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1 | 5390-5393 | 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 What is the emf developed across the
cut if the velocity of the loop is 1 cm s–1 in a direction normal to the
(a) longer side, (b) shorter side of the loop |
1 | 5391-5394 | 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 What is the emf developed across the
cut if the velocity of the loop is 1 cm s–1 in a direction normal to the
(a) longer side, (b) shorter side of the loop For how long does the
induced voltage last in each case |
1 | 5392-5395 | 3 T
directed normal to the loop What is the emf developed across the
cut if the velocity of the loop is 1 cm s–1 in a direction normal to the
(a) longer side, (b) shorter side of the loop For how long does the
induced voltage last in each case 6 |
1 | 5393-5396 | What is the emf developed across the
cut if the velocity of the loop is 1 cm s–1 in a direction normal to the
(a) longer side, (b) shorter side of the loop For how long does the
induced voltage last in each case 6 5
A 1 |
1 | 5394-5397 | For how long does the
induced voltage last in each case 6 5
A 1 0 m long metallic rod is rotated with an angular frequency of
400 rad s–1
about an axis normal to the rod passing through its one
end |
1 | 5395-5398 | 6 5
A 1 0 m long metallic rod is rotated with an angular frequency of
400 rad s–1
about an axis normal to the rod passing through its one
end The other end of the rod is in contact with a circular metallic
ring |
1 | 5396-5399 | 5
A 1 0 m long metallic rod is rotated with an angular frequency of
400 rad s–1
about an axis normal to the rod passing through its one
end The other end of the rod is in contact with a circular metallic
ring A constant and uniform magnetic field of 0 |
1 | 5397-5400 | 0 m long metallic rod is rotated with an angular frequency of
400 rad s–1
about an axis normal to the rod passing through its one
end The other end of the rod is in contact with a circular metallic
ring A constant and uniform magnetic field of 0 5 T parallel to the
axis exists everywhere |
1 | 5398-5401 | The other end of the rod is in contact with a circular metallic
ring A constant and uniform magnetic field of 0 5 T parallel to the
axis exists everywhere Calculate the emf developed between the
centre and the ring |
1 | 5399-5402 | A constant and uniform magnetic field of 0 5 T parallel to the
axis exists everywhere Calculate the emf developed between the
centre and the ring 6 |
1 | 5400-5403 | 5 T parallel to the
axis exists everywhere Calculate the emf developed between the
centre and the ring 6 6
A horizontal straight wire 10 m long extending from east to west is
falling with a speed of 5 |
1 | 5401-5404 | Calculate the emf developed between the
centre and the ring 6 6
A horizontal straight wire 10 m long extending from east to west is
falling with a speed of 5 0 m s–1, at right angles to the horizontal
component of the earth’s magnetic field, 0 |
1 | 5402-5405 | 6 6
A horizontal straight wire 10 m long extending from east to west is
falling with a speed of 5 0 m s–1, at right angles to the horizontal
component of the earth’s magnetic field, 0 30 ´ 10–4 Wb m–2 |
1 | 5403-5406 | 6
A horizontal straight wire 10 m long extending from east to west is
falling with a speed of 5 0 m s–1, at right angles to the horizontal
component of the earth’s magnetic field, 0 30 ´ 10–4 Wb m–2 (a) What is the instantaneous value of the emf induced in the wire |
1 | 5404-5407 | 0 m s–1, at right angles to the horizontal
component of the earth’s magnetic field, 0 30 ´ 10–4 Wb m–2 (a) What is the instantaneous value of the emf induced in the wire (b) What is the direction of the emf |
1 | 5405-5408 | 30 ´ 10–4 Wb m–2 (a) What is the instantaneous value of the emf induced in the wire (b) What is the direction of the emf (c) Which end of the wire is at the higher electrical potential |
1 | 5406-5409 | (a) What is the instantaneous value of the emf induced in the wire (b) What is the direction of the emf (c) Which end of the wire is at the higher electrical potential 6 |
