Draw the graphs for instantaneous voltage and current against time for RC series circuit.. a How does the impedance of an L-R circuit change with the increase in the frequency of the ac
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VERY SHORT AND SHORT-ANSWERS QUESTIONS
46 Draw the graphs for instantaneous voltage and current against time for RC series circuit.
47. What is the impedence of a coil having resistance of 3 Ω and reactance of 4 Ω ?
48. Show graphically the variation of inductive and capacitive reactances with the frequency of
49. A bulb and a capacitor are connected in series to a source of alternating current What will happen on increasing the frequency of the source ?
50. A choke coil and a bulb are connected in series to an a.c source If an iron core is inserted in the choke coil, is there any change in the brightness of the bulb ? (AISSCE 1995)
51. In the above problem if the a.c source is replaced by a d.c source, than what is the change in brightness of the source ?
52. (a) How does the impedance of an L-R circuit change with the increase in the frequency of the
ac source ?
(b) What is the effect of increase in the source frequency on the impedence of a C-R circuit.
53. A coil of area A and number of turns N rotates with a constant angular velocity ω, with its axis
perpendicular to a magnetic field B Write an expression for the instantaneous e.m.f induced in
the coil
54 What is the working principle of an a.c generator ?
55 What is the source of electrical energy obtained from a dynamo ?
56 For which position of a coil, rotating in a uniform magnetic field, is the induced e.m.f
maxi-mum ?
57 What is the use of a starter with a motor ?
58 What is the efficiency of a d.c motor ?
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59. What is the principle of induction coil ?
60 An ordinary moving coil ammeter used for d.c can not be used to measure a.c even if its
61. What is the condition of resonance in an LCR series circuit ? What is the impedance of the
circuit at resonance ?
62. What is a choke ?
63. The domestic electric supply is 220 V-50 Hz What are the rms and the peak values of the voltage ?
64. Name the effect on which a.c meters are based ?
65. The instantaneous value of alternating current in a circuit is I = 14.14 sin 2πt What is the
rms value of the current ?
66. A pure inductance is connected to a 220 V, 50 Hz source What is the phase difference between the current and the e.m.f in the circuit ?
67. What is the minimum value of power factor ? When does it occur ?
68. An ideal inductor, when connected to an a.c source, does not produce any heating effect Yet it reduces the current in the circuit Explain
69. A capacitor is used in the primary circuit of an induction coil Why ?
70. In an inductor, does the current rise to a steady value at a constant rate
71. When the current through an inductor is switched off, will the induced current be in the direc-tion of the main current or opposite to it ?
72. Can we use a capacitor of suitable capacitance instead of a choke coil in an a c circuit ?
73. What is the effect on the current if the frequency of the a.c source is increased in the following circuits ?
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(a) An a.c circuit containing a pure inductor
(b) An a.c circuit containing a pure capacitor.
74 Why is a choke preferred to a rheostat in controlling the current in an a.c circuit ?
75. How will the inductive reactance change on doubling the frequency of a.c ?
76. Give the phase difference between the applied a.c voltage and the current in an LCR circuit at
resonance
77. The impedence of a coil in an a.c circuit is 141.4 Ω and its resistance is 100 Ω What is its
78. What is the flux linked with a coil rotating in a magnetic field when the current induced in the coil is maximum
79. What energy transformation occurs in a d.c motor?
80. What is the function of a starter in a motor ?
81. (a) Show that the average value of a.c over one complete cycle is zero.
(b) Find the average value of a.c for the positive half cycle.
82. Show that for an alternating current rms 0 .
2
I
83 An a.c voltage E = E0 sin ωt is applied across an inductance L Obtain an expression for the
84 An a.c voltage E = E0 sin ωt is applied across a capacitor of capacitance C Obtain an
expres-sion for the current
85. Derive an expression for the impedance of an a.c circuit with a capacitor and a resistor in
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86. An inductor L and a resistor R are connected in series in an a.c circuit Derive an expression for the impedance Z of circuit Also derive an expression for the phase angle.
87 In an a.c circuit, an inductance L, a capacitance C and a resistance R are connected in series.
Derive an expression for the impedance and the phase angle
88. Show that average power transferred to an a.c circuit is in general given by
P = Erms Irms cos φ where the symbols have their usual meanings
89 What is a choke ? Explain its action in a.c circuits (AISSCE 1992 C)
90 What do you understand by wattless current ? Show that the current flowing in an ideal choke
coil is wattless ?
ANSWERS
46 V = V0 sin ωt
I = I0 sin (ωt + φ)
V
and
I
V
I
t
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47.
