ALTERNATING CURRENT. Lesson-1. Alternating Current and Voltage

Transcription

1 esson- ATENATING UENT Alternating urrent and oltage An alternating current or voltage is that variation of current or voltage respectively whose magnitude and direction vary periodically and continuously with time. For current or voltage to be alternating (i) magnitude should be constant (ii) alternate half cycle is positive and half negative. The simplest type of alternating current I and voltage have a sinusoidal variation whose instantaneous value is given by I I sin (t + ) sin (t + ) where is initial phase graphs of some of the alternating current and voltage cases: Where I and denote the peak values of the current and e.m.f. respectively and is the angular frequency. The time period T of alternating current is given by T. The frequency f, of alternating current is given by f T. Average value of alternating current: The average value of current over time t is defined as I avg t t i dt dt where i is intantaneous value of the current.

2 For sinusoidal variation of current and voltages ase I: Average value over complete cycle Similarly avg ase II: Average value over half cycle T i sin( t ) dt T dt i avg T/ i sin( t ) dt T/ dt I Similarly avg Effective urrent and oltage : Since the average value over a time period of sinusoidal alternating current or voltage is zero, a D.. ammeter or voltmeter does not show any deflection for alternating current circuits. Therefore alternating current ammeter or voltmeters are used which measure the average or effective values of the current and voltage and are based on measuring the root mean square values of current and voltage. The root mean square value of an alternating current is that steady current which when passed through a resistor will produce the same amount of heat as the actual alternating current shall develop when passed for the same time. It is denoted by I rms The magnitude of I rms is given by I eff I rms. I T I dt I sin ( t) dt I I rms T T dt dt T.77 I Where I is the peak value of the current. Similarly eff or rms..77 E Power in Alternating urrent ircuit: Average power in alternating current circuit over time t is defined as P avg t i dt t dt

3 where and i are instantaneous values of voltage and current respectively et sin t i i sin (t ) Average power over a cycle P avg T i sin t sin( t ) dt T T dt i sin t cos sin t sin dt T i cos rms i rms cos The term cos is known as power factor. It is said to be leading if current leads voltage, lagging if current lags voltage. Thus, a power factor of.5 laging means current lags the voltage by 6 (as cos.5 6 ). The product of rms and i rms gives the apparent power. While the true power is obtained by multiplying the apparent power by the power factor cos. Thus, and Apparent power rms i rms True power apparent power power factor For, the current and voltage are in phase. The power is thus, maximum ( rms i rms ). For 9, the powr is zero. The current is then stated wattless. Such a case will arise when resistance in the circuit is zero. The circuit is purely inductive or capacitive. The case is similar to that of a frictionless pendulum, where the total work done by gravity upon the pendulum in a cycle is zero. Alternating current through different circuits () Purely esistive ircuit : If sin (t) i I sin( t) I sin( ) t Both voltage and curent are in same phase. Instantancous Power dissipated P I I sin (t) Average power dissipated over a cycle T I P P dt T P rms. I rms. I. I or O t

4 () Purely Inductive ircuit I t et sin t d I dt I I sin (t /) I ω rms ()I rms i 9 Therefore, in purely inductive circuit (i) the current lags behind the voltage in phase by /, and (ii) the quantity is a measure of the effective opposition offered to the flow of an alternating current by an inductor. It is denoted by X and is called inductive reactance : X X constant X constant or f Thus inductance offers larger opposition to a.c. of higher frequency than to a.c. of lower frequency. Also the average power consumed by an purely inductive circuit in a cycle is zero : P 9 Since the current flows without any power loss, it is called wattless current. During one quarter cycle, when the current increases, energy is stored in the inductance in the form of magnetic energy and during the next quarter, when the current decreases, this energy is transferred back to the source. An inductance coil with a high value of reactance and low resistance has got the property of opposing alternating current without any significant power loss. Such a coil is called a choke coil. (3) Purely apacitive ircuit i If sin t, Q sin (t) dq d I ( dt dt sin t) cos t

5 I I sin (t +/) ~ I t where I ( ) rms It shows that in a purely capacitor circuit (i) the current leads the voltage in phase by /, (ii) the quantity (/) is a measure of the effective opposition offered to the flow of an alternating current by capacitor. It is denoted by X and is called capacitive reactance. X / I rms constant constant X X or f As X /, a capacitor offers smaller opposition to a.c. of higher frequency than to a.c. of lower frequency. The average power is zero. During one quarter cycle, energy is stored in the capacitor in the form of electrostatic field, and this energy is delivered back to the source during the next quarter cycle. (4) The Series ircuit ~ S S I S If, and S are the r.m.s. voltages across, and the a.c. source respectively. Then S S X I where I S is r.m.s. value of source current. The total opposition to the current is called impedance and it is denoted by Z. Z I S S X ω

6 The phase angle by which the applied voltage leads the current is tan X tan ω (5) The Series ircuit S S IS If S, and are r.m.s. voltage across source, resistance and capacitor respectively S I S X S Impedence of circuit, Z X I ω S leads I S by S tan X tan The current leads the applied voltage by angle. (6) The Series ircuit For seeries circuit, ω S ~ S I S S ( ) Impedence of circuit, Z (X X ) ω ω S leads I S by Power in ircuit φ tan X X tan ω ω P rms I rms cos f I rms rms I Z rms Where cos is called the power factor of the circuit.

