38 Concepts of Physics. A series AC circuit contains an inductor ( mh), a capacitor ( (JF), a resistor ( ft) and an AC source of V, Hz. Find the energy dissipated in the circuit in s. Solution : The time period of the source is T - /v ms. The given time s is much larger than the time period. Hence we can write the average power dissipated as Pav V rms i rm. C S C P where coscp /Z is the power factor. Thus, V V > V' r rms it rms z z" z v ( ft) ( V) 7 ft-v' The capacitive reactance The inductive reactance co f>c N X x x " ft. N x x x The net reactance is X - w mc ft - K ft - - ft. Thus, Z ( ft) + (- ft) 3 ft :.. (i) ft ft k ft. 7 ft-v ' From (i), average power Pal, -8 W. 3 ft The energy dissipated in s P * s - 3 x 3 J. 7. An inductor of inductance mh is connected in series with a resistance, a variable capacitance and an AC source of frequency ' khz. What should be the value of the capacitance so that maximum current may be drawn into the circuit? Solution : This is an C series circuit. The current will be maximum when the net reactance is zero. For this, or, ~ co co C C CO 4n X (* X ;> s "') (O'l H) 3 nf. 8. An inductor coil joined to a V battery draws a steady current of A. This coil is connected to a capacitor and an AC source of rms voltage V in series. If the current in the circuit is in phase with the emf, find the rms current. Solution : The resistance of the coil is V A ' ft. In the AC circuit, the current is in phase with the emf. This means that the net reactance of the circuit is zero. The impedance is equal to the resistance, i.e., Z - ft. rms voltage _ V The rms current A. ' ft QUSTIONS FO SHOT A N S W. What is the reactance of a capacitor connected to a constant DC source?. The voltage and current in a series AC circuit are given by V V cos cot and i i sin cot. What is the power dissipated in the circuit? 3. Two alternating currents are given by, i sin u)t and i., - i sirj co + j. I i Will the rms values of the currents be equal or different? 4. Can the peak voltage across the inductor be greater than the peak voltage of the source in an C circuit?. In a circuit containing a capacitor and an AC source, the current is zero at the instant the source voltage is maximum. Is it consistent with Ohm's law?. An AC source is connected to a capacitor. Will the rms current increase, decrease or remain constant if a dielectric slab is inserted into the capacitor? 7. When the frequency of the AC source in an C circuit equals the resonant frequency, the reactance of the circuit is zero. Does it mean that there is no current through the inductor or the capacitor? 8. When an AC source is connected to a capacitor there is a steady-state current in the circuit. Does it mean that
39 the charges jump from one plate to the other to complete the circuit? 9. A current i, ia sin cot passes through a resistor of resistance. How much thermal energy is produced in one time period? A current ^ i sin cot passes through the resistor. How much thermal energy is produced in one time period? If t, and i both pass through the resistor simultaneously, how much thermal energy is produced? Is the principle of superposition obeyed in this case?. Is energy produced when a transformer steps up the voltage?. A transformer is designed to convert an AC voltage of V to an AC voltage of V. If the input terminals are connected to a DC voltage of V, the transformer usually burns. xplain.. Can you have an AC series circuit in which there is a phase difference of 8 between the emf and the current?? 3. A resistance is connected to an AC source. If a capacitor is included in the series circuit, will the average power absorbed by the resistance increase or decrease? If an inductor of small inductance is also included in the series circuit, will the average power absorbed increase or decrease further? 4. Can a hot-wire ammeter be used to measure a direct current having a constant value? Do we have to change the graduations? OBJCTIV I. A capacitor acts as an infinite resistance for (a) DC (b) AC (c) DC as well as AC (d) neither AC nor DC.. An AC source producing emf [cos( n s ' l )t + cos( n s " ')t] is connected in series with a capacitor and a resistor. The steady-state current in the circuit is found to be i - i, cos[( 7t s "')«+ cp,] + i, cos[( jt s " ')t + 4>] (a) i, > i (b) i, - i (c) i, < i (d) the information is insufficient to find the relation between i, and i. 3. The peak voltage in a V AC source is (a) V (b) about V (c) about 3 V (d) 44 V. 4. An AC source is rated V, Hz. The average voltage is calculated in a time interval of ' s. It (a) must be zero (b) may be zero - c) is never zero (d) is (//)V.. The magnetic field energy in an inductor changes from /'maximum value to minimum value in ' ms when connected to an AC source. The frequency of the source is (a) Hz (b) Hz (c) Hz (d) Hz.. Which of the following plots may represent the reactance of a series C combination? Figure 39-Q frequency 7. A series AC circuit has a resistance of 4 Q and a reactance of 3.C. The impedance of the circuit is (a) a (b) 7 Q (c) / Q (d) 7/ ft. 8. Transformers are used (a) in DC circuits only (b) in AC circuits only (c) in both DC and AC circuits (d) neither in DC nor in AC circuits. 