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1
5290-5293
21) where n is the frequency of revolution of the generator’s coil Note that Eq (6 20) and (6
1
5291-5294
Note that Eq (6 20) and (6 21) give the instantaneous value of the emf and e varies between +e0 and –e0 periodically
1
5292-5295
(6 20) and (6 21) give the instantaneous value of the emf and e varies between +e0 and –e0 periodically We shall learn how to determine the time-averaged value for the alternating voltage and current in the next chapter
1
5293-5296
20) and (6 21) give the instantaneous value of the emf and e varies between +e0 and –e0 periodically We shall learn how to determine the time-averaged value for the alternating voltage and current in the next chapter In commercial generators, the mechanical energy required for rotation of the armature is provided by water falling from a height, for example, from dams
1
5294-5297
21) give the instantaneous value of the emf and e varies between +e0 and –e0 periodically We shall learn how to determine the time-averaged value for the alternating voltage and current in the next chapter In commercial generators, the mechanical energy required for rotation of the armature is provided by water falling from a height, for example, from dams These are called hydro-electric generators
1
5295-5298
We shall learn how to determine the time-averaged value for the alternating voltage and current in the next chapter In commercial generators, the mechanical energy required for rotation of the armature is provided by water falling from a height, for example, from dams These are called hydro-electric generators Alternatively, water is heated to produce steam using coal or other sources
1
5296-5299
In commercial generators, the mechanical energy required for rotation of the armature is provided by water falling from a height, for example, from dams These are called hydro-electric generators Alternatively, water is heated to produce steam using coal or other sources The steam at high pressure produces the rotation of the armature
1
5297-5300
These are called hydro-electric generators Alternatively, water is heated to produce steam using coal or other sources The steam at high pressure produces the rotation of the armature These are called thermal generators
1
5298-5301
Alternatively, water is heated to produce steam using coal or other sources The steam at high pressure produces the rotation of the armature These are called thermal generators Instead of coal, if a nuclear fuel is used, we get nuclear power generators
1
5299-5302
The steam at high pressure produces the rotation of the armature These are called thermal generators Instead of coal, if a nuclear fuel is used, we get nuclear power generators Modern day generators produce electric power as high as 500 MW, i
1
5300-5303
These are called thermal generators Instead of coal, if a nuclear fuel is used, we get nuclear power generators Modern day generators produce electric power as high as 500 MW, i e
1
5301-5304
Instead of coal, if a nuclear fuel is used, we get nuclear power generators Modern day generators produce electric power as high as 500 MW, i e , one can light Rationalised 2023-24 Physics 172 EXAMPLE 6
1
5302-5305
Modern day generators produce electric power as high as 500 MW, i e , one can light Rationalised 2023-24 Physics 172 EXAMPLE 6 10 Example 6
1
5303-5306
e , one can light Rationalised 2023-24 Physics 172 EXAMPLE 6 10 Example 6 10 Kamla peddles a stationary bicycle
1
5304-5307
, one can light Rationalised 2023-24 Physics 172 EXAMPLE 6 10 Example 6 10 Kamla peddles a stationary bicycle The pedals of the bicycle are attached to a 100 turn coil of area 0
1
5305-5308
10 Example 6 10 Kamla peddles a stationary bicycle The pedals of the bicycle are attached to a 100 turn coil of area 0 10 m2
1
5306-5309
10 Kamla peddles a stationary bicycle The pedals of the bicycle are attached to a 100 turn coil of area 0 10 m2 The coil rotates at half a revolution per second and it is placed in a uniform magnetic field of 0
1
5307-5310