7 esonance in series circuit At a particular angular frequency of the source, when X X or, the impedance of ω circuit becomes minimum and equal to and, therefore, the current will be maximum. The circuit is then said to be in resonance. The resonant angular frequency and frequency v are given by, v π The variation of r.m.s. current with the frequency of the applied voltage is shown in the figure. If the applied voltage consists of a number of frequency components, the current will be large for the component having frequency v. ω The Q factor of an series circuit is given by Q A direct current flows uniformly throughout the cross-section of the conductor. An alternating current, on the other hand, flows mainly along the surface of the conductor. This effect is known as skin effect. The reason is that when a.c. flows through a conductor, the flux changes in the inner part of the conductor are higher. I rms v o v

8 SOED EXAMPES Ex. What is average and r.m.s. current over half cycle if instantaneous current is given by i 4 sin t + 3 cos t. Sol.: Given i 4 sin t + 3 cos t sin t cos t sin (t + ) where cos 4 5 omparing with i i sin(t + ) i 5 A 5 i rms A i avg A Ex. A. H inductor and a resistor are connected in series to a, 5 Hz ac source. alculate the current in the circuit and the phase angle between the current and the source voltage. Sol.: Impedance urrent I rms Z ω Z rms 3.8 A 67.H S, 5Hz S Phase angle tan ω I tan I I tan (5.495) Ex.3 A resistance, an inductance and capacitor are connected in series with an A supply where 6, inductive reactance X 4 and capacitive reactance X. If the current in the circuit is 5 ampere find (a) P.D. across, and (b) impedance of circuit, (c) voltage of A supply and (d) phase angle.

9 I Sol.: (a) 5 6 8, I 5 4, 5 6 ω ω (b) Z ω 6 4 (c) E IZ 5 S I (d) tan ω ω tan 4 6 tan (.75) 3646 ' Ex.4 A current of 4 A flows in a coil when connected to a dc source. If the same coil is connected to a, 5 rad/s ac source, a current of.4 A flows in the circuit. Determine the inductance of the coil. Also find the power developed in the circuit if a 5 F condenser is connected in series with the coil. Sol.: esistance of the coil, 3 4 Impedance of the coil, Z.4 5Ω ( esistance of inductor in de circuit is zero) Now Z² ² + ²² or Z ω 4.8H 5 eactance of the capacitor 8 ω 5 5 X c 6 When capacitor is connected in series, Z (X X Power factor, cos 3 Z 5 ) 3 (8 4) 5 Ω Power developed P I Z cos (.4)² W. rms Ex.5 A a.c source of frequency 5 Hz is connected to a series circuit with 8. mh,.5 µf and. Find the potential difference across the resistance. Sol.: Inductive reactance, X apacitive reactance, X 6 X X This is the condition of resonance. This means that total potential drop occurs across the resistance only. Now.

10 Ex.6 Two inductances of 5. H and. H are connceted in parallel circuit. Find the equivalent inductance and r.m.s. voltage in each inductor and in mains circuit when connected to a source of A. Sol.: et E E sin t I I sin(t ) E sin t (since current legs by ) where is equivalent inductance of circuit. I E π sinωt, I ω E ω π sinωt I I I 5. H. H I I + I E sin ωt ω π E ω sin ωt π E ω sin ωt π A i.e., 3 H I rms in, I rms in 5 I rms in circuit 5π π 3 π Ex.7 A 75 hertz, source is connected to a resistance of ohm, an inductance of.83 henry and a capacitance of microfarad all in series. alculate the time in which the resistance (thermal capacity joule / ) will get heated by? Sol.: Inductive reactance, X apacitive reactance, ω X 6.3Ω Z X X Power dissipated rms i rms cos rms Heat required by wire to get heated by H. J, H P t Z. 575 W. 834 t H 348 second. P.574