9. An alternating current is given by i i, cos coi + i sin cot. The rms current is given by i, + i. i, + i, (a) (b) (c) (d)v^ +, / ^ / /. An alternating current having peak value 4 A is used to heat a metal wire. To produce the same heating effect, a constant current i can be used where i is (a) 4 A (b) about A (c) 7 A (d) about A.. A constant current of '8 A exists in a resistor. The rms current is (a) '8 A (b) about A (c) '4 A (d) undefined for a direct current. OBJCTIV II ". An inductor, a resistor and a capacitor are joined in series with an AC source. As the frequency of the source is slightly increased from a very low value, the reactance (a) of the inductor increases (b) of the resistor increases (c) of the capacitor increases (d) of the circuit increases.. The reactance of a circuit is zero. It is possible that the circuit contains (a) an inductor and a capacitor (b) an inductor but no capacitor
33 Concepts of Physics (c) a capacitor but no inductor (d) neither an inductor nor a capacitor, f 3^ In an AC series circuit, the instantaneous current is zero when the instantaneous voltage is maximum. Connected to the source may be a (a) pure inductor (b) pure capacitor (c) pure resistor (d) combination of an inductor and a capacitor. An inductor-coil having some resistance is connected to an AC source. Which of the following quantities have zero average value over a cycle? (a) current (b) induced emf in the inductor (c) Joule heat (d) magnetic energy stored in the inductcr.. The AC voltage across a resistance can be measured using (a) a potentiometer (b) a hot-wire voltmeter (c) a moving-coil galvanometer (d) a moving-magnet galvanometer.. To convert mechanical energy into electrical energy, one can use (a) DC dynamo (b) AC dynamo (c) motor (d) transformer. ^7?) An AC source rated V (rms) supplies a current of A (rms) to a circuit. The average power delivered by the source (a) must be W (b) may be W (c) may be greater than W (d) may be less than W. XCISS. Find the time required for a Hz alternating current to change its value from zero to the rms value.. The household supply of electricity is at V (rms value) and Hz. Find the peak voltage and the least possible time in which the voltage can change from the rms value to zero. 3. A bulb rated W at V is connected across a household supply of alternating voltage of V. Calculate the maximum instantaneous current through the filament. 4. An electric bulb is designed to operate at volts DC. If this bulb is connected to an AC source and gives normal brightness, what would be the peak voltage of the source?. The peak power consumed by a resistive coil when connected to an AC source is 8 W. Find the energy consumed by the coil in seconds which is many times larger than the time period of the source. Gy The dielectric strength of air is 3' x 8 V/m. A ^ parallel-plate air-capacitor has area cm and plate separation O'lO mm. Find the maximum rms voltage of an AC source which can be safely connected to this capacitor. 7. The current in a discharging circuit is given by i - ie" where T is the time constant of the circuit. Calculate the rms current for the period t to t - T. 8. A capacitor of capacitance jif is connected to an oscillator giving an output voltage ( V)sin cof. Find the peak currents in the circuit for co - s ", s ", s"', s"'. 9. A coil of inductance ' mh and negligible resistance is connected to the oscillator of the previous problem. Find the peak currents in the circuit for co-loos", ^ s", s". J A coil has a resistance of ft and an inductance of '4 3 henry. It is connected to an AC source of ' V, Hz. K Find the average power consumed in the circuit.. A resistor of resistance ft is connected to an AC source ( V) sin ( n s ~')i. Find the energy dissipated as heat during t - to t - * ms.. In a series C circuit with an AC source, - 3, C - JJF, V and v - /rc Hz. Find the peak current and the average power dissipated in the circuit 3. An electric bulb is designed to consume W when operated at volts. It is connected to a V, Hz line through a choke coil in series. What should be the inductance of the coil for which the bulb gets correct voltage? 