Alternating Current Class 12 Physics Notes

• Average value over a full cycle is zero for a pure sine wave.
• Peak value:
• $V_0=\sqrt{2}{V}_{\text{rms}}$
• .
• Angular frequency:
• $\omega =2\pi f$
• .

• In a pure resistor, voltage and current are in phase.
• In a pure inductor, current lags voltage by 90°.
• In a pure capacitor, current leads voltage by 90°.

• Quality factor:
• $Q=\frac{{\omega }_{0}L}{R}$
• — sharpness of resonance.
• At resonance the inductor and capacitor voltages can exceed the source voltage (Q times).
• Used in radio tuning — matching the resonant frequency to the desired station.

In an LC circuit, energy alternately stores in the capacitor (electric) and inductor (magnetic), oscillating at the natural frequency $\omega =\frac{1}{\sqrt{LC}}$. In an ideal circuit with no resistance, the oscillations persist.

Example: A series LCR circuit has $R=6\Omega$, $X_L=10\Omega$, $X_C=2\Omega$. Find the impedance.

Solution: $Z=\sqrt{6^2+(10-2{)}^2}=\sqrt{36+64}=\sqrt{100}=10\Omega$.

Exercises (NCERT Q7.1–Q7.8)

• 7.1) A 100 Ω resistor is connected to a 220 V, 50 Hz ac supply. (a) What is the rms value of current in the circuit? (b) What is the net power consumed over a full cycle?
• 7.2) (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?
• 7.3) A 44 mH inductor is connected to 220 V, 50 Hz ac supply. Determine the rms value of the current in the circuit.
• 7.4) A 60 µF capacitor is connected to a 110 V, 60 Hz ac supply. Determine the rms value of the current in the circuit.
• 7.5) In Exercises 7.3 and 7.4, what is the net power absorbed by each circuit over a complete cycle. Explain your answer.
• 7.6) A charged 30 µF capacitor is connected to a 27 mH inductor. What is the angular frequency of free oscillations of the circuit?
• 7.7) A series LCR circuit with R = 20 Ω, L = 1.5 H and C = 35 µF is connected to a variable-frequency 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?
• 7.8) Figure 7.17 shows a series LCR circuit connected to a variable frequency 230 V source. L = 5.0 H, C = 80 µF, R = 40 Ω. (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.

Answers at a glance

1. 1
   Q7.1 → (a) I_rms = 2.2 A. (b) Net power = 484 W.
2. 2
   Q7.2 → (a) V_rms ≈ 212 V. (b) I₀ ≈ 14.1 A.
3. 3
   Q7.3 → I_rms ≈ 15.9 A ≈ 16 A.
4. 4
   Q7.4 → I_rms ≈ 2.5 A.
5. 5
   Q7.5 → Net power = 0 in each circuit, because current and voltage are 90° out of phase (cosφ = 0).
6. 6
   Q7.6 → ω ≈ 1.1 × 10³ rad s⁻¹.
7. 7
   Q7.7 → Average power at resonance = 2000 W.
8. 8
   Q7.8 → (a) f ≈ 8 Hz. (b) Z = 40 Ω; current amplitude ≈ 8.1 A. (c) V_R = 230 V, V_L = V_C ≈ 1.4 kV; LC combination drop = 0.

MCQ Questions

1 Mark Questions

1. 1
   Define the rms value of an alternating current.
2. 2
   State the condition for resonance in a series LCR circuit.
3. 3
   What is the phase difference between the voltage and the current in a purely inductive AC circuit?
4. 4
   What is the principal use of a transformer?

2 Mark Questions

1. 1
   An AC source of peak voltage 311 V and frequency 50 Hz is connected to a 220 Ω resistor. Find the rms current in the circuit. (Ans: Irms = Vrms/R = 220/220 = 1 A)
2. 2
   A series LCR circuit has L = 0.1 H and C = 10 μF. Find its resonant frequency. (Ans: f0 = 1/(2π√LC) ≈ 159 Hz)
3. 3
   The primary of a step-down transformer has 4000 turns and works on 2200 V AC; the secondary gives 220 V. Find the number of secondary turns. (Ans: Ns = Np × Vs/Vp = 400 turns)

3 Mark Questions

1. 1
   Derive the expression for the average power consumed in a series LCR circuit over one complete cycle and define the power factor. (Ans: P = Vrms Irms cos φ)
2. 2
   Explain, with a circuit diagram, why the current in a series LCR circuit is maximum at resonance and obtain the expression for the resonant frequency. (Ans: f0 = 1/(2π√LC))
3. 3
   Describe the principle, construction and working of a step-up transformer; mention two sources of energy loss. (Ans: mutual induction; copper, iron, flux and hysteresis losses)

5 Mark Questions

1. 1
   With a neat circuit diagram, derive the impedance of a series LCR circuit driven by an AC source, obtain the resonance condition and derive the power dissipated as P = Vrms Irms cos φ. (Ans: Z = √(R² + (Xl − Xc)²), resonance when Xl = Xc)
2. 2
   Explain the working of a transformer on the principle of mutual induction. Derive the voltage and current ratios in terms of turns, and discuss the main sources of energy loss and how they are minimized.