L
Z= R +X
For R = 3 Ω and X L = 4 Ω,
3 4
48. We know that
X L= ωL = 2πν L
⇒ X L∝ν [Fig 4 B.5]
2 1 [Fig 4 B.6]
C
C
X
X
ν
49. The impedance of the circuit is
2
2 1 )
C
(ω
With the increase in the frequency of the source, Z will decrease, which will result in the increase
of current So the bulb will glow more brightly.
50 When an iron core is inserted in the choke coil, its inductance L increases, so the current in the
circuit decreases Therefore the brightness of the bulb decreases
51. No change Choke coil offers reactance to a.c only
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52. (a) Impedance of L-R circuit is given by
2 2 2
Z = R + ω L
With increase in frequency, ω increases and hence the impedance will increase.
(b) Impedance of C-R circuit is 2 ( )2 2
1/
With increase in frequency of a.c impedance will decrease.
53. Instantaneous induced e.m.f is given by E = E0 sin ωt,
where E0 = NBA ω.
54 Working principle of a.c generator
When a coil is rotated in a magnetic field about an axis perpendicular to the field, the flux linked with the coil changes Due to this change in the magnetic flux, an e.m.f is induced and current flows in the coil
55. The mechanical energy expanded in rotating the coil is obtained in the form of electrical energy
56. The induced e.m.f in a rotating a coil is maximum when the plane of the coil is parallel to the magnetic field
57. When a motor is switched on, the back e.m.f is negligible initially and so a large current would flow This would burn the windings To control this current, the starter, which is a variable resistance, is put in series with the motor
58. Efficiency of d.c motor, Back e.m.f.
Applied e.m.f.
η =
59 Induction coil is based on the principle of mutual induction.
60. Due to inertia the needle will not follow the variation of alternating current even if the frequency
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is low
61 Resonance in LCR circuit takes place when the inductive reactance becomes equal to the
capaci-tive reactance, i.e.,
X L = X C
or ωL = 1/ωC
At resonance, the impedance is purely resistive, i.e., Z = R.
62 A choke is an inductor coil having large value of self inductance.
63. RMS value of voltage = 220 V; Peak value of voltage =200 2 V.
64. Heating effect of current is used in a.c meters because it does not depend on the direction of current
65. I = 14.14 sin 2π t
Peak value of current, I0 = 14.14 A
RMS value of current, rms 0 14.14
14.14 2
I
66 In an a.c circuit containing a pure inductor, the current lags behind the voltage in phase by π/2
67 The minimum value of power factor is zero It occurs in a pure inductive or pure capacitive
circuit
68 Heat is produced due to resistance An inductor offers nonresistive opposition to the flow of
current, called reactance Therefore no heat is produced
69 The high induced voltage when the circuit is broken charges the capacitor This avoids sparking
in the circuit
70 No, in an inductor the current rises as I = (E/R) (1– e –Rt/L)
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71. The induced current will flow in the direction of the main current
72 Yes, like a choke coil, a pure capacitor consumes no power Therefore, it can be used to control
the current without any appreciable power loss
73. (a) The current will decrease on increasing the frequency of a.c.
(b) The current will increase on increasing the frequency of a.c.
74 A choke is preferred over a rheostat to control the current in an a.c circuit because it reduces the
current without consuming any power
75 Inductive reactance X L = ωL = 2πν L, i.e., X L∝ν
Thus, on doubling the frequency, the reactance will be doubled
76 The phase difference is zero.
77. Impedence 2 2
L
Z= R +X
Z = 141.4 Ω, R = 100 Ω
∴ X L2 = Z L2 – R2 = (141.4)2 – 1002
2 2
( 2 100) 100
= 1002
or X L = 100 Ω
78 The flux is zero.
79. In a d.c motor, electrical energy is converted into mechanical energy
80. When the motor is switched on, the starter, which is a veriable resistance, does not allow the current to increase beyond safe limit Thus it prevents the burning of the armature winding of a motor
81. (a) Average value of a.c for one complete cycle : The instantaneous value of a.c is given by
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I = I0 sin ωt
0
0 0
0 0
0
1 Average value
1 sin
cos
2
1 cos
[1 1]
Zero
T
T
T
T
T
T
I
T
I T
= ∫
ω
π
ω
ω
=
Thus, the average value of a.c over one complete cycle is zero.