The pedals of the bicycle are attached to a 100 turn coil of area 0 10 m2 The coil rotates at half a revolution per second and it is placed in a uniform magnetic field of 0 01 T perpendicular to the axis of rotation of the coil
1
5308-5311
10 m2 The coil rotates at half a revolution per second and it is placed in a uniform magnetic field of 0 01 T perpendicular to the axis of rotation of the coil What is the maximum voltage generated in the coil
1
5309-5312
The coil rotates at half a revolution per second and it is placed in a uniform magnetic field of 0 01 T perpendicular to the axis of rotation of the coil What is the maximum voltage generated in the coil Solution Here n = 0
1
5310-5313
01 T perpendicular to the axis of rotation of the coil What is the maximum voltage generated in the coil Solution Here n = 0 5 Hz; N =100, A = 0
1
5311-5314
What is the maximum voltage generated in the coil Solution Here n = 0 5 Hz; N =100, A = 0 1 m2 and B = 0
1
5312-5315
Solution Here n = 0 5 Hz; N =100, A = 0 1 m2 and B = 0 01 T
1
5313-5316
5 Hz; N =100, A = 0 1 m2 and B = 0 01 T Employing Eq
1
5314-5317
1 m2 and B = 0 01 T Employing Eq (6
1
5315-5318
01 T Employing Eq (6 19) e0 = NBA (2 p n) = 100 × 0
1
5316-5319
Employing Eq (6 19) e0 = NBA (2 p n) = 100 × 0 01 × 0
1
5317-5320
(6 19) e0 = NBA (2 p n) = 100 × 0 01 × 0 1 × 2 × 3
1
5318-5321
19) e0 = NBA (2 p n) = 100 × 0 01 × 0 1 × 2 × 3 14 × 0
1
5319-5322
01 × 0 1 × 2 × 3 14 × 0 5 = 0
1
5320-5323
1 × 2 × 3 14 × 0 5 = 0 314 V The maximum voltage is 0
1
5321-5324
14 × 0 5 = 0 314 V The maximum voltage is 0 314 V
1
5322-5325
5 = 0 314 V The maximum voltage is 0 314 V We urge you to explore such alternative possibilities for power generation
1
5323-5326
314 V The maximum voltage is 0 314 V We urge you to explore such alternative possibilities for power generation FIGURE 6
1
5324-5327
314 V We urge you to explore such alternative possibilities for power generation FIGURE 6 14 An alternating emf is generated by a loop of wire rotating in a magnetic field
1
5325-5328
We urge you to explore such alternative possibilities for power generation FIGURE 6 14 An alternating emf is generated by a loop of wire rotating in a magnetic field up 5 million 100 W bulbs
1
5326-5329
FIGURE 6 14 An alternating emf is generated by a loop of wire rotating in a magnetic field up 5 million 100 W bulbs In most generators, the coils are held stationary and it is the electromagnets which are rotated
1
5327-5330
14 An alternating emf is generated by a loop of wire rotating in a magnetic field up 5 million 100 W bulbs In most generators, the coils are held stationary and it is the electromagnets which are rotated The frequency of rotation is 50 Hz in India
1
5328-5331
up 5 million 100 W bulbs In most generators, the coils are held stationary and it is the electromagnets which are rotated The frequency of rotation is 50 Hz in India In certain countries such as USA, it is 60 Hz
1
5329-5332
In most generators, the coils are held stationary and it is the electromagnets which are rotated The frequency of rotation is 50 Hz in India In certain countries such as USA, it is 60 Hz Rationalised 2023-24 Electromagnetic Induction 173 SUMMARY 1
1
5330-5333
The frequency of rotation is 50 Hz in India In certain countries such as USA, it is 60 Hz Rationalised 2023-24 Electromagnetic Induction 173 SUMMARY 1 The magnetic flux through a surface of area A placed in a uniform magnetic field B is defined as, FB = B
1
5331-5334
In certain countries such as USA, it is 60 Hz Rationalised 2023-24 Electromagnetic Induction 173 SUMMARY 1 The magnetic flux through a surface of area A placed in a uniform magnetic field B is defined as, FB = B A = BA cos q where q is the angle between B and A
1
5332-5335
Rationalised 2023-24 Electromagnetic Induction 173 SUMMARY 1 The magnetic flux through a surface of area A placed in a uniform magnetic field B is defined as, FB = B A = BA cos q where q is the angle between B and A 2
1
5333-5336
The magnetic flux through a surface of area A placed in a uniform magnetic field B is defined as, FB = B A = BA cos q where q is the angle between B and A 2 Faraday’s laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it, dB Nd Φt ε = − Here FB is the flux linked with one turn of the coil
