CBSE Class 12 Physics VBQs Alternating Current

ALTERNATING CURRENT

2 Mark Questions:nn

Question. A 15 μF capacitor is connected to a 220 V, 50 Hz source. Find the capacitive reactance and the currents (rms and peak) in the circuit. If the frequency is doubled, what happens to the capacitive reactance and the current?
Answer. Vrms = 220 V , 𝝂 = 50 Hz, C= 15 μF
XC = 1/ωC = 1/2𝝅 𝝂 C
= 1/(2 × 𝝅 ×50 × 15 ×10-6)
= 212 ohm
Irms = V_rms/XC = 220/212= 1.04 A
Io= √2 I_rms = 1.414× 1.04 = 1.47 A

Question. Why is the use of a.c. voltage preferred over d.c. voltage? Give two reasons.
Answer. (i) A.C. can be transmitted to distant places without any significant line loss by using step up transformers.
(ii) A.C. can be reduced by using a choke coil without any significant loss of energy. As average power consumed by a choke coil in one cycle is zero.

Question. Calculate the current drawn by the primary coil of a transformer which steps down 200 V to 20 V to operate a device of resistance 20 Ω. Assume the efficiency of the transformer to be 80%.
Answer. E_P = 200 V, E_S= 20 V, R_L = 20 ohm
Efficiency= η= 80 %
I_S= E_S/ R_L= 20/20 = 1A
η = P_o/ P_i = E_S I_S/ E_P I_P
So I_P =E_S I_S / η I_P = (20×1×100) / (80 ×200) = 0.125 A

Question. An AC voltage V= Vm sin ωt is applied across:
(i) A series RC circuit in which capacitive impedance is ‘a’ time the resistance in the circuit.
(ii) A series RL circuit in which inductive impedance is ‘b’ times the resistance in the circuit. Find the value of power factor of the circuit in each case.
Answer. 
a) Power factor cos ϕ= R/Z where Z= [R² + X²]^1/2
(i) X= XC= aR
So, Z= [R² + (aR)²]^1/2
= R [a² +1]^1/2
So, cos ϕ= R/ R [a² +1]^1/2 = [a² +1]^-1/2
(ii) X= XL= bR
So, Z= [R² + (bR)²]^1/2
== R [b² +1]^1/2
So, cos ϕ= R/ R [b² +1]^1/2 = [b² +1]^-1/2

Question. A resistor ‘R’ and an element’ are connected in series to an ac source of voltage. The voltage is found to lead the current in phase by 𝝅/4. If ‘X’ is replaced by another element ‘Y’, the voltage lags behind the current by 𝝅/4.
(i) Identify elements ‘X’ and ‘Y’.
(ii) When both ‘X’ and ‘Y’ are connected in series with R to the same source, will the power dissipated in the circuit be maximum or minimum? Justify your answer.
Answer. (i) X is an inductor, because in series R-L circuit voltage leads current.
Y is a capacitor because in series R-C circuit voltage lags behind current.
(ii) In series LCR circuit Pav= Vrms Irms cos ϕ
If inductive reactance is not equal to capacitive reactance, so V and I are not in phase i.e. cos ϕ<1 so power dissipated is minimum.
If XC = XL then ϕ=0 then power dissipated will maximum.

Question. An ac voltage of 100 V, 50 Hz is connected across a 20-ohm resistor and a 2mH inductor in series. Calculate (i) impedance of the circuit (ii) wattless current of the given ac circuit.
Answer. Given V= 100 V, 𝝂 =50 Hz , R= 20 ohm, L= 2 mH
(i) Impedance of R-L circuits ,Z = [R² + XL 2]^1/2
= [R² + (ωL) ²]^1/2 = [R² + (2𝝅 𝝂L) ²]^1/2
=[20² + (2×3.14 ×50 ×2 ×10^-3) ²]^1/2
= [400 + (0.628) ²]^1/2 ≈ 20 ohm
(ii) rms current = I_rms = V/Z = 100/20 = 5A

Question. How is the large-scale transmissions of electric energy over long distances done with the use of transformers?
Answer. At the power station, the voltage of AC is stepped up by using step up transformer. This reduces the current that flows through the cable. As energy lost as heat is I2Rt so it reduces the resistive power loss. Then at the consumer end, the voltage is decreased using step down transformer.