1 | 5407-5410 | (b) What is the direction of the emf (c) Which end of the wire is at the higher electrical potential 6 7
Current in a circuit falls from 5 |
1 | 5408-5411 | (c) Which end of the wire is at the higher electrical potential 6 7
Current in a circuit falls from 5 0 A to 0 |
1 | 5409-5412 | 6 7
Current in a circuit falls from 5 0 A to 0 0 A in 0 |
1 | 5410-5413 | 7
Current in a circuit falls from 5 0 A to 0 0 A in 0 1 s |
1 | 5411-5414 | 0 A to 0 0 A in 0 1 s If an average emf
of 200 V induced, give an estimate of the self-inductance of the circuit |
1 | 5412-5415 | 0 A in 0 1 s If an average emf
of 200 V induced, give an estimate of the self-inductance of the circuit 6 |
1 | 5413-5416 | 1 s If an average emf
of 200 V induced, give an estimate of the self-inductance of the circuit 6 8
A pair of adjacent coils has a mutual inductance of 1 |
1 | 5414-5417 | If an average emf
of 200 V induced, give an estimate of the self-inductance of the circuit 6 8
A pair of adjacent coils has a mutual inductance of 1 5 H |
1 | 5415-5418 | 6 8
A pair of adjacent coils has a mutual inductance of 1 5 H If the
current in one coil changes from 0 to 20 A in 0 |
1 | 5416-5419 | 8
A pair of adjacent coils has a mutual inductance of 1 5 H If the
current in one coil changes from 0 to 20 A in 0 5 s, what is the
change of flux linkage with the other coil |
1 | 5417-5420 | 5 H If the
current in one coil changes from 0 to 20 A in 0 5 s, what is the
change of flux linkage with the other coil Rationalised 2023-24
7 |
1 | 5418-5421 | If the
current in one coil changes from 0 to 20 A in 0 5 s, what is the
change of flux linkage with the other coil Rationalised 2023-24
7 1 INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources |
1 | 5419-5422 | 5 s, what is the
change of flux linkage with the other coil Rationalised 2023-24
7 1 INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources These currents do not change direction with time |
1 | 5420-5423 | Rationalised 2023-24
7 1 INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources These currents do not change direction with time But voltages
and currents that vary with time are very common |
1 | 5421-5424 | 1 INTRODUCTION
We have so far considered direct current (dc) sources and circuits with dc
sources These currents do not change direction with time But voltages
and currents that vary with time are very common The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time |
1 | 5422-5425 | These currents do not change direction with time But voltages
and currents that vary with time are very common The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)* |
1 | 5423-5426 | But voltages
and currents that vary with time are very common The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)* Today, most of the electrical devices we use require ac voltage |
1 | 5424-5427 | The electric mains
supply in our homes and offices is a voltage that varies like a sine function
with time Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)* Today, most of the electrical devices we use require ac voltage This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current |
1 | 5425-5428 | Such a voltage is called alternating voltage (ac voltage) and
the current driven by it in a circuit is called the alternating current (ac
current)* Today, most of the electrical devices we use require ac voltage This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers |
1 | 5426-5429 | Today, most of the electrical devices we use require ac voltage This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers Further, electrical energy can also be transmitted
economically over long distances |
1 | 5427-5430 | This is mainly because most of the electrical energy sold by power
companies is transmitted and distributed as alternating current The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers Further, electrical energy can also be transmitted
economically over long distances AC circuits exhibit characteristics which
