Exercise
Question 1. A 100 Q resistor is connected to a 220 V, S0Hz ac supply.
(a) What is the rms value of current in the circuit?
(b) What is the net power consumed over a full cycle?
Question 2. (a) The peak voltage of an ac supply is 300 V. What is the rms voltage?
(b) The rms value of current in an ac circuit is 10 A. What is the peak current?
Question 3. A 44 mH inductor is connected to 220 V, 50 Hz ac supply. Determine the rms value of the current in the circuit.
Question 4. A 60µF capacitor is connected to a 110 V, 60 Hz ac supply. Determine the rms value of the current in the circuit.
Note: Refer Chapter at a Glance (3)
Question 5. In Qs. 7.3 and 7.4, what is the net power absorbed by each circuit over a complete cycle. Explain your answer.
Question 6. Obtain the resonant frequency cor of a series LCR circuit with L = 2.0H, C = 32 µF and R = 10 Q What is the Q-value of this circuit?
Question 7. A charged 30 µF capacitor is connected to a 27 mH inductor. What is the angular frequency of free oscillations of the circuit?
Question 8. Suppose the initial charge on the capacitor in q0• 7.7 is 6 mC. What is the total energy stored in the circuit initially? What is the total energy at later time?
Note: If the capacitor is connected to an inductor, discharging of the capacitor takes place in such a way that loss in electrical energy equals the gain in magnetic energy stored in the inductor.
Question 9. A series LCR circuit with R = 20 Q, L = 1.5 Hand C = 35 µFis connected to a variable-frequency of 200 V ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle?
Question 10. A radio can tune over the frequency range of a portion of MW broadcast band: (800 kHz to 1200 kHz). If its LC circuit has an effective inductance of 200 µH, what must be the range of its variable capacitor?
[Hint: For tuning, the natural frequency i.e., the frequency of free oscillations of the LC circuit should be equal to the frequency of the radiowave.]
Question 11. Figure shows a series LCR circuit connected to a variable frequency 230 V source. L = 5.0 H, C = 80µF, R = 40 Q.
(a) Determine the source frequency which drives the circuit in resonance.
(b) Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.
(c) Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the LC combination is zero at the resonating frequency.
ADDITIONAL NCERT EXERCISES
Question 12. An LC circuit contains a 20 mH inductor and a 50 µF capacitor with initial charge of lO mC. The resistance of the circuit is negligible. Let the instant the circuit is closed be t = 0.
(a) What is the total energy stored initially? Is it conserved during LC oscillations?
(b) What is the natural frequency of the circuit?
(c) At what time is the energy stored
(i) completely electrical (i.e. stored in the capacitor)?
(ii) completely magnetic (i.e. stored in the inductor)?
(d) At what times is the total energy shared equally between the inductor and capacitor?
(e) If a resistor is inserted in the circuit, how much energy is eventually dissipated as heat?
Note: If the resistance of circuit is negligible, there is no loss of energy into joules heat, So, total energy remains conserved during LC oscillation.
Question 13. A coil of inductance 0.50 Hand resistance lO0Q is connected to a 240 V, 50 Hz ac supply.
(a) What is the maximum current in the coil?
(b) What is the time lag between the voltage maximum and the current maximum?
Question 14. Obtain the answers to (a) and (b) in Q. 7.13, if the circuit is connected to a high frequency supply (240 V, 10 kHz). Hence, explain the statement that at very high frequency, an inductor in a circuit nearly amounts to an open circuit. How does an inductor behave in a d.c circuit after the steady state?
Question 15. A 100 µF capacitor in series with a 40 Q resistance is connected to a 110 V, 60 Hz supply.
(a) What is the maximum current in the circuit?
(b) What is the time lag between the current maximum and the voltage maximum?
Question 16. Obtain the answers to (a) and (b) in Q. 7.15 if the circuit is connected to a 110 V, 12 kHz supply? Hence, explain the statement that a capacitor is a conductor at very high frequencies. Compare this behaviour with that of a capacitor in a de circuit after the steady state.
Question 17. Keeping the source frequency equal to the resonating frequency of the series LCR circuit, if the three elements, L, C and R are arranged in parallel, show that the total current in the parallel LCR circuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for this frequency. Source has e.m.f. 230 V and L = 5.0 H, C = 80 µF,R=40Q.
Question 18. A circuit containing a 80 mH inductor and a 60 µF capacitor in series is connected to a 230 V, 50 Hz supply. The resistance of the circuit is negligible.
(a) Obtain the current amplitude and rms values.
(b) Obtain the rms values of potential drops across each element.
(c) What is the average power transferred to the inductor?
(d) What is the average power transferred to the capacitor.
(e) What is the total average power absorbed by the circuit? [‘Average’ implies ‘averaged over one cycle’.]
Question 19. Suppose the circuit in Q.7.18 has a resistance of is Q. Obtain the average power transferred to each element of the circuit and the total power absorbed.
Question 20. A series LCR circuit with L = 0.12 H, C= 480 µF, R = 23 Q is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum. Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?
(d) What is the Q-factor of the given circuit?
Question 21. Obtain the resonant frequency and Q-factor of a series LCR circuit with L=3.0 H, C= 27 µF, and R= 7.4Q. It is desired to improve the sharpness of the resonance of the circuit by reducing its full width at half maximum by a factor of 2. Suggest a suitable way.
Question 22. Answer the following questions.
- (a) In any ac circuit, is the applied instantaneous voltage equal to the algebraic sum of the instantaneous voltages across the series elements of the circuit? Is the same true for rms voltage?
- (b) A capacitor is used in the primary circuit of an induction coil.
- (c) An applied voltage signal consists ofa superposition ofa de voltage and an ac voltage of high frequency. The circuit consists of an inductor and a capacitor in series. Show that the de signal will appear across C and the ac signal across L.
- (d) A choke coil in series with a lamp is connected to a de line. The lamp is seen to shine brightly. Insertion of an iron core in the choke causes no change in the lamp’s brightness. Predict the corresponding observations if the connection is to an ac line.
- (e) Why is choke coil needed in the use of fluorescent tubes with ac mains? Why can we not use an ordinary resistor instead of the choke coil?
Question 23. A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns. What should be the number of turns in the secondary in order to get output power at 230V?
Question 24. At a hydroelectric power plant, the water pressure head is at a height of 300 m and the water flow available is 100 m3s-1. If the turbine generator efficiency is 60%, estimate the electric power available from the plant(g = 9.8 ms2).
Question 25. A small town with a demand of 800 kW of electric power at 220 Vis situated 15 km away from an electric plant generating power at 440V. The resistance of the two wire line carrying power is 0.5 Q per km. The town gets power from the line through a 4000-220 V step-down transformer at a sub station in the town.
(a) Estimate the line power loss in the form of heat.
(b) How much power must the plant supply, assuming there is negligible power loss due to leakage?
(c) Characterise the step up transformer at the plant.
Question 26. Do the same exercise as above with the replacement of the earlier transformer by a 40,000 – 200 V step down transformer. (Neglect, as before, leakage losses through this may not be a good assumption any longer because of the very high voltage transmission involved) Hence, explain why high voltage transmission is preferred?
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