
From Eq (6 17), the magnetic energy is
2
21
UB
LI
=
=
=
(
)
1
2
2
L
B
n
nI
µ
µ
0
0
sinceB
for a solenoid
,
EXAMPLE 6 |
1 | 5238-5241 | Solution
(a)
From Eq (6 17), the magnetic energy is
2
21
UB
LI
=
=
=
(
)
1
2
2
L
B
n
nI
µ
µ
0
0
sinceB
for a solenoid
,
EXAMPLE 6 9
Rationalised 2023-24
Physics
170
EXAMPLE 6 |
1 | 5239-5242 | (6 17), the magnetic energy is
2
21
UB
LI
=
=
=
(
)
1
2
2
L
B
n
nI
µ
µ
0
0
sinceB
for a solenoid
,
EXAMPLE 6 9
Rationalised 2023-24
Physics
170
EXAMPLE 6 9
=
21
0
2
0
2
(
)
µ
µ
n Al
B
n
[from Eq |
1 | 5240-5243 | 17), the magnetic energy is
2
21
UB
LI
=
=
=
(
)
1
2
2
L
B
n
nI
µ
µ
0
0
sinceB
for a solenoid
,
EXAMPLE 6 9
Rationalised 2023-24
Physics
170
EXAMPLE 6 9
=
21
0
2
0
2
(
)
µ
µ
n Al
B
n
[from Eq (6 |
1 | 5241-5244 | 9
Rationalised 2023-24
Physics
170
EXAMPLE 6 9
=
21
0
2
0
2
(
)
µ
µ
n Al
B
n
[from Eq (6 15)]
2
0
21
µB Al
=
(b)
The magnetic energy per unit volume is,
B
B
U
u
=V
(where V is volume that contains flux)
UB
Al
=
2
20
B
µ
=
(6 |
1 | 5242-5245 | 9
=
21
0
2
0
2
(
)
µ
µ
n Al
B
n
[from Eq (6 15)]
2
0
21
µB Al
=
(b)
The magnetic energy per unit volume is,
B
B
U
u
=V
(where V is volume that contains flux)
UB
Al
=
2
20
B
µ
=
(6 18)
We have already obtained the relation for the electrostatic energy
stored per unit volume in a parallel plate capacitor (refer to Chapter 2,
Eq |
1 | 5243-5246 | (6 15)]
2
0
21
µB Al
=
(b)
The magnetic energy per unit volume is,
B
B
U
u
=V
(where V is volume that contains flux)
UB
Al
=
2
20
B
µ
=
(6 18)
We have already obtained the relation for the electrostatic energy
stored per unit volume in a parallel plate capacitor (refer to Chapter 2,
Eq 2 |
1 | 5244-5247 | 15)]
2
0
21
µB Al
=
(b)
The magnetic energy per unit volume is,
B
B
U
u
=V
(where V is volume that contains flux)
UB
Al
=
2
20
B
µ
=
(6 18)
We have already obtained the relation for the electrostatic energy
stored per unit volume in a parallel plate capacitor (refer to Chapter 2,
Eq 2 73),
2
210
u
E
Ε
ε
=
(2 |
1 | 5245-5248 | 18)
We have already obtained the relation for the electrostatic energy
stored per unit volume in a parallel plate capacitor (refer to Chapter 2,
Eq 2 73),
2
210
u
E
Ε
ε
=
(2 73)
In both the cases energy is proportional to the square of the field
strength |
1 | 5246-5249 | 2 73),
2
210
u
E
Ε
ε
=
(2 73)
In both the cases energy is proportional to the square of the field
strength Equations (6 |
1 | 5247-5250 | 73),
2
210
u
E
Ε
ε
=
(2 73)
In both the cases energy is proportional to the square of the field
strength Equations (6 18) and (2 |
1 | 5248-5251 | 73)
In both the cases energy is proportional to the square of the field
strength Equations (6 18) and (2 73) have been derived for special
cases: a solenoid and a parallel plate capacitor, respectively |
1 | 5249-5252 | Equations (6 18) and (2 73) have been derived for special
cases: a solenoid and a parallel plate capacitor, respectively But they
are general and valid for any region of space in which a magnetic field
or/and an electric field exist |
1 | 5250-5253 | 18) and (2 73) have been derived for special
cases: a solenoid and a parallel plate capacitor, respectively But they
are general and valid for any region of space in which a magnetic field
or/and an electric field exist FIGURE 6 |
1 | 5251-5254 | 73) have been derived for special
cases: a solenoid and a parallel plate capacitor, respectively But they
are general and valid for any region of space in which a magnetic field
or/and an electric field exist FIGURE 6 13 AC Generator
Interactive animation on ac generator:
http://micro |
1 | 5252-5255 | But they
are general and valid for any region of space in which a magnetic field
or/and an electric field exist FIGURE 6 13 AC Generator
Interactive animation on ac generator:
http://micro magnet |
1 | 5253-5256 | FIGURE 6 13 AC Generator
Interactive animation on ac generator:
http://micro magnet fsu |
1 | 5254-5257 | 13 AC Generator
Interactive animation on ac generator:
http://micro magnet fsu edu/electromag/java/generator/ac |
1 | 5255-5258 | magnet fsu edu/electromag/java/generator/ac html
6 |
1 | 5256-5259 | fsu edu/electromag/java/generator/ac html
6 8 AC GENERATOR
The phenomenon of electromagnetic induction
has been technologically exploited in many ways |
1 | 5257-5260 | edu/electromag/java/generator/ac html
6 8 AC GENERATOR
The phenomenon of electromagnetic induction
has been technologically exploited in many ways An exceptionally important application is the
generation of alternating currents (ac) |
1 | 5258-5261 | html
6 8 AC GENERATOR
The phenomenon of electromagnetic induction
has been technologically exploited in many ways An exceptionally important application is the
generation of alternating currents (ac) The
modern ac generator with a typical output
capacity of 100 MW is a highly evolved machine |
1 | 5259-5262 | 8 AC GENERATOR
The phenomenon of electromagnetic induction
has been technologically exploited in many ways An exceptionally important application is the
generation of alternating currents (ac) The
modern ac generator with a typical output
capacity of 100 MW is a highly evolved machine In this section, we shall describe the basic
principles behind this machine |
1 | 5260-5263 | An exceptionally important application is the
generation of alternating currents (ac) The
modern ac generator with a typical output
capacity of 100 MW is a highly evolved machine In this section, we shall describe the basic
principles behind this machine The Yugoslav
inventor Nicola Tesla is credited with the
development of the machine |
1 | 5261-5264 | The
modern ac generator with a typical output
capacity of 100 MW is a highly evolved machine In this section, we shall describe the basic
principles behind this machine The Yugoslav
inventor Nicola Tesla is credited with the
development of the machine As was pointed out
in Section 6 |
1 | 5262-5265 | In this section, we shall describe the basic
principles behind this machine The Yugoslav
inventor Nicola Tesla is credited with the
development of the machine As was pointed out
in Section 6 3, one method to induce an emf or
current in a loop is through a change in the
loop’s orientation or a change in its effective area |
1 | 5263-5266 | The Yugoslav
inventor Nicola Tesla is credited with the
development of the machine As was pointed out
in Section 6 3, one method to induce an emf or
current in a loop is through a change in the
loop’s orientation or a change in its effective area As the coil rotates in a magnetic field B, the
effective area of the loop (the face perpendicular
to the field) is A cos q, where q is the angle
between A and B |
1 | 5264-5267 | As was pointed out
in Section 6 3, one method to induce an emf or
current in a loop is through a change in the
loop’s orientation or a change in its effective area As the coil rotates in a magnetic field B, the
effective area of the loop (the face perpendicular
to the field) is A cos q, where q is the angle
between A and B This method of producing a
flux change is the principle of operation of a
Rationalised 2023-24
Electromagnetic
Induction
171
simple ac generator |
1 | 5265-5268 | 3, one method to induce an emf or
current in a loop is through a change in the
loop’s orientation or a change in its effective area As the coil rotates in a magnetic field B, the
effective area of the loop (the face perpendicular
to the field) is A cos q, where q is the angle
between A and B This method of producing a
flux change is the principle of operation of a
Rationalised 2023-24
Electromagnetic
Induction
171
simple ac generator An ac generator converts mechanical energy into
electrical energy |
1 | 5266-5269 | As the coil rotates in a magnetic field B, the