3 Mark Questions:

Question. The primary coil of an ideal step-up transformer has 100 turns and transformation ratio is also 100. The input voltage and power are 220 V and 1100 W respectively. Calculate:(i) The current in the primary coil (ii) The voltage across the secondary coil (iii) The power in the secondary coil
Answer. N_p = 100, k=100 , E_P= 220 V, P_in = 1100 W
(i) P_in = E_P I_P or I_P = P_in / E_P = 1100/220 = 5A
(ii) k = N_S/ N_p= E_S/E_P or E_S= kE_P = 100 × 220 = 22000V
(iii) P_O = E_S I_S = 22000 × 5/100 = 1100 W

Question. (a) For given a.c. i=im sin ωt, show that the average power dissipated in a resistor R over a complete cycle is 1/2 im² R.
(b) A light bulb is rated at 100 W for a 220 V a.c. supply. Calculate the resistance of the bulb.
Answer. 
(i) When a.c. source is connected across a resistor, current and potential are in phase so
V=V_m sin ωt
Instantaneous power P= dW/dt
dW= Pdt = VI dt =Vm sin ωt Im sin ωt dt
W=∫ dW = ₀∫^T V_m I_m sin² ωt dt
= V_m  I_m 0∫T (1- cos 2 ωt)/2 dt
= V_m  I_m T/2
Pav= W/T= V_m  I_m/2 = I_m²R/2
(ii) P= 100 W, V_rms =220 V
Pav= V_rms²/ R or R=V_rms²/Pav= (220)²/100 = 484 ohm

MCQ

Question. The phase difference between instantaneous voltage and current of purely capacitive circuit is
a. 180
b. 90
c. 0
d. 360
Answer. B

Question. The current flowing in purely capacitive circuit connected with ac source
a. conventional current
b. electronic current
c. wattless current
d. none
Answer. C

Question. For unity power factor the circuit should be
a. purely capacitive
b. purely resistive
c. LCR series circuit
d. purely inductive
Answer. B

Question. An inductor of reactance 1 Ω and a resistor of 2 Ω are connected in series to the terminals of a 6 V (rms) AC source. The power dissipated in the circuit is
a. 8 W
b. 12 W
c. 14.4 W
d. 0 W
Answer. C

Question. The output of a step-down transformer is measured to be 24 V when connected to a 12 W light bulb. The value of the peak current is
a. 1/√2 A
b. 2 A
c. 2 √2 A
d. 18 A
Answer. A

Question. As frequency of an ac circuit increases, the current first increases and then decreases. What combination of circuit elements is most likely to comprise the circuit?
a. Inductor and capacitor
b. Resistor and inductor
c. Resistor and capacitor
d. Resistor, inductor and capacitor
Answer. A

Question. Electrical energy is transmitted over large distances at high alternating voltages. Which of the following statements is (are) correct?
a. For a given power level, there is a lower current
b. Lower current implies less power loss
c. Transmission lines can be made thinner
d. It is easy to reduce the voltage at the receiving end using step-down transformers
Answer. B

Question. When an AC voltage of 220 V is applied to the capacitor C
a. the maximum voltage between plates is 220 V.
b. the current is in phase with the applied voltage.
c. the charge on the plates is in phase with the applied voltage.
d. power delivered to the capacitor is zero
Answer. D

Question. In an AC circuit V and I are given by V = 50 sin 50t Volt and I = 100 sin (50t + π/3) mA. The power dissipated in the circuit
a. 2.5 kW
b. 1.25 kW
c. 5.0 kW
d. 500 W
Answer. B

Question. A coil of resistance 2000Ω and self-inductance 1.0 Henry has been connected to an AC source of frequency 2000/2π Hz. The phase difference between voltage and current is
a. 30^o
b. 60^o
c. 75^o
d. 45^o
Answer. D

Assertions & Reasons Questions
a) Both A and R are true and R is the correct explanation of A.
b) Both A and R are true and R is not the correct explanation of A.
c)A is true but R is false. d.)A is false and R is also false

Question. Assertion(A): A capacitor blocks DC and offers an easy path to AC.
Reason (R): Capacitive reactance is inversely proportional to frequency.
Answer. A

Question. Assertion(A): It is advantageous to transmit electric power at high voltage.
Reason(R): High voltage implies high current.
Answer. C

Question. Assertion(A): In series LCR circuit,resonance occurs at one frequency only.
Reason(R): At resonance,the inductive reactance is equal to capacitive reactance.
Answer. A

Question. Assertion(A): When the capacitive reactance is less than the inductive reactance in a series LCR circuit, the emf leads the current.
Reason(R): The angle by which the alternating voltage leads the alternating current in a series LCR circuit is given by tanφ=(XL-XC)/R
Answer. A

Question. Assertion(A): A transformer doesn’t work on DC.
Reason(R): The magnitude and direction of DC keeps changing periodically.
Answer. C