11 Ex.8 Sol.: A series circuit containing a resistance of has angular resonance frequency 4 5 rad s. At resonance, the voltage across resistance and inductance are 6 and 4 respectively. Find the values of and. At what frequency the current in the circuit lags the voltage by 45? For resistance; I rms or I rms 6.5 A For inductor I rms or or 4 H at resonance, X X or (/3)F ω (4 ) 5 4 When the current lags behind the voltage by 45, tan 45 (/ ) Source oltage ω ω ω 45 i ω ω ω ² 4 ² (4 5 ) 4 ² 6 5 (4 5 ) Taking roots, 8 5, 5 ejecting the negative root, 8 5 Hz Ex.9 A circuit has mh, 3 ohm and F connected in series to a source of 5 cos (t) volt. alculate the current amplitude and the average power dissipated per cycle at a frequency that is % lower than the resonant frequency. Sol.: esonant frequency /. Here, mh 3 H and F 6 F.

12 3 6 ( )( ) 4 /sec. Now, % less frequency will be / sec. At this frequency, X 9 3 ( 3 ) 9 ohm X ω X X ]. ohm Z [ [(3) (9. )].3 ohm urrent amplitude Average power, E Z I amp. 3 E I cos ; where cos. 4 Z.3 P P watt.

13 OBJETIE QUESTIONS. A F capacitor is connected across a, 5 Hz a.c. supply. The peak current through the circuit is (a).6 amp (b).6 amp (c) (.6 / ) amp (d) (.6 /) amp.. In a circuit the a.c. source has voltage and the potential difference across the inductance is 76 volts. The potential difference across the resistance will be (a) 44 (b) 396 (c) 3 (d) [( 576)] 3. An a.c. voltage source E sin t is connected across a circuit containing an a.c. ammeter and capacitor of capacity µf. the reading of ammeter is (a) ma (b) ma (c) 4 ma (d) 8 ma 4. A resistor and a. henry inductor are connected in series to an a.c. source operating at volt, 5 cycle. The phase angle between the current and source voltage is (a) tan 3 (b) tan (c) tan (5.5) 3 (d) tan () 5. In order to obtain a time constant of seconds in an.. circuit containing a resistance of 3, the capacity of the condenser should be (a) µf (b) µf (c) µf (d) µf 6. An A.. series circuit contains 4 of resistance and 3 of inductive reactance. Then the impedance of circuit is (a) 7 (b) (c) 5 (d) 7 7. A group of electric lamps having a total power rating of watt is supplied by an a.c. voltage E sin (3 t + 6 ). Then the r.m.s. value of the circuit current is (a) amp (b) amp (c) amp (d) amp. 8. In the alternating current circuit shown in following figure, the currents through the inductor and the capacitor are. amp and. amp. respectively. The current drawn from the generator is (a).4 amp. (b). amp. (c). amp. (d). amp. 9. An alternating voltage E sin t, where is in volt and t in seconds, is connected to a series combination of µf capacitor and a k resistor through an ac ammeter. The reading of the ammeter will be (a) ma (b) ma (c) ma (d) ma

14 . An inductor of henry is connected across a, 5 Hz supply. The peak value of the curent is approximately (a).5 A (b).7 A (c) A (d).4 A. An inductive coil has a resistance of. When an ac signal of frequency Hz is applied to the coil, the voltage leads the current by 45. The inductance of the coil is (a) (b). An series circuit consists of a resistance of, a capacitance of reactance 6 and an inductor coil. The circuit is found to resonate when put across a 3, Hz supply. The inductance of the coil is (take 3). (a). H (b). H (c). H (d). H 3. When dc is applied across a coil, a current of A flows through it. When ac of 5 Hz is applied across the same coil, only.5 A flows. the resistance and inductance of the coil are (take ² ). (a) 5,.3 H (b) 5,. 3 H (c),.3 H (d),. 3 H 4. In an ac circuit sin ( t) volt and I sin ( t + /3) ma. The power dissipated in the circuit is (a) 4 W (b) W (c).5 W (d) 5 W 5. An electric bulb which runs at 8 dc and consumes A current is connected across a, 5 Hz ac supply. The inductance of the choke required so that it consumes rated power, is (take 3). (a). H (b). H (c).4 H (d).8 H 6. The tuning circuit of a radio receiver has a resistance of 5, an inductor of mh and a variable capacitor. A MHz radio wave produces a potential difference of. m. The value of the capacitor to produce resonance is (take ²). (a).5 F (b) 5. F (c).5 F (d) 5. F 7. The impedance of a circuit consists of 3 resistance and 4 reactance. The power factor of the circuit is (a).4 (b).6 (c).8 (d). 8. Two coils A and B are connected in series across a 4, 5 Hz supply. The resistance of A is 5 and the inductance of B is. H. The power consumed is 3 kw and the power factor is.75. The impedance of the circuit is (a).44 (b).44 (c) 4.4 (d) A resistance, an inductance and a capacitance are connected in series across an ac source of angular frequency. If the resonant frequency is, then the current will lag behind the voltage if (a) < (b) > (c) (d) <. An series circuit containing a resistance of has angular resonance frequency 4 5 rad s. At resonance the voltage across resistance and inductance are 6 and 4 respectively. The values of and are (a). m H, /3 µf (b).4 m H, /6 µf (c). mh, /6 µf (d).4 mh, /3 µf (c) 4 (d) 6