are exploited in many devices of daily use |
1 | 5428-5431 | The main
reason for preferring use of ac voltage over dc voltage is that ac voltages
can be easily and efficiently converted from one voltage to the other by
means of transformers Further, electrical energy can also be transmitted
economically over long distances AC circuits exhibit characteristics which
are exploited in many devices of daily use For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter |
1 | 5429-5432 | Further, electrical energy can also be transmitted
economically over long distances AC circuits exhibit characteristics which
are exploited in many devices of daily use For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter Chapter Seven
ALTERNATING
CURRENT
*
The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current |
1 | 5430-5433 | AC circuits exhibit characteristics which
are exploited in many devices of daily use For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter Chapter Seven
ALTERNATING
CURRENT
*
The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use |
1 | 5431-5434 | For example, whenever we
tune our radio to a favourite station, we are taking advantage of a special
property of ac circuits – one of many that you will study in this chapter Chapter Seven
ALTERNATING
CURRENT
*
The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use Further, voltage – another phrase commonly
used means potential difference between two points |
1 | 5432-5435 | Chapter Seven
ALTERNATING
CURRENT
*
The phrases ac voltage and ac current are contradictory and redundant,
respectively, since they mean, literally, alternating current voltage and alternating
current current Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use Further, voltage – another phrase commonly
used means potential difference between two points Rationalised 2023-24
Physics
178
NICOLA TESLA (1856 – 1943)
Nicola Tesla (1856 –
1943) Serbian-American
scientist, inventor and
genius |
1 | 5433-5436 | Still, the abbreviation ac to designate an electrical quantity
displaying simple harmonic time dependance has become so universally accepted
that we follow others in its use Further, voltage – another phrase commonly
used means potential difference between two points Rationalised 2023-24
Physics
178
NICOLA TESLA (1856 – 1943)
Nicola Tesla (1856 –
1943) Serbian-American
scientist, inventor and
genius He conceived the
idea
of
the
rotating
magnetic field, which is the
basis of practically all
alternating
current
machinery, and which
helped usher in the age of
electric power |
1 | 5434-5437 | Further, voltage – another phrase commonly
used means potential difference between two points Rationalised 2023-24
Physics
178
NICOLA TESLA (1856 – 1943)
Nicola Tesla (1856 –
1943) Serbian-American
scientist, inventor and
genius He conceived the
idea
of
the
rotating
magnetic field, which is the
basis of practically all
alternating
current
machinery, and which
helped usher in the age of
electric power He also
invented among other
things the induction motor,
the polyphase system of ac
power,
and
the
high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment |
1 | 5435-5438 | Rationalised 2023-24
Physics
178
NICOLA TESLA (1856 – 1943)
Nicola Tesla (1856 –
1943) Serbian-American
scientist, inventor and
genius He conceived the
idea
of
the
rotating
magnetic field, which is the
basis of practically all
alternating
current
machinery, and which
helped usher in the age of
electric power He also
invented among other
things the induction motor,
the polyphase system of ac
power,
and
the
high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment The SI unit of magnetic field
is named in his honour |
1 | 5436-5439 | He conceived the
idea
of
the
rotating
magnetic field, which is the
basis of practically all
alternating
current
machinery, and which
helped usher in the age of
electric power He also
invented among other
things the induction motor,
the polyphase system of ac
power,
and
the