1
5334-5337
A = BA cos q where q is the angle between B and A 2 Faraday’s laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it, dB Nd Φt ε = − Here FB is the flux linked with one turn of the coil If the circuit is closed, a current I = e/R is set up in it, where R is the resistance of the circuit
1
5335-5338
2 Faraday’s laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it, dB Nd Φt ε = − Here FB is the flux linked with one turn of the coil If the circuit is closed, a current I = e/R is set up in it, where R is the resistance of the circuit 3
1
5336-5339
Faraday’s laws of induction imply that the emf induced in a coil of N turns is directly related to the rate of change of flux through it, dB Nd Φt ε = − Here FB is the flux linked with one turn of the coil If the circuit is closed, a current I = e/R is set up in it, where R is the resistance of the circuit 3 Lenz’s law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it
1
5337-5340
If the circuit is closed, a current I = e/R is set up in it, where R is the resistance of the circuit 3 Lenz’s law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it The negative sign in the expression for Faraday’s law indicates this fact
1
5338-5341
3 Lenz’s law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it The negative sign in the expression for Faraday’s law indicates this fact 4
1
5339-5342
Lenz’s law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it The negative sign in the expression for Faraday’s law indicates this fact 4 When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is e = Bl v 5
1
5340-5343
The negative sign in the expression for Faraday’s law indicates this fact 4 When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is e = Bl v 5 Inductance is the ratio of the flux-linkage to current
1
5341-5344
4 When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is e = Bl v 5 Inductance is the ratio of the flux-linkage to current It is equal to NF/I
1
5342-5345
When a metal rod of length l is placed normal to a uniform magnetic field B and moved with a velocity v perpendicular to the field, the induced emf (called motional emf) across its ends is e = Bl v 5 Inductance is the ratio of the flux-linkage to current It is equal to NF/I 6
1
5343-5346
Inductance is the ratio of the flux-linkage to current It is equal to NF/I 6 A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1)
1
5344-5347
It is equal to NF/I 6 A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1) This relation is given by, 2 1 12 d Id M t ε = − The quantity M12 is called mutual inductance of coil 1 with respect to coil 2
1
5345-5348
6 A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1) This relation is given by, 2 1 12 d Id M t ε = − The quantity M12 is called mutual inductance of coil 1 with respect to coil 2 One can similarly define M21
1
5346-5349
A changing current in a coil (coil 2) can induce an emf in a nearby coil (coil 1) This relation is given by, 2 1 12 d Id M t ε = − The quantity M12 is called mutual inductance of coil 1 with respect to coil 2 One can similarly define M21 There exists a general equality, M12 = M21 7
1
5347-5350
This relation is given by, 2 1 12 d Id M t ε = − The quantity M12 is called mutual inductance of coil 1 with respect to coil 2 One can similarly define M21 There exists a general equality, M12 = M21 7 When a current in a coil changes, it induces a back emf in the same coil
1
5348-5351
One can similarly define M21 There exists a general equality, M12 = M21 7 When a current in a coil changes, it induces a back emf in the same coil The self-induced emf is given by, d d I L t ε = − L is the self-inductance of the coil
1
5349-5352
There exists a general equality, M12 = M21 7 When a current in a coil changes, it induces a back emf in the same coil The self-induced emf is given by, d d I L t ε = − L is the self-inductance of the coil It is a measure of the inertia of the coil against the change of current through it
1
5350-5353