15 . In an - circuit, the value of is (.4/) henry and the value of is 3 ohm. If in the circuit, an alternating emf of volt at 5 cycles per second is connected, the impedance of the circuit and current will be : (a).4 ohm, 7.5 ampere (c) 4.4 ohm, 5 ampere (b) 3.7 ohm, 6.5 ampere (d) 5 ohm, 4 ampere.. An inductor 3 Henry, a capacitor µf and a resistor 5 are connected in series across a source of EMF sin 34t. If resistance is removed from the circuit and the value of inductance is doubled, then the variation of current with time in the new circuit is (a).5 cos 34 t (b).5 sin 34 t (c).5 sin (34 t + /3) (d) None of these 3. A coil having an inductance of / henry is connected in series with a resistance of 3. If volt from a cycle source are impressed across the combination, the value of the phase angle between the voltage and the current is : (a) tan (5/4) (b) tan (4/5) (c) tan (3/4) (d) tan (4/3) 4. An A.. source is in series with and. If respective potential drops are and 5 and 5, what is the applied voltage? (a) 5 (b) 5 (c) 5 (d) 5. The value of current in two series circuits at resonance is same when connected across a sinusoidal voltage source. Then (a) both circuits must be having same value of capacitance and inductor (b) in both circuits ratio of and will be same (c) for both the circuits X /X must be same at that frequency (d) both circuits must have same impedance at all frequencies. 6. When D.. is applied across a solenoid a current of A flows in it. When A.. is applied across the same coil, the current drops to.5 A. If the frequency of the A.. source is 5 Hz the impedance and inductance of the solenoid are (a),.93 H (b),. H (c),.86 H (d),.55 H 7. If in a series -- circuit, the voltage across, and are, and respectively, then the voltage of applied A source must be : (a) + + (b) [ ] (c) + (d) / [( ) ] 8. The p.d. across an instrument in an a.c. circuit of frequency f is and the current flowing through it is I such that 5 cos ft(b) volt and I sin ( ft) amp. The power dissipate in the instrument is : (a) zero (b) watt (c) 5 watt (d).5 watt. 9. In an A.. circuit, maximum value of voltage is 43 volt. Its effective voltage is : (a) 33 (b) 34 (c) 4 (d) esonance frequency of a circuit is f. If the capacitance is made 4 times the initial value, then the resonance frequency will become : (a) f/ (b) f (c) f (d) f/4

16 MOE THAN ONE OET HOIE 3. The symbols,, represent inductance, capacitance and resistance respectively. Dimension of frequency are given by the combination : (a) / (b) / (c) / (d) / 3. In the circuit shown in the figure, if both the bulbs B and B are identical (a) their brightness will be the same (b) B will be brighter than B (c) as frequency of supply voltage is increased, brightness of B will increase and that of B will decrease (d) only B will glow because the capacitor has infinite impedance 33. In the circuit shown, resistance, inductance H and capacitance 8 F are connected in series with an ac source of volt and frequency f. If the readings of the hot wire voltmeters and are same then (a) f 5 Hz (c) current through is A ~ (b) f 5 Hz (d) volt 34. In the A circuit shown below, the supply voltage has a constant rms value but variable frequency f. At resonance, the circuit : (a) has a current i given by I / (b) has a resonance frequency 5 Hz (c) has a voltage across the capacitor which is 8 out of phase with that across the inductor, f s / H (d) has a current given by I 35. An A.. source producing sint + sint is connected in series with a box, containing either capacitor or inductor and resistance. The current found in the circuit is : i i sin(t + ) + i sin(t + ). Here and may be positive or negative. (a) if i > i, box has inductor and resistor (b) if i > i, box has capacitor and resistor (c) if i > i box has inductor and resistor (d) if i > i box has capacitor and resistor