high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment The SI unit of magnetic field
is named in his honour 7 |
1 | 5437-5440 | He also
invented among other
things the induction motor,
the polyphase system of ac
power,
and
the
high
frequency induction coil
(the Tesla coil) used in radio
and television sets and
other electronic equipment The SI unit of magnetic field
is named in his honour 7 2 AC VOLTAGE APPLIED TO A RESISTOR
Figure 7 |
1 | 5438-5441 | The SI unit of magnetic field
is named in his honour 7 2 AC VOLTAGE APPLIED TO A RESISTOR
Figure 7 1 shows a resistor connected to a source e of
ac voltage |
1 | 5439-5442 | 7 2 AC VOLTAGE APPLIED TO A RESISTOR
Figure 7 1 shows a resistor connected to a source e of
ac voltage The symbol for an ac source in a circuit
diagram is |
1 | 5440-5443 | 2 AC VOLTAGE APPLIED TO A RESISTOR
Figure 7 1 shows a resistor connected to a source e of
ac voltage The symbol for an ac source in a circuit
diagram is We consider a source which produces
sinusoidally varying potential difference across its
terminals |
1 | 5441-5444 | 1 shows a resistor connected to a source e of
ac voltage The symbol for an ac source in a circuit
diagram is We consider a source which produces
sinusoidally varying potential difference across its
terminals Let this potential difference, also called ac
voltage, be given by
msin
v
v
ωt
=
(7 |
1 | 5442-5445 | The symbol for an ac source in a circuit
diagram is We consider a source which produces
sinusoidally varying potential difference across its
terminals Let this potential difference, also called ac
voltage, be given by
msin
v
v
ωt
=
(7 1)
where vm is the amplitude of the oscillating potential
difference and w is its angular frequency |
1 | 5443-5446 | We consider a source which produces
sinusoidally varying potential difference across its
terminals Let this potential difference, also called ac
voltage, be given by
msin
v
v
ωt
=
(7 1)
where vm is the amplitude of the oscillating potential
difference and w is its angular frequency To find the value of current through the resistor, we
apply Kirchhoff’s loop rule
∑ε( )t =
0 (refer to Section
3 |
1 | 5444-5447 | Let this potential difference, also called ac
voltage, be given by
msin
v
v
ωt
=
(7 1)
where vm is the amplitude of the oscillating potential
difference and w is its angular frequency To find the value of current through the resistor, we
apply Kirchhoff’s loop rule
∑ε( )t =
0 (refer to Section
3 13), to the circuit shown in Fig |
1 | 5445-5448 | 1)
where vm is the amplitude of the oscillating potential
difference and w is its angular frequency To find the value of current through the resistor, we
apply Kirchhoff’s loop rule
∑ε( )t =
0 (refer to Section
3 13), to the circuit shown in Fig 7 |
1 | 5446-5449 | To find the value of current through the resistor, we
apply Kirchhoff’s loop rule
∑ε( )t =
0 (refer to Section
3 13), to the circuit shown in Fig 7 1 to get
=
vmsin
t
i R
ω
or
vmsin
i
t
R
ω
=
Since R is a constant, we can write this equation as
msin
i
i
ωt
=
(7 |
1 | 5447-5450 | 13), to the circuit shown in Fig 7 1 to get
=
vmsin
t
i R
ω
or
vmsin
i
t
R
ω
=
Since R is a constant, we can write this equation as
msin
i
i
ωt
=
(7 2)
where the current amplitude im is given by
m
m
v
i
=R
(7 |
1 | 5448-5451 | 7 1 to get
=
vmsin
t
i R
ω
or
vmsin
i
t
R
ω
=
Since R is a constant, we can write this equation as
msin
i
i
ωt
=
(7 2)
where the current amplitude im is given by
m
m
v
i
=R
(7 3)
Equation (7 |
1 | 5449-5452 | 1 to get
=
vmsin
t
i R
ω
or
vmsin
i
t
R
ω
=
Since R is a constant, we can write this equation as
msin
i
i
ωt
=
(7 2)
where the current amplitude im is given by
m
m
v
i
=R
(7 3)
Equation (7 3) is Ohm’s law, which for resistors, works equally
well for both ac and dc voltages |
1 | 5450-5453 | 2)
where the current amplitude im is given by
m
m
v
i
=R
(7 3)
Equation (7 3) is Ohm’s law, which for resistors, works equally
well for both ac and dc voltages The voltage across a pure resistor
and the current through it, given by Eqs |
1 | 5451-5454 | 3)