When a current in a coil changes, it induces a back emf in the same coil The self-induced emf is given by, d d I L t ε = − L is the self-inductance of the coil It is a measure of the inertia of the coil against the change of current through it 8
1
5351-5354
The self-induced emf is given by, d d I L t ε = − L is the self-inductance of the coil It is a measure of the inertia of the coil against the change of current through it 8 The self-inductance of a long solenoid, the core of which consists of a magnetic material of relative permeability mr, is given by L = mr m0 n2 Al where A is the area of cross-section of the solenoid, l its length and n the number of turns per unit length
1
5352-5355
It is a measure of the inertia of the coil against the change of current through it 8 The self-inductance of a long solenoid, the core of which consists of a magnetic material of relative permeability mr, is given by L = mr m0 n2 Al where A is the area of cross-section of the solenoid, l its length and n the number of turns per unit length 9
1
5353-5356
8 The self-inductance of a long solenoid, the core of which consists of a magnetic material of relative permeability mr, is given by L = mr m0 n2 Al where A is the area of cross-section of the solenoid, l its length and n the number of turns per unit length 9 In an ac generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction
1
5354-5357
The self-inductance of a long solenoid, the core of which consists of a magnetic material of relative permeability mr, is given by L = mr m0 n2 Al where A is the area of cross-section of the solenoid, l its length and n the number of turns per unit length 9 In an ac generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction If coil of N turn and area A is rotated at n revolutions per second in a uniform magnetic field B, then the motional emf produced is e = NBA (2pn) sin (2pnt) where we have assumed that at time t = 0 s, the coil is perpendicular to the field
1
5355-5358
9 In an ac generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction If coil of N turn and area A is rotated at n revolutions per second in a uniform magnetic field B, then the motional emf produced is e = NBA (2pn) sin (2pnt) where we have assumed that at time t = 0 s, the coil is perpendicular to the field Rationalised 2023-24 Physics 174 POINTS TO PONDER 1
1
5356-5359
In an ac generator, mechanical energy is converted to electrical energy by virtue of electromagnetic induction If coil of N turn and area A is rotated at n revolutions per second in a uniform magnetic field B, then the motional emf produced is e = NBA (2pn) sin (2pnt) where we have assumed that at time t = 0 s, the coil is perpendicular to the field Rationalised 2023-24 Physics 174 POINTS TO PONDER 1 Electricity and magnetism are intimately related
1
5357-5360
If coil of N turn and area A is rotated at n revolutions per second in a uniform magnetic field B, then the motional emf produced is e = NBA (2pn) sin (2pnt) where we have assumed that at time t = 0 s, the coil is perpendicular to the field Rationalised 2023-24 Physics 174 POINTS TO PONDER 1 Electricity and magnetism are intimately related In the early part of the nineteenth century, the experiments of Oersted, Ampere and others established that moving charges (currents) produce a magnetic field
1
5358-5361
Rationalised 2023-24 Physics 174 POINTS TO PONDER 1 Electricity and magnetism are intimately related In the early part of the nineteenth century, the experiments of Oersted, Ampere and others established that moving charges (currents) produce a magnetic field Somewhat later, around 1830, the experiments of Faraday and Henry demonstrated that a moving magnet can induce electric current
1
5359-5362
Electricity and magnetism are intimately related In the early part of the nineteenth century, the experiments of Oersted, Ampere and others established that moving charges (currents) produce a magnetic field Somewhat later, around 1830, the experiments of Faraday and Henry demonstrated that a moving magnet can induce electric current 2
1
5360-5363