17 36. Graph shows variation of source emf and current i in a series circuit, with time (a) To increase the rate at which energy is transferred to the resistive load, should be decreased /i (b) To increase the rate at which energy is transferred to the resistive load, should be decreased i t (c) The circuit is more inductive than capacitive (d) The current leads the emf in the circuit 37. A light bulb is rated at W for a supply. Then the resistance of the bulb and the peak voltage of the source are (a) 4 (b) 5 (c) (d) An circuit has mh, 3 and F connected in series to a source of 5 cos t. Then the current amplitude and the average power dissipated per cycle at a frequency that is % lower than the resonance frequency will be (a).74 A (b).4 A (c) J/cycle (d) 3. 4 J/cycle 39. A resistor of and a capacitor of 5. F are connected in series to a, 5 Hz ac source. Then the impedance and rms current will be (a).74 A (b).755 A (c) 9.5 (d) A pure inductor of 5. mh is connected to a source of. Find the inductive reactance and rms current in the circuit if the frequency of the source is 5 Hz. (a) 4. A (b).755 A (c) 5.7 (d) 7.5

18 MISEANEOUS ASSIGNMENT omprehension- The a.c. generator which is one of the most important applications of the phenomenon of electromagnetic induction converts mechanical energy into electrical energy. A rectangular coil consisting of a large number of turns of copper wire would over a soft iron core is rotated between the pole pieces of a permanent strong magnet. The magnetic flux through the coil changes continuously with time, thus producing induced emf given by E E sin t When a load resistor is connected across the terminals, a current I flows through the circuit. E E I sin t I sin t Where I E /. Such a current is called a.c. or alternating current.. In an a.c. generator, a coil of area A and having N turns rotates in a magnetic field B. The magnetic flux through the coil is (a) maximum equal to NAB when the plane of the coil is perpendicular to the magnetic field. (b) zero when the plane of the coil is parallel to the field. (c) / NAB when the plane of the coil makes an angle of 6 with the field. (d) /4 NAB when the plane of the coil makes an angle of 3 with the field.. In an a.c. generator, initially (i.e. at t ) the plane of the coil is normal to the magnetic field. Which graph shown in the figure represents the variation of induced emf E with time t? (a) (b) (c) (d) 3. In an a.c. generator, the peak value of the induced emf depends upon the (a) frequency of rotation of the coil (b) area of the coil (c) number of turns in the coil (d) strength of the magnetic field 4. In an a.c. generator (a) the coil is wound over a soft iron core in order to increase the flux. (b) an electromagnetic field with an alternating current in used. (c) the output is always taken across a load resistor (d) the mechanical energy of the rotating coil is converted into electrical energy

19 omprehension- In a series circuit with an ideal ac source of peak voltage E 5, frequency v 5 Hz and 3. The average electric field energy stored in the capacitor and average magnetic energy stored in the coil are 5 mj and 5 mj respectively. The value of MS current in the circuit is. A. Then find : 5. apacitance () of capacitor- (a) F (b) 5F (c) F (d) None of these 6. Inductance () of inductor- (a).5 Henry (b).5 Henry (c) Henry (d) Henry 7. The sum of rms potential difference across each of the three elements- (a) 5 volt (b) 5 volt (c) 5 volt (d) None of these 8. A circuit containing a. H inductor and a 5 µf capacitor in series is connected to a 3 volt, / Hz supply. The resistance of the circuit is negligible. olumn I olumn II A. urrent amplitude (p) 3 B. Average power transferred to inductor (q) zero. Average power transferred to capacitor (r) 3 D. rms value of current (s) For the circuit shown column II give data for quantities given in column I match them correctly. olumn I olumn II A. eactance of circuit (p) B. Impedance of circuit (q). urrent (r) 3 D. reading (s) 4 In an -- series circuit 5 W, X 9W and X 7W. If applied voltage in the circuit is 5 volt then find impedance of the circuit in ohm. 5. The resistance () in a series circuit is 4, while the reactance of the inductor () and the capacitor () are and 8 respectively at a certain frequency. The total impedance (Z) of the circuit, if the frequency is halved, is nearly n. Find the value of n.

20 . A choke coil is needed to operate an arc lamp at 5 (rms) and 5 Hz. The lamp has an effective resistance of 5 when running at A (rms). The inductance of the choke coil is (n/) H. Find the value of n. 3. The sinusoidal voltage wave changes from to maximum value of volt. The voltage when the phase angle is 3º is n volt. Find the value of n. 4. A circuit draws a power of 55 watt from a source of volt, 5 Hz. The power factor of the circuit is.8 and the current lags in phase behind the potential difference. To make the power factor of the circuit as., capacitance 5n µf will have to be connected with it. Find the value of n. 5. A series circuit containing a resistance of has angular resonance frequency 4 5 rad s. At resonance the voltages across resistance and inductance are 6 and 4 respectively. At n 5 rad/s frequency the current in the circuit lags the voltage by 45º. Find the value of n. 6. A series -- circuit is connected to an A source of and 5 Hz as shown in figure. If the readings of the three voltmeters, and 3 are 65, 45 and 4 respectively, the value of the capacitor is n µf. Find the value of n. 5 Hz 3 7. A resistance and inductance and a capacitor all are connected in series with an A supply. The resistance of is 6 ohm and for a given frequency, the inductive reactance of is 4 ohm and capacitive reactance of is ohm. If the current in the circuit is 5 amp. The impedance of the circuit is 5n ohm. Find the value of n. 8. A 75 Hz., source is connected to a resistance of ohm, an inductance of.83 henry and a capacitance of microfarad all in series. After.9 n minutes time the resistor (thermal capacity J/º) is heated by º. Find the value of n. 9. A sinusoidal voltage of peak value 83 and frequency 5 Hz is applied to a series circuit in which 3, 5.48 mh, and 796 µf. The impedance of the circuit is n. Find the value of n.