Equation (7 3) is Ohm’s law, which for resistors, works equally
well for both ac and dc voltages The voltage across a pure resistor
and the current through it, given by Eqs (7 |
1 | 5452-5455 | 3) is Ohm’s law, which for resistors, works equally
well for both ac and dc voltages The voltage across a pure resistor
and the current through it, given by Eqs (7 1) and (7 |
1 | 5453-5456 | The voltage across a pure resistor
and the current through it, given by Eqs (7 1) and (7 2) are
plotted as a function of time in Fig |
1 | 5454-5457 | (7 1) and (7 2) are
plotted as a function of time in Fig 7 |
1 | 5455-5458 | 1) and (7 2) are
plotted as a function of time in Fig 7 2 |
1 | 5456-5459 | 2) are
plotted as a function of time in Fig 7 2 Note, in particular that
both v and i reach zero, minimum and maximum values at the
same time |
1 | 5457-5460 | 7 2 Note, in particular that
both v and i reach zero, minimum and maximum values at the
same time Clearly, the voltage and current are in phase with
each other |
1 | 5458-5461 | 2 Note, in particular that
both v and i reach zero, minimum and maximum values at the
same time Clearly, the voltage and current are in phase with
each other We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle |
1 | 5459-5462 | Note, in particular that
both v and i reach zero, minimum and maximum values at the
same time Clearly, the voltage and current are in phase with
each other We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero |
1 | 5460-5463 | Clearly, the voltage and current are in phase with
each other We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero The fact that the average current is zero, however, does
FIGURE 7 |
1 | 5461-5464 | We see that, like the applied voltage, the current varies
sinusoidally and has corresponding positive and negative values
during each cycle Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero The fact that the average current is zero, however, does
FIGURE 7 1 AC voltage applied to a resistor |
1 | 5462-5465 | Thus, the sum of the instantaneous current
values over one complete cycle is zero, and the average current
is zero The fact that the average current is zero, however, does
FIGURE 7 1 AC voltage applied to a resistor FIGURE 7 |
1 | 5463-5466 | The fact that the average current is zero, however, does
FIGURE 7 1 AC voltage applied to a resistor FIGURE 7 2 In a pure
resistor, the voltage and
current are in phase |
1 | 5464-5467 | 1 AC voltage applied to a resistor FIGURE 7 2 In a pure
resistor, the voltage and
current are in phase The
minima, zero and maxima
occur at the same
respective times |
1 | 5465-5468 | FIGURE 7 2 In a pure
resistor, the voltage and
current are in phase The
minima, zero and maxima
occur at the same
respective times Rationalised 2023-24
179
Alternating Current
GEORGE WESTINGHOUSE (1846 – 1914)
George
Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct
current |
1 | 5466-5469 | 2 In a pure
resistor, the voltage and
current are in phase The
minima, zero and maxima
occur at the same
respective times Rationalised 2023-24
179
Alternating Current
GEORGE WESTINGHOUSE (1846 – 1914)
George
Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct
current Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current |
1 | 5467-5470 | The
minima, zero and maxima
occur at the same
respective times Rationalised 2023-24
179
Alternating Current
GEORGE WESTINGHOUSE (1846 – 1914)
George
Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct
current Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current Westinghouse
was convinced that the
technology of alternating
current was the key to
the
electrical
future |
1 | 5468-5471 | Rationalised 2023-24
179
Alternating Current
GEORGE WESTINGHOUSE (1846 – 1914)
George
Westinghouse
(1846 – 1914) A leading
proponent of the use of
alternating current over
direct
current Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current Westinghouse
was convinced that the
technology of alternating
current was the key to
the
electrical
future He founded the famous