(b) Average value of a.c over half cycle :
/ 2 0 0
/ 2
0 0
1
sin ( / 2)
2
T
T
T
T
I
−
= π
∫
82. Irms = 2
I
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0
2 2
0 0 2 0 0 2 0
2 0 0
0
0
1 Now
1 sin
2
cos 2 2
sin 2 ( )
2
0.707 2
T
T
T
T T
T T
T
T
dt T
I
T
t T
T T I
= ∫
ω
2ω
83. E = E0 sin ωt
If I is the current in the circuit at some instant then the induced e.m.f is – L dI.
dt
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0
0
0
0
cos
dI
dt
E
L E
t L E
t L
∫
or I = I0 sin (ωt – π / 2),
where I0 = E0 / ωL
This is the required expression
84. E = E0 sin ωt
Let I be the current in the circuit and q be the charge on the capacitor at some instant At that instant
the potential difference across the capacitor is .
q
C Thus,
q E
C
=
or q = CE = CE0 sin ωt
Differentiating both sides w.r.t time
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0
( sin )
= CE0ω cos ωt
or I = CE0ω sin (ωt + π / 2)
or I = I0 sin (ωt + π / 2)
where I0 = ωC E0
85. The fig 4 B.7 shows a source of alternating e.m.f
connected to a series combination of a capacitor C
and a resistor R.
Let Irms be root mean square value of current in the
circuit, V R be the rms voltage across the resistance
which is in phase with the current, and V C be the rms
value of voltage across the capacitor which lags
be-hind the current in phase by π/2.
In the given phase diagram (Fig 4 B.8), OA and
OB represent V R and V C respectively, and OC gives
the resultant of V R and V C , i.e., Vrms Thus,
rms
rms rms
1
C
V
ω
V
A
O
V
rms
φ
V C
V R
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Vrms / Irms is the impedance Z of the circuit Thus 2 2
1
C
2
ω
86. Fig 4B.9 shows a source of alternating e.m.f
connected to a series combination of an
in-ductor L and a resistor R Let Irms be the rms
value of the current in the circuit, V R and V L
be the rms values of voltages across R and L
respectively In the phase diagram (Fig 4
B.10) V R = Irms R is the voltage across the
resistor It is in phase with the current and is
represented by OA
V L = Irms X L is the voltage across the inductor It leads the
current in phase by
2
π
and hence it is represents by OB
The resultant of OA and OB is represented by OC Thus
OC represents Vrms So,
L
V = V + V = I R + I X
where X L = ωL Thus,
R L
Fig 4 B.9
V R
V L V rm s
φ
Fig 4 B.10
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Vrms / Irms is the impedance of circuit Z.So,
Z= R2+ ω2L2
Also, from the phase diagram, tan L
R
V AC
OA V
R
ω
87. Fig 4 B.11 shows a source of alternating e.m.f
connected to a series combination of an
induc-tor L, a capaciinduc-tor C and a resisinduc-tor R Let Irms be
the root mean square value of the current in the
circuit and V L , V C and V R be the rms values of
voltages across L, C and R, respectively.
V R and Irms are in same phase, so V R is
repre-sented by OA
V L leads the current by π /2, so it is represented
by OB
Fig 4 B.11
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V C lags behind the current by π /2, so it is
represented by OC
Let V L > V C Then OD represents (V L – V C)
OF is the resultant of OD and OA From
the figure we have
where V R = Irms R,
V L = Irms X L, V C = Irms X C,
X L = ωL and X C = 1
C
ω
rms
rms rms
So, V /I = R + ω − ω( L 1 / C)
Vrms / Irms is called the impedance of the circuit, Z Thus,
φ
D
O
F
C
A
V R
V C
V L (V – V ) L C
Fig 4 B.12
B
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Z= R + ω − ωL C
If φ is the phase angle, then
tan
1/
or tan
R
AF
R
−
φ = =
ω − ω
φ =
88. The instantaneous e.m.f in an a.c circuit is given by E = E0 sin ωt Instantaneous current I = I0 sin (ωt – φ)
Instantaneous Power P = EI
Average power
0
1 T
T
= ∫
0 0 0
1
T
T
E I T
T
∫
0 0
T
E I
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0
2
0
sin ( 2 sin
(1 cos 2 )
sin 2 2
sin
2 sin cos
2
cos 2
T T
T T
T
t t
t
d
t dt
ω ω
=
φ
φ
φ
∫
0
0 0
0 0
rms rms
cos ( )
Eq (1) becomes
cos 2
since
T t
P
E I
T T
=
=
φ
=
89 A coil having a high value of inductance and low resistance is called a choke It is made of a large
number of turns of thick insultated coper wire
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It is used in a.c circuits to reduce the value of current because it consumes very little power We know that the power consumed by a pure inductor is zero
If instead of a choke coil we use a resistor to reduce the current, then there will be substantial power loss due to joule heating
90 Wattless current - See Q 41.
Average power consumed in an ideal choke coil
Instantaneous Power P = VI
In a pure inductor the current lage behind the voltage by π/2 Therefore,
P = V0 I0 sin ωt sin (ωt – π/2)
= – V0 I0 sin ωt cos ωt
= – 0 0 sin 2
2
V I
t
ω
Average power
0
0 0
1
1 2
T
T
tdt
T
V I T
∫
This shows that the current in an ideal choke is wattless.