effective area of the loop (the face perpendicular
to the field) is A cos q, where q is the angle
between A and B This method of producing a
flux change is the principle of operation of a
Rationalised 2023-24
Electromagnetic
Induction
171
simple ac generator An ac generator converts mechanical energy into
electrical energy The basic elements of an ac generator are shown in Fig |
1 | 5267-5270 | This method of producing a
flux change is the principle of operation of a
Rationalised 2023-24
Electromagnetic
Induction
171
simple ac generator An ac generator converts mechanical energy into
electrical energy The basic elements of an ac generator are shown in Fig 6 |
1 | 5268-5271 | An ac generator converts mechanical energy into
electrical energy The basic elements of an ac generator are shown in Fig 6 13 |
1 | 5269-5272 | The basic elements of an ac generator are shown in Fig 6 13 It consists
of a coil mounted on a rotor shaft |
1 | 5270-5273 | 6 13 It consists
of a coil mounted on a rotor shaft The axis of rotation of the coil is
perpendicular to the direction of the magnetic field |
1 | 5271-5274 | 13 It consists
of a coil mounted on a rotor shaft The axis of rotation of the coil is
perpendicular to the direction of the magnetic field The coil (called
armature) is mechanically rotated in the uniform magnetic field by some
external means |
1 | 5272-5275 | It consists
of a coil mounted on a rotor shaft The axis of rotation of the coil is
perpendicular to the direction of the magnetic field The coil (called
armature) is mechanically rotated in the uniform magnetic field by some
external means The rotation of the coil causes the magnetic flux through
it to change, so an emf is induced in the coil |
1 | 5273-5276 | The axis of rotation of the coil is
perpendicular to the direction of the magnetic field The coil (called
armature) is mechanically rotated in the uniform magnetic field by some
external means The rotation of the coil causes the magnetic flux through
it to change, so an emf is induced in the coil The ends of the
coil are connected to an external circuit by means of slip rings
and brushes |
1 | 5274-5277 | The coil (called
armature) is mechanically rotated in the uniform magnetic field by some
external means The rotation of the coil causes the magnetic flux through
it to change, so an emf is induced in the coil The ends of the
coil are connected to an external circuit by means of slip rings
and brushes When the coil is rotated with a constant angular speed w, the angle q
between the magnetic field vector B and the area vector A of the coil at any
instant t is q = wt (assuming q = 0° at t = 0) |
1 | 5275-5278 | The rotation of the coil causes the magnetic flux through
it to change, so an emf is induced in the coil The ends of the
coil are connected to an external circuit by means of slip rings
and brushes When the coil is rotated with a constant angular speed w, the angle q
between the magnetic field vector B and the area vector A of the coil at any
instant t is q = wt (assuming q = 0° at t = 0) As a result, the effective area
of the coil exposed to the magnetic field lines changes with time, and from
Eq |
1 | 5276-5279 | The ends of the
coil are connected to an external circuit by means of slip rings
and brushes When the coil is rotated with a constant angular speed w, the angle q
between the magnetic field vector B and the area vector A of the coil at any
instant t is q = wt (assuming q = 0° at t = 0) As a result, the effective area
of the coil exposed to the magnetic field lines changes with time, and from
Eq (6 |
1 | 5277-5280 | When the coil is rotated with a constant angular speed w, the angle q
between the magnetic field vector B and the area vector A of the coil at any
instant t is q = wt (assuming q = 0° at t = 0) As a result, the effective area