Company named after him
and enlisted the services
of
Nicola
Tesla
and
other inventors in the
development of alternating
current
motors
and
apparatus
for
the
transmission
of
high
tension current, pioneering
in large scale lighting |
1 | 5469-5472 | Thus,
he came into conflict
with Thomas Alva Edison,
an advocate of direct
current Westinghouse
was convinced that the
technology of alternating
current was the key to
the
electrical
future He founded the famous
Company named after him
and enlisted the services
of
Nicola
Tesla
and
other inventors in the
development of alternating
current
motors
and
apparatus
for
the
transmission
of
high
tension current, pioneering
in large scale lighting not mean that the average power consumed is zero and
that there is no dissipation of electrical energy |
1 | 5470-5473 | Westinghouse
was convinced that the
technology of alternating
current was the key to
the
electrical
future He founded the famous
Company named after him
and enlisted the services
of
Nicola
Tesla
and
other inventors in the
development of alternating
current
motors
and
apparatus
for
the
transmission
of
high
tension current, pioneering
in large scale lighting not mean that the average power consumed is zero and
that there is no dissipation of electrical energy As you
know, Joule heating is given by i2R and depends on i2
(which is always positive whether i is positive or negative)
and not on i |
1 | 5471-5474 | He founded the famous
Company named after him
and enlisted the services
of
Nicola
Tesla
and
other inventors in the
development of alternating
current
motors
and
apparatus
for
the
transmission
of
high
tension current, pioneering
in large scale lighting not mean that the average power consumed is zero and
that there is no dissipation of electrical energy As you
know, Joule heating is given by i2R and depends on i2
(which is always positive whether i is positive or negative)
and not on i Thus, there is Joule heating and
dissipation
of
electrical
energy
when
an
ac current passes through a resistor |
1 | 5472-5475 | not mean that the average power consumed is zero and
that there is no dissipation of electrical energy As you
know, Joule heating is given by i2R and depends on i2
(which is always positive whether i is positive or negative)
and not on i Thus, there is Joule heating and
dissipation
of
electrical
energy
when
an
ac current passes through a resistor The instantaneous power dissipated in the resistor is
2
2
sin2
m
p
i R
i R
ωt
=
=
(7 |
1 | 5473-5476 | As you
know, Joule heating is given by i2R and depends on i2
(which is always positive whether i is positive or negative)
and not on i Thus, there is Joule heating and
dissipation
of
electrical
energy
when
an
ac current passes through a resistor The instantaneous power dissipated in the resistor is
2
2
sin2
m
p
i R
i R
ωt
=
=
(7 4)
The average value of p over a cycle is*
2
2
sin2
m
p
i R
i R
ωt
= <
> = <
>
[7 |
1 | 5474-5477 | Thus, there is Joule heating and
dissipation
of
electrical
energy
when
an
ac current passes through a resistor The instantaneous power dissipated in the resistor is
2
2
sin2
m
p
i R
i R
ωt
=
=
(7 4)
The average value of p over a cycle is*
2
2
sin2
m
p
i R
i R
ωt
= <
> = <
>
[7 5(a)]
where the bar over a letter (here, p) denotes its average
value and < |
1 | 5475-5478 | The instantaneous power dissipated in the resistor is
2
2
sin2
m
p
i R
i R
ωt
=
=
(7 4)
The average value of p over a cycle is*
2
2
sin2
m
p
i R
i R
ωt
= <
> = <
>
[7 5(a)]
where the bar over a letter (here, p) denotes its average
value and < > denotes taking average of the quantity
inside the bracket |
1 | 5476-5479 | 4)
The average value of p over a cycle is*
2
2
sin2
m
p
i R
i R
ωt
= <
> = <
>
[7 5(a)]
where the bar over a letter (here, p) denotes its average
value and < > denotes taking average of the quantity
inside the bracket Since, i2
m and R are constants,
2
sin2
m
p
i R
ωt
=
<
>
[7 |
1 | 5477-5480 | 5(a)]
where the bar over a letter (here, p) denotes its average
value and < > denotes taking average of the quantity