21 PEIOUS YEA QUESTIONS IIT-JEE/JEE-ADANE QUESTIONS. A varying voltage is applied to the terminals AB as shown in figure, such that the voltage across capacitor plates varies as shown in the figure below. The time dependence of voltage across the terminals and D is (a) (b) (c) (d). For the circuit shown in the figure the ammeter A reads.6 A, ammeter A 3 reads.4. Then (a) (b) (c) The ammeter A reads A The ammeter A reads. A f (d) 3 3. A charged capacitor discharges through a resistance with time constant. The two are now placed in series across an A source of angular frequency. (a) The impedance of the circuit will be (b) (c) (d) 4. An A.. supply of 5, 5 Hz is connected to an inductance, capacitance and resistance in series. and are A.. voltmeters connected across, and respectively. If their readings are 3 and 5 respectively, the reading of the voltmeter 3 across will be (a) (b) 5 (c) (d) 5

22 5. The mains electrical supply is at 5 Hz, the light bulb is rated at, W. What max is required if the rate of energy dissipation in the light bulb is to be reduced by a factor of 5 from its upper limit of W? (a).69 H (b).8 H (c).38 H (d).56 H 6. An electric lamp designed for operation on A is connected to a A supply, through a choke coil of inductance H, for proper operation. The angular frequency of the A is rad/s. If a capacitor is to be used in place of the choke coil, its capacitance must be (a) µf (b) µf (c) 5 µf (d) µf 7. A resistance draws P power when connected to an A source. If an inductance is now placed in series with the resistance, such that the impedance of the circuit becomes Z, the power drawn will be (a) P Z (b) P Z (c) P (d) P Z 8. If the instantaneous current in a circuit is given by i cos (t ) amperes, the r.m.s. value of the current is (a) ampere (b) ampere (c) ampere (d) zero ampere 9. The figure shows (in part) the variation with time of periodic current, the root mean square current is (a) A (b) (c) A (d) A 3 A 3. An inductor and a capacitor are joined in series to an A source. The frequency of A is gradually increased. The phase difference between the emf and the current is plotted against the angular frequency. Which of the following best represents the resulting curve? (a) (b) (c) (d). An A voltage source of variable angular frequency and fixed amplitude is connected in series with a capacitance and an electric bulb of resistance (inductance zero). When is increased (a) the bulb glows dimmer (b) the bulb glows brighter (c) total impedance of the circuit is unchanged (d) total impedance of the circuit increases

23 . A series - circuit is connected to A voltage source. onsider two cases; (A) when is without a dielectric medium and (B) when is filled with dielectric of constant 4. The current I through the resistor and voltage across the capacitor are compared in the two cases. Which of the following is/ are true? (A) A B A B A B I I (B) I I () (D) A 3. A series - combination is connected to an A voltage of angular frequency 5 radian/s. If the impedance of the - circuit is.5, the time constant (in millisecond) of the circuit is B 4. In the given circuit, the A source has rad/s. onsidering the inductor and capacitor to be ideal, the correct choice (s) is (are) µf (a) The current through the circuit I is.3 A (b) The current through the circuit, I is.3 A.5H (c) The voltage across resistor I (d) The voltage across 5 resistor Paragraph for Question 5 to 6 A thermal power plant produces electric power of 6 kw at 4, which is to be transported to a place km away from the power plant for consumers usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformation at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers end, a step down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values. 5. If the direct transmission method with a cable of resistance.4 km is used, the power dissipation (in %) during transmission is (a) (b) 3 (c) 4 (d) 5 6. In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is :. If the power to the consumers has to be supplied at, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is (a) : (b) 5 : (c) : (d) 5 :