of the coil exposed to the magnetic field lines changes with time, and from
Eq (6 1), the flux at any time t is
FB = BA cos q = BA cos wt
From Faraday’s law, the induced emf for the rotating coil of N turns
is then,
d
d
–
–
(cos
)
dt
d
B
N
NBA
t
t
Φ
ε
ω
=
=
εThus, the instantaneous value of the emf is
ω
ω
= NBA
sin
t
(6 |
1 | 5278-5281 | As a result, the effective area
of the coil exposed to the magnetic field lines changes with time, and from
Eq (6 1), the flux at any time t is
FB = BA cos q = BA cos wt
From Faraday’s law, the induced emf for the rotating coil of N turns
is then,
d
d
–
–
(cos
)
dt
d
B
N
NBA
t
t
Φ
ε
ω
=
=
εThus, the instantaneous value of the emf is
ω
ω
= NBA
sin
t
(6 19)
where NBAw is the maximum value of the emf, which occurs when
sin wt = ±1 |
1 | 5279-5282 | (6 1), the flux at any time t is
FB = BA cos q = BA cos wt
From Faraday’s law, the induced emf for the rotating coil of N turns
is then,
d
d
–
–
(cos
)
dt
d
B
N
NBA
t
t
Φ
ε
ω
=
=
εThus, the instantaneous value of the emf is
ω
ω
= NBA
sin
t
(6 19)
where NBAw is the maximum value of the emf, which occurs when
sin wt = ±1 If we denote NBAw as e0, then
e = e0 sin wt
(6 |
1 | 5280-5283 | 1), the flux at any time t is
FB = BA cos q = BA cos wt
From Faraday’s law, the induced emf for the rotating coil of N turns
is then,
d
d
–
–
(cos
)
dt
d
B
N
NBA
t
t
Φ
ε
ω
=
=
εThus, the instantaneous value of the emf is
ω
ω
= NBA
sin
t
(6 19)
where NBAw is the maximum value of the emf, which occurs when
sin wt = ±1 If we denote NBAw as e0, then
e = e0 sin wt
(6 20)
Since the value of the sine fuction varies between +1 and –1, the sign, or
polarity of the emf changes with time |
1 | 5281-5284 | 19)
where NBAw is the maximum value of the emf, which occurs when
sin wt = ±1 If we denote NBAw as e0, then
e = e0 sin wt
(6 20)
Since the value of the sine fuction varies between +1 and –1, the sign, or
polarity of the emf changes with time Note from Fig |
1 | 5282-5285 | If we denote NBAw as e0, then
e = e0 sin wt
(6 20)
Since the value of the sine fuction varies between +1 and –1, the sign, or
polarity of the emf changes with time Note from Fig 6 |
1 | 5283-5286 | 20)
Since the value of the sine fuction varies between +1 and –1, the sign, or
polarity of the emf changes with time Note from Fig 6 14 that the emf
has its extremum value when q = 90° or q = 270°, as the change of flux is
greatest at these points |
1 | 5284-5287 | Note from Fig 6 14 that the emf
has its extremum value when q = 90° or q = 270°, as the change of flux is
greatest at these points The direction of the current changes periodically and therefore the current
is called alternating current (ac) |
1 | 5285-5288 | 6 14 that the emf
has its extremum value when q = 90° or q = 270°, as the change of flux is
greatest at these points The direction of the current changes periodically and therefore the current
is called alternating current (ac) Since w = 2pn, Eq (6 |
1 | 5286-5289 | 14 that the emf
has its extremum value when q = 90° or q = 270°, as the change of flux is
greatest at these points The direction of the current changes periodically and therefore the current
is called alternating current (ac) Since w = 2pn, Eq (6 20) can be written as
e = e0sin 2p n t
(6 |
1 | 5287-5290 | The direction of the current changes periodically and therefore the current
is called alternating current (ac) Since w = 2pn, Eq (6 20) can be written as
e = e0sin 2p n t
(6 21)
where n is the frequency of revolution of the generator’s coil |
1 | 5288-5291 | Since w = 2pn, Eq (6 20) can be written as
e = e0sin 2p n t
(6 21)
where n is the frequency of revolution of the generator’s coil Note that Eq |
1 | 5289-5292 | 20) can be written as
e = e0sin 2p n t
(6 21)
where n is the frequency of revolution of the generator’s coil Note that Eq (6 |
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