In the early part of the nineteenth century, the experiments of Oersted, Ampere and others established that moving charges (currents) produce a magnetic field Somewhat later, around 1830, the experiments of Faraday and Henry demonstrated that a moving magnet can induce electric current 2 In a closed circuit, electric currents are induced so as to oppose the changing magnetic flux
1
5361-5364
Somewhat later, around 1830, the experiments of Faraday and Henry demonstrated that a moving magnet can induce electric current 2 In a closed circuit, electric currents are induced so as to oppose the changing magnetic flux It is as per the law of conservation of energy
1
5362-5365
2 In a closed circuit, electric currents are induced so as to oppose the changing magnetic flux It is as per the law of conservation of energy However, in case of an open circuit, an emf is induced across its ends
1
5363-5366
In a closed circuit, electric currents are induced so as to oppose the changing magnetic flux It is as per the law of conservation of energy However, in case of an open circuit, an emf is induced across its ends How is it related to the flux change
1
5364-5367
It is as per the law of conservation of energy However, in case of an open circuit, an emf is induced across its ends How is it related to the flux change 3
1
5365-5368
However, in case of an open circuit, an emf is induced across its ends How is it related to the flux change 3 The motional emf discussed in Section 6
1
5366-5369
How is it related to the flux change 3 The motional emf discussed in Section 6 5 can be argued independently from Faraday’s law using the Lorentz force on moving charges
1
5367-5370
3 The motional emf discussed in Section 6 5 can be argued independently from Faraday’s law using the Lorentz force on moving charges However, even if the charges are stationary [and the q (v × B) term of the Lorentz force is not operative], an emf is nevertheless induced in the presence of a time-varying magnetic field
1
5368-5371
The motional emf discussed in Section 6 5 can be argued independently from Faraday’s law using the Lorentz force on moving charges However, even if the charges are stationary [and the q (v × B) term of the Lorentz force is not operative], an emf is nevertheless induced in the presence of a time-varying magnetic field Thus, moving charges in static field and static charges in a time-varying field seem to be symmetric situation for Faraday’s law
1
5369-5372
5 can be argued independently from Faraday’s law using the Lorentz force on moving charges However, even if the charges are stationary [and the q (v × B) term of the Lorentz force is not operative], an emf is nevertheless induced in the presence of a time-varying magnetic field Thus, moving charges in static field and static charges in a time-varying field seem to be symmetric situation for Faraday’s law This gives a tantalising hint on the relevance of the principle of relativity for Faraday’s law
1
5370-5373
However, even if the charges are stationary [and the q (v × B) term of the Lorentz force is not operative], an emf is nevertheless induced in the presence of a time-varying magnetic field Thus, moving charges in static field and static charges in a time-varying field seem to be symmetric situation for Faraday’s law This gives a tantalising hint on the relevance of the principle of relativity for Faraday’s law EXERCISES 6
1
5371-5374
Thus, moving charges in static field and static charges in a time-varying field seem to be symmetric situation for Faraday’s law This gives a tantalising hint on the relevance of the principle of relativity for Faraday’s law EXERCISES 6 1 Predict the direction of induced current in the situations described by the following Figs
1
5372-5375
This gives a tantalising hint on the relevance of the principle of relativity for Faraday’s law EXERCISES 6 1 Predict the direction of induced current in the situations described by the following Figs 6
1
5373-5376
EXERCISES 6 1 Predict the direction of induced current in the situations described by the following Figs 6 15(a) to (f )
1
5374-5377