inside the bracket Since, i2
m and R are constants,
2
sin2
m
p
i R
ωt
=
<
>
[7 5(b)]
Using the trigonometric identity, sin2 wt =
1/2 (1– cos 2wt), we have < sin2 wt > = (1/2) (1– < cos 2wt >)
and since < cos2wt > = 0**, we have,
2
1
sin
2
ωt
<
> =
Thus,
212
m
p
i R
=
[7 |
1 | 5478-5481 | > denotes taking average of the quantity
inside the bracket Since, i2
m and R are constants,
2
sin2
m
p
i R
ωt
=
<
>
[7 5(b)]
Using the trigonometric identity, sin2 wt =
1/2 (1– cos 2wt), we have < sin2 wt > = (1/2) (1– < cos 2wt >)
and since < cos2wt > = 0**, we have,
2
1
sin
2
ωt
<
> =
Thus,
212
m
p
i R
=
[7 5(c)]
To express ac power in the same form as dc power
(P = I2R), a special value of current is defined and used |
1 | 5479-5482 | Since, i2
m and R are constants,
2
sin2
m
p
i R
ωt
=
<
>
[7 5(b)]
Using the trigonometric identity, sin2 wt =
1/2 (1– cos 2wt), we have < sin2 wt > = (1/2) (1– < cos 2wt >)
and since < cos2wt > = 0**, we have,
2
1
sin
2
ωt
<
> =
Thus,
212
m
p
i R
=
[7 5(c)]
To express ac power in the same form as dc power
(P = I2R), a special value of current is defined and used It is called, root mean square (rms) or effective current
(Fig |
1 | 5480-5483 | 5(b)]
Using the trigonometric identity, sin2 wt =
1/2 (1– cos 2wt), we have < sin2 wt > = (1/2) (1– < cos 2wt >)
and since < cos2wt > = 0**, we have,
2
1
sin
2
ωt
<
> =
Thus,
212
m
p
i R
=
[7 5(c)]
To express ac power in the same form as dc power
(P = I2R), a special value of current is defined and used It is called, root mean square (rms) or effective current
(Fig 7 |
1 | 5481-5484 | 5(c)]
To express ac power in the same form as dc power
(P = I2R), a special value of current is defined and used It is called, root mean square (rms) or effective current
(Fig 7 3) and is denoted by Irms or I |
1 | 5482-5485 | It is called, root mean square (rms) or effective current
(Fig 7 3) and is denoted by Irms or I *
The average value of a function F (t) over a period T is given by F t
T
F t
t
T
( )
( )
=
∫
1
0
d
** <
> =
∫
=
=
−
[
] =
cos
cos
sin
sin
2
1
2
1
22
21
2
0
0
0
0
ω
ω
ωω
ω
ω
t
T
t dt
T
t
T
T
T
T
FIGURE 7 |
1 | 5483-5486 | 7 3) and is denoted by Irms or I *
The average value of a function F (t) over a period T is given by F t
T
F t
t
T
( )
( )
=
∫
1
0
d
** <
> =
∫
=
=
−
[
] =
cos
cos
sin
sin
2
1
2
1
22
21
2
0
0
0
0
ω
ω
ωω
ω
ω
t
T
t dt
T
t
T
T
T
T
FIGURE 7 3 The rms current I is related to the
peak current im by I =
mi/ 2
= 0 |
1 | 5484-5487 | 3) and is denoted by Irms or I *
The average value of a function F (t) over a period T is given by F t
T
F t
t
T
( )
( )
=
∫
1
0
d
** <
> =
∫
=
=
−
[
] =
cos
cos
sin
sin
2
1
2
1
22
21
2
0
0
0
0
ω
ω
ωω
ω
ω
t
T
t dt
T
t
T
T
T
T
FIGURE 7 3 The rms current I is related to the
peak current im by I =
mi/ 2
= 0 707 im |
1 | 5485-5488 | *
The average value of a function F (t) over a period T is given by F t
T
F t
t
T
( )
( )
=
∫
1
0
d
** <
> =
∫
=
=
−
[
] =
cos
cos
sin
sin
2
1
2
1
22
21
2
0
0
0
0
ω
ω
ωω
ω
ω
t
T
t dt
T
t
T
T
T
T
FIGURE 7 3 The rms current I is related to the
peak current im by I =
mi/ 2
= 0 707 im Rationalised 2023-24
Physics
180
It is defined by
2
212
2
m
m
i
I
i
i
=
=
=
= 0 |
1 | 5486-5489 | 3 The rms current I is related to the
peak current im by I =
mi/ 2
= 0 707 im Rationalised 2023-24
Physics
180
It is defined by
2
212
2
m
m
i
I
i
i
=
=
=
= 0 707 im
(7 |
1 | 5487-5490 | 707 im Rationalised 2023-24
Physics
180
It is defined by
2
212
2
m
m
i
I
i
i
=
=
=
= 0 707 im
(7 6)
In terms of I, the average power, denoted by P is
2
2
21
m
p
P
i R
I R
=
=
=
(7 |
1 | 5488-5491 | Rationalised 2023-24
Physics
180
It is defined by
2
212
2
m
m
i
I
i
i
=
=
=
= 0 707 im
(7 6)
In terms of I, the average power, denoted by P is
2
2
21
m
p
P
i R
I R
=
=
=
(7 7)
Similarly, we define the rms voltage or effective voltage by
V =
2
m
v
= 0 |
1 | 5489-5492 | 707 im
(7 6)
In terms of I, the average power, denoted by P is
2
2
21
m
p
P
i R
I R
=
=
=
(7 7)
Similarly, we define the rms voltage or effective voltage by
V =
2
m
v
= 0 707 vm
(7 |
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