4. In a series C circuit with an AC source, 3 Q, C pf, l'o henry, V and v /n Hz. Find, (a) the rms current in the circuit and (b) the rms potential differences across the capacitor, the resistor and the inductor. Note that the sum of the rms potential differences across the three elements is greater than the rms voltage of the source.. Consider the situation of the previous problem. Find the average electric field energy stored in the capacitor and the average magnetic field energy stored in the coil.. An inductance of ' H, a capacitance of 8 fjf and a resistance of kft are connected to an AC source of V with adjustable frequency, (a) What frequency should be chosen to maximise the current in the circuit? (b) What is the value of this maximum current? 7. An inductor-coil, a capacitor and an AC source of rms voltage 4 V are connected in series. When the frequency of the source is varied, a maximum rms current of A is observed. If this inductor coil is connected to a battery of emf V and internal resistance 4' ft, what will be the current? 8. Figure (39-) shows a typical circuit for low-pass filter. An AC input Vi mv is applied at the left end and T V;. kq -AAV nf T T l T I i Figure 39-
the output V is received at the right end. Find the output voltages for v khz, khz, MHz and lo'o MHz. Note that as the frequency is increased the output decreases and hence the name low-pass filter. A transformer has turns in the primary and in the secondary. If the primary is connected to a V DC supply, what will be the voltage across the secondary? ANSWS OBJCTIV I. V. (a). (c) 3. (c) 4. (b). (b). (d) 7. {e* - D/ 7. (a) 8. (b) 9. (c). (d). (a) e 8. - x A, A, - A, A OBJCTIV II 9 - A - 4- A ' - A. /8 W Ma). (a), (d) 3. (a), (b), (d) 4. (a), (b). (b). (a), (b). x J 7. (b), (d). - A, - W 3. - H XCISS 4 ' (a) ' A (b) V ' 3 V ' V. mj, mj - - ms. (a) 7 Hz (b) ma. 3 V, ' ms 7. - a 3- -39 A 8. 8- mv, mv, mv, ^V 4. 7 volts 9. zer.. 4- kj
CHAPT 39 ATNATING CUNT. Hz Sin Wt Peak value Sin Wt Sin Wt Sin 4 Wt. or, t 4 4. rms V Frequency Hz 4 8 8. s. ms (a) rms rms.44 3.8 V 3 V (b) Time taken for the current to reach the peak value Time taken to reach the value from r.m.s t 4 Sin t t 4 4 f 8 4 3. P W V V P v 8.7.44 3.8 4. volts i t i rmst 8.7 3.8.38.39 A rms 44 44.97 7 V. P 8 W (given). ms P rms P 4 W nergy consumed P t 4 4 J 4. KJ. 3 V/m, A cm, d. mm Potential diff. across the capacitor d 3. 3 3 V Max. rms Voltage V 3 V 39.
7. i i e ur i t / i e dt i t e dt / t / e i i e i i e i e e 8. C F F F ( V) Sin t a) 3 A Xc C b) s A. A C c) s A. A C d) s C 9. Inductance. mh. H a) s X. Ω A. A i X b) s A. X i X c) s 4 A. X. Ω Ω i X A. Ω,.4 Henry. V, 3 Hz Z X () Power V rms rms cos.... ().. 3.4... 7 8 39.
V. H T, V,, Ω H H dh Sin t dt 44 sin t dt cos t.44 dt 3 3.44 3 3 Sin t dt Cost dt.7 ( ).7. 7.4. 4 J. 3Ω, C F F, V, Hz X c Z c c X (3) 4 4 ( 3) (4) (a) Peak current. A Z (b) Average Power dissipitated, rms rms Cos Z 3 3.. 3. Power W, Voltage V, esistance frequency () Hz, V V Current in the circuit Z ( ) Voltage drop across the resistor ir () () V ( ) V P ( ) () () + () (44) 484 + 4 93 4 93 484 4 4.47.3. Hz 4. 3 Ω, C F F Henry, V V Hz (a), Z Z (3) c (X X ) ( 3) C (3) Ω 4 V V. A Z 39.3
(b) Potential across the capacitor i X c. V Potential difference across the resistor i. 3 3 V Potential difference across the inductor i X. V ms. potential V Net sum of all potential drops V + 3 V + V 9 V Sum or potential drops >.M.S potential applied.. 3 Ω C F F H, Z (from 4) V,. A Z lectric nergy stored in Capacitor (/) CV (/) 3 J mj Magnetic field energy stored in the coil (/) (/) (.) 3 J mj. (a)for current to be maximum in a circuit X l X c (esonant Condition) W WC W C 8 W 3 3 3.37 Hz 7 Hz (b) Maximum Current (in resonance and) 3 7. rms 4 V r 4 Ω, rms A 3 A ma 4 4 Ω Internal esistance 4 Ω Hence net resistance 4 + 4 8 Ω Current. A 8 8. V 3 V 3 Ω C 9 F V Ω nf V (a) X c WC Z C Xc 3 3 9 4 4 V 3 39.4
(b) X c WC Z C Xc V V X c 3 (c) MHz Hz X c WC Z C 3 3 Xc V V X c 9 3.4 V. mv 3 (d) MHz 7 Hz X c WC C Z 3 3 3 Xc V V X c 3 9. mv 7 3 V 9 3 9. Transformer works upon the principle of induction which is only possible in case of AC. Hence when DC is supplied to it, the primary coil blocks the Current supplied to it and hence induced current supplied to it and hence induced Current in the secondary coil is zero. P Sec 39.