1 Predict the direction of induced current in the situations described by the following Figs 6 15(a) to (f ) Quantity Symbol Units Dimensions Equations Magnetic Flux FB Wb (weber) [M L2 T –2 A–1] FB = B A i EMF e V (volt) [M L2 T –3 A–1] e = B d( )/d N t Φ − Mutual Inductance M H (henry) [M L2 T –2 A–2] e1 ( ) 12 d2 /d M I t = − Self Inductance L H (henry) [M L2 T –2 A–2] ( Ld /d) I t ε = − Rationalised 2023-24 Electromagnetic Induction 175 FIGURE 6
1
5375-5378
6 15(a) to (f ) Quantity Symbol Units Dimensions Equations Magnetic Flux FB Wb (weber) [M L2 T –2 A–1] FB = B A i EMF e V (volt) [M L2 T –3 A–1] e = B d( )/d N t Φ − Mutual Inductance M H (henry) [M L2 T –2 A–2] e1 ( ) 12 d2 /d M I t = − Self Inductance L H (henry) [M L2 T –2 A–2] ( Ld /d) I t ε = − Rationalised 2023-24 Electromagnetic Induction 175 FIGURE 6 15 6
1
5376-5379
15(a) to (f ) Quantity Symbol Units Dimensions Equations Magnetic Flux FB Wb (weber) [M L2 T –2 A–1] FB = B A i EMF e V (volt) [M L2 T –3 A–1] e = B d( )/d N t Φ − Mutual Inductance M H (henry) [M L2 T –2 A–2] e1 ( ) 12 d2 /d M I t = − Self Inductance L H (henry) [M L2 T –2 A–2] ( Ld /d) I t ε = − Rationalised 2023-24 Electromagnetic Induction 175 FIGURE 6 15 6 2 Use Lenz’s law to determine the direction of induced current in the situations described by Fig
1
5377-5380
Quantity Symbol Units Dimensions Equations Magnetic Flux FB Wb (weber) [M L2 T –2 A–1] FB = B A i EMF e V (volt) [M L2 T –3 A–1] e = B d( )/d N t Φ − Mutual Inductance M H (henry) [M L2 T –2 A–2] e1 ( ) 12 d2 /d M I t = − Self Inductance L H (henry) [M L2 T –2 A–2] ( Ld /d) I t ε = − Rationalised 2023-24 Electromagnetic Induction 175 FIGURE 6 15 6 2 Use Lenz’s law to determine the direction of induced current in the situations described by Fig 6
1
5378-5381
15 6 2 Use Lenz’s law to determine the direction of induced current in the situations described by Fig 6 16: (a) A wire of irregular shape turning into a circular shape; Rationalised 2023-24 Physics 176 (b) A circular loop being deformed into a narrow straight wire
1
5379-5382
2 Use Lenz’s law to determine the direction of induced current in the situations described by Fig 6 16: (a) A wire of irregular shape turning into a circular shape; Rationalised 2023-24 Physics 176 (b) A circular loop being deformed into a narrow straight wire FIGURE 6
1
5380-5383
6 16: (a) A wire of irregular shape turning into a circular shape; Rationalised 2023-24 Physics 176 (b) A circular loop being deformed into a narrow straight wire FIGURE 6 16 6
1
5381-5384
16: (a) A wire of irregular shape turning into a circular shape; Rationalised 2023-24 Physics 176 (b) A circular loop being deformed into a narrow straight wire FIGURE 6 16 6 3 A long solenoid with 15 turns per cm has a small loop of area 2
1
5382-5385
FIGURE 6 16 6 3 A long solenoid with 15 turns per cm has a small loop of area 2 0 cm2 placed inside the solenoid normal to its axis
1
5383-5386
16 6 3 A long solenoid with 15 turns per cm has a small loop of area 2 0 cm2 placed inside the solenoid normal to its axis If the current carried by the solenoid changes steadily from 2
1
5384-5387
3 A long solenoid with 15 turns per cm has a small loop of area 2 0 cm2 placed inside the solenoid normal to its axis If the current carried by the solenoid changes steadily from 2 0 A to 4
1
5385-5388
0 cm2 placed inside the solenoid normal to its axis If the current carried by the solenoid changes steadily from 2 0 A to 4 0 A in 0
1
5386-5389
If the current carried by the solenoid changes steadily from 2 0 A to 4 0 A in 0 1 s, what is the induced emf in the loop while the current is changing
1
5387-5390
0 A to 4 0 A in 0 1 s, what is the induced emf in the loop while the current is changing 6
1
5388-5391
0 A in 0 1 s, what is the induced emf in the loop while the current is changing 6 4 A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0
1
5389-5392
1 s, what is the induced emf in the loop while the current is changing 6 4 A rectangular wire loop of sides 8 cm and 2 cm with a small cut is moving out of a region of uniform magnetic field of magnitude 0 3 T directed normal to the loop