class 12 notes

Class 12 Physics Notes Chapter 7 (Alternating Current) – Examplar Problems (English) Book

Philoid

Philoid

— 6 min read

Examplar Problems (English)
Alright class, let's dive deep into Chapter 7: Alternating Current. This is a crucial chapter, not just for your board exams but also extensively tested in various government entrance examinations. We'll focus on the core concepts, formulas, and applications as presented in the NCERT framework, keeping exam patterns in mind.

Chapter 7: Alternating Current (AC) - Detailed Notes for Exam Preparation

1. Introduction to Alternating Current & Voltage

  • Definition: An alternating current (or voltage) is one whose magnitude changes continuously with time and whose direction reverses periodically.
  • Sinusoidal AC: The most common form of AC is sinusoidal:
    • Instantaneous Voltage: v = Vm sin(ωt)
    • Instantaneous Current: i = Im sin(ωt + φ)
    • Where:
      • v, i: Instantaneous values of voltage and current.
      • Vm, Im: Peak values or amplitudes of voltage and current.
      • ω: Angular frequency (ω = 2πf = 2π/T).
      • f: Frequency (in Hz).
      • T: Time period.
      • t: Instantaneous time.
      • φ: Phase difference between voltage and current.

2. Important AC Values

  • Peak Value (Vm, Im): Maximum value attained by the AC during a cycle.
  • Mean or Average Value (Vavg, Iavg):
    • Over a full cycle, the average value of sinusoidal AC is zero.
    • Over a half cycle:
      • Iavg = (2/π) Im ≈ 0.637 Im
      • Vavg = (2/π) Vm ≈ 0.637 Vm
  • Root Mean Square (RMS) Value (Vrms, Irms):
    • Also known as the 'effective' or 'virtual' value. It's the equivalent DC value that would produce the same heating effect in a given resistor.
    • Irms = Im / √2 ≈ 0.707 Im
    • Vrms = Vm / √2 ≈ 0.707 Vm
    • Significance: AC measuring instruments (ammeters, voltmeters) usually measure RMS values. The voltage rating of domestic supply (e.g., 220V in India) is the RMS value. Power calculations typically use RMS values.

3. Phasors

  • A phasor is a rotating vector representing a sinusoidally varying quantity (like voltage or current).
  • Representation:
    • Length of the vector represents the peak value (Vm or Im).
    • Angular speed of rotation is ω (same as the AC).
    • Projection of the vector onto the vertical axis gives the instantaneous value.
    • The angle between phasors represents the phase difference (φ).
  • Use: Simplifies the analysis of AC circuits, especially when dealing with phase differences in L, C, R combinations.

4. AC Circuits with Single Components

  • (a) AC Applied to a Resistor (R):

    • Voltage: v = Vm sin(ωt)
    • Current: i = (Vm/R) sin(ωt) = Im sin(ωt) where Im = Vm/R.
    • Phase: Voltage and Current are in phase (φ = 0).
    • Phasor Diagram: V and I phasors are parallel.
    • Power Dissipation: Pavg = Vrms * Irms = Irms² * R = Vrms² / R.

  • (b) AC Applied to an Inductor (L):

    • Voltage: v = Vm sin(ωt)
    • Current: i = (Vm / ωL) sin(ωt - π/2) = Im sin(ωt - π/2)
    • Inductive Reactance (XL): XL = ωL = 2πfL. It's the opposition offered by the inductor to AC flow (Unit: Ohm). XL increases with frequency.
    • Im = Vm / XL
    • Phase: Current lags behind the voltage by π/2 (or 90°). (Mnemonic: CIVIL - in Capacitor, Current leads Voltage; in Inductor, Voltage leads Current).
    • Phasor Diagram: I phasor is π/2 behind the V phasor.
    • Power Dissipation: Average power consumed by a pure inductor over a full cycle is zero.

  • (c) AC Applied to a Capacitor (C):

    • Voltage: v = Vm sin(ωt)
    • Current: i = (Vm / (1/ωC)) sin(ωt + π/2) = Im sin(ωt + π/2)
    • Capacitive Reactance (XC): XC = 1 / (ωC) = 1 / (2πfC). It's the opposition offered by the capacitor to AC flow (Unit: Ohm). XC decreases with