24 7. At time t, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current I(t) I cos(t), with I A and 5 rad s starts flowing in it with the initial direction 7 B D shown in the figure. At t, the key is switched from 6 A B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. 5 If F, and the battery is ideal with emf of 5, identify the correct statement (s) 7 (a) Magnitude of the maximum charge on the capacitor before t is (b) The current in the left part of the circuit just before t is clockwise 6 (c) Immediately after A is connected to D, the current in is A (d) Q 3 DE QUESTIONS A. A ~ A 3 at resonance which ammeter reads. (a) A (b) A (c) A 3 (d) all three A, A & A 3. In an circuit, the graph between resonance frequency, w r and impedance z is (a) (b) (c) (d) 3. In a series circuit, resonant frequency depends on (a) (b) (c) (d) 4. In a circuit, H, E 5 volts,, what is the energy stored in inductor? (a) 5 J (b) 5 J (c) J (d) none of these 5. Average power generated in an inductor connected to an A source is (a) i (b) i (c) zero (d) none of these

25 6. For EM wave propagating along x axis has E max 3 /m. What is maximum value of magnetic field? (a) 7 tesla (b) 8 tesla (c) 9 tesla (d) 6 tesla 7. In circuit if resistance increases quality factor (a) increases finitely (b) decreases finitely (c) remains constant (d) none of these 8. Direction of EM wave is given by (a) E B (b) B E (c) E (d) B MAINS QUESTIONS. Alternating current can not be measured by D.. ammeter because (a) (b) Average value of current for complete cycle is zero A.. changes direction (c) A.. can not pass through D.. Ammeter (d) D.. Ammeter will get damaged. In an series a.c. circuit, the voltage across each of the components,, and is 5. The voltage across the combination will be (a) (b) 5 (c) 5 (d) 3. In a circuit capacitance is changed from to. For the resonant frequency to remain unchanged, the inductance should be changed from to (a) / (b) (c) 4 (d) /4 4. In a series resonant circuit, the voltage across is volts and k with μf. The resonant frequency is rad/s. At resonance the voltage across is (a).5 (b) 4 (c) 5 (d) In an A generator, a coil with N turns, all of the same area A and total resistance, rotates with frequency in a magnetic field B. The maximum value of emf generated in the coil is (a) N.A.B.. (b) N.A.B (c) N.A.B. (d) N.A.B. 6. In an a.c. circuit the voltage applied is E E sin t. The resulting current in the circuit is sin I I. t The power consumption in the circuit is given by (a) E I P (b) P E I (c) E I P (d) P zero 7. In a series circuit and the voltage and the frequency of the main supply is and 5 Hz respectively. On taking out the capacitance from the circuit the current lags behind the voltage by 3. On taking out the inductor from the circuit the current leads the voltage by 3. The power dissipated in the circuit is (a) zero W (b) 4 W (c) 35 W (d) W

26 8. A fully charged capacitor with initial charge q is connected to a coil of self inductane at t. The time at which the energy is stored equally between the electric and the magnetic field is (a) (b) (c) (d) 4 9. In an circuit as shown below both switches are open initially. Now switch S is closed, S kept open. (q is charge on the capacitor and is capacitive time constant). Which of the following statement is correct? (a) At t, q ( e ) S S (b) At t /, q ( e ) (c) Work done by the battery is half of the energy dissipated in the resistor (d) At t, q /. In the circuit shown here, the point is kept connected to point A till the current flowing through the circuit becomes constant. Afterward, suddenly, point is disconnected from point A and connected to point B at time t. atio of the voltage across resistance and the inductor at t / will be equal to (a) (b) e e (c) e e (d)

27 SUBJETIE POBEMS. Find the value of inductance which should be connected in series with a capacitance of.5 µf, a resistance of ohms and an A.. source of 5 caps so that the power factor of the circuit is unity.. A coil of resistance 6 ohms and inductance 3 henry is connected in series with a condenser of 4 µf and an a.c. supply of volts and 5 c/s. alculate (a) the impedance of the circuit (b) phase difference between current and voltage (c) p.d. across the inductance coil, and (d) p.d. across the capacitor 3. A km long telegraph wire has a capacity of.4 F/km. If it carries an alternating current of frequency 5 KHz, find the value of the inductance which should be connected in series so that the impedance is minimum. 4. A circuit draws a power of 55 watt from a source of olt, 5 Hertz. The power factor of the circuit is.8 and the current lags in phase behind the potential difference. If the power factor of the circuit is to be increased to., find the value of capacitance which should be connected. 5. Prove that in a series circuit, the frequencies f and f at which the current amplitude falls to / of the current at resonance are separated by an interval equal to (/) 6. A series circuit consists of a resistance, inductance and capacitance. The applied voltage and the current at any time t are respectively given by E4.4 cos (3t ) and I5 cos(3t 55). If the inductance is. henry, find the values of the resistance and capacitance. 7. A F capacitor in series with a 4 resistance is connected to a, 6 Hz supply. Find the maximum current in the circuit. Also find the time lag between the maximum current and maximum voltage. 8. A volts 5 watt lamp is used on a.c. mains of volts and 5 Hz. Find (a) the values of (i) capacitor and (ii) inductor to be put in series with the lamp to run it. (b) the value of pure resistance which should be connected in series with the lamp to run it. 9. A choke coil is required to operate an arc lamp at 6 r.m.s. and 5 Hz. The arc lamp has an effective resistance of 5 when running at A r.m.s. Find the inductance of the choke coil. If the same arc lamp is operated on 6 d.c., find the additional resistance required and power loss if d.c. is used.