Question. Assertion(A): Most of the electrical energy sold by power companies is transmitted and distributed in the form of alternating voltage.
Reasoning(R): AC voltage cannot be easily and efficiently converted from one value to the other by means of transformers.
Answer. C

Question.. Assertion(A): An AC voltage source of variable angular frequency ω and fixed amplitude V0 is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When ω is increased, the bulb glows brighter.
Reason(R): When frequency is increased the inductive reactance increases.
Answer. B

Question. Assertion(A) : At resonant frequency the current amplitude in series LCR circuit is minimum.
Reasoning (R): The impedance of the circuit is maximum at resonant frequency.
Answer. D

Question. Assertion(A) : An alternating current doesn’t show any magnetic effect.
Reason (R): Alternating current doesn’t vary with time.
Answer. D

Question. Assertion (A): A bulb connected in series with a solenoid is connected to an ac source. If a soft iron core is introduced in the solenoid, the bulb will glow dimmer.
Reason (R) : On introducing soft iron core in the solenoid, the inductance decreases.
Answer. C
 

Case Study

1. When a resistor of resistance R, inductor of inductance L, and a capacitor of capacitance C are connected in series to an AC source. The current at any instant through the three elements has the same amplitude and is represented as I = Io sin ωt. However, voltage across each element has a different phase relationship with the current as shown in the phasor diagram.
The effective resistance of LCR circuit is called impedance (Z) of the circuit and the voltage leads the current by a phase angle 𝜑.
A resistor of 12Ω, a capacitor of reactance 14 Ω and an inductor of inductance 0.1 H are connected in series with a source 200V, 50 Hz ac supply.

Question. The value of inductive reactance is
(a) 0.1 Ω
(b) 62.8 Ω
(c) 14 Ω
(d) 31.4
Answer. D

Question. Impedance of the circuit is
(a) 21.13 Ω
(b) 42.26 Ω
(c) 15 Ω
(d) 30 Ω
Answer. A

Question. The value of current in the circuit is
(a) 14 A
(b) 5 A
(c) 9.46 A
(d) 6.28 A
Answer. C

Question. The phase angle between current and voltage is
(a) 50^o
(b) 53^o4’
(c) 55^o4’
(d) 63^o9’
Answer. C

Question. Find the correct relationship from the phasor diagram
(a) VL > VC
(b) VL < VC
(c) VL = VC
(d) VL ≥ VC
Answer. A

2. In an LCR circuit for a particular value of frequency of ac supply, the inductive reactance and capacitive reactance become equal, the impedance of circuit is equal to the ohmic resistance in the circuit only. This condition is called resonance and the frequency of the ac supply is known as resonant frequency. Unlike the LCR circuit, resonance cannot be observed in an RL or RC circuit.
A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V as supply.

Question. The value of source frequency for which current amplitude is maximum.
(a) 222.32 Hz
(b) 550.52 Hz
(c) 770Hz
(d) 663.48Hz
Answer. D

Question. At resonance, find the value of current
(a) 22.52 A
(b) 14.14 A
(c) 47.41 A
(d) 50.25 A
Answer. B

Question. Find the Q factor of this circuit
(a) 42.21
(b) 35.42
(c) 21.74
(d) 25
Answer. C

Question. Find the maximum power dissipated in the circuit
(a) 2299.3 W
(b) 5500 W
(c) 2200 W
(d) 4700 W
Answer. A

Question. At resonance, which of the physical quantity(s) is/are maximum?
(a) Current
(b) Impedance
(c) both a and b
(d) None of these
Answer. A

3. Step-down transformers reduce the voltage level of an input ac. They are used to supply required voltage to domestic use as well as factories. A small town has a demand of 800 kW of electric power at 220 V is situated 15 km away from an electric plant generating power at 440 V. The resistance of the two wires line carrying power is 0.5Ω per km. The town gets power from the line through a 4000 – 220 V step-down transformer at a sub-station in the town.

Question. The total resistance of the wire is
(a) 30 Ω
(b) 35 Ω
(c) 25 Ω
(d) 15 Ω
Answer. D

Question. Find the power loss in the line as heat
(a) 700 kW
(b) 600 kW
(c) 650 kW
(d) 550 kW
Answer. B

Question. Find the power suppled by the electric plant, if the power loss is negligible
(a) 1400 kW
(b) 500 kW
(c) 1600 kW
(d) 600 kW
Answer. A

Question. The total voltage transmitted from the plant is
(a) 7 kV
(b) 3 kV
(c) 4 kV
(d) 0.5 kV
Answer. A

Question. The voltage drop in the power line is
(a) 3 kV
(b) 2 kV
(c) 1.7 kV
(d) 2.8 kV
Answer. A