28 . A series circuit with. H, 48 nf and 3 is connected to a 3 variable frequency supply. (a) (b) (c) (d) What is the source frequency for which current amplitude is maximum? Find this maximum value of current. What is the source frequency for which average power absorbed by the circuit is maximum? Obtain the value of maximum power. For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the Q factor of the circuit?. A box P and a coil Q are connected in series with an ac source of variable frequency. The e.m.f. of source is constant at. Box P contains a capacitance of µf in series with a resistance of 3. oil Q has a self inductance 4.9 mh and a resistance of 68. The frequency is adjusted so that the maximum current flows in P and Q. Find the impedance of P and Q at this frequency. Also find the voltage across P and Q respectively.. An inductor 3 henry, a capacitor F and a resistor 5 W are connected in series across a source of e.m.f. sin 34t. Find the energy dissipated in the circuit in minutes. If the resistance is removed from the circuit and the value of inductance is doubled then find the variation of current with time in the new circuit? 3. Two alternating-current generators in series are connected to a coil and a resistor as shown in the figure. The voltages across generators and are 6 and 8, respectively. If the voltage between points a and b is, and the current in the circuit is A, calculate the phase difference between the voltages supplied by the two generators. (a) What is the impedance of the circuit? (b) If the power supplied to the coil is W, calculate the resistance and the inductive reactance of the coil X. a b 4. A coil of radius cm has 49 turns and it lies in a vertical plane. The end of the coil are connected to an external resistor of 7. If the coil rotates about its vertical diameter at a speed of 36 rpm, calculate the average power delivered to the resistor. The horizontal component of the earth s magnetic field is. oersted. Neglect the resistance and the inductance of the coil. 5. An alternating emf of angular frequency 4 radius per second is applied to a coil of inductance mh and resistance 6. (a) What should be the capacitance of a condenser placed in the circuit so as to make the power factor unity? (b) alculate the change in current due to the insertion of this condenser.

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30 ANSWES Objectives Problems. (b). (c) 3. (b) 4. (c) 5. (c) 6. (c) 7. (b) 8. (b) 9. (b). (c). (b). (a) 3. (d) 4. (c) 5. (b) 6. (a) 7. (b) 8. (c) 9. (b). (a). (d). (a) 3. (d) 4. (a) 5. (c) 6. (d) 7. (b) 8. (a) 9. (d) 3. (a) 3. (a,b,c) 3. (b,c) 33. (b,c,d) 34. (a,b,c) 35. (a,d) 36. (a,b,c) 37. (a,d) 38. (a,c) 39. (b,c) 4. (a,c) Miscellaneous Assignment. (a,b,c). (c) 3. (a,b,c,d) 4. (a,c,d) 5. (c) 6. (c) 7. (d) 8. A-(p); B-(q); -(q); D-(r) 9. A-(q); B-(p); -(s); D-(p). (3). (6). () 3. (5) 4. (5) 5. (8) 6. (5) 7. (4) 8. () 9. (5 Previous Year Questions IIT-JEE/JEE-ADANE QUESTIONS. (b). (b) 3. (c) 4. (c) 5. (b) 6. (c) 7. (a) 8. (b) 9. (c). (a). (b). (b,c) 3. (4) 4. (a,c) 5. (b) 6. (a) 7. (c,d) DE QUESTIONS. (c). (c) 3. (c) 4. (d) 5. (c) 6. (d) 7. (b) 8. (a) MAINS QUESTIONS. (b). (d) 3. (a) 4. (c) 5. (d) 6. (d) 7. (b) 8. (b) 9. (a). (a) Subjective Problems.. H. (a) 57.8 (b) 67.7 (c) 93 (d) mh F. 6., 33.3 F amp,.55 3 second. 8. (a) (i) 4. F (ii).53 H (b) H,, W.. (a) 467 rad/sec., 4.4 amp (b) 467 rad/sec, 3 watt (c) 463 rad/sec, 47 rad/sec. (d).74. P 76, Q 98 P 7.6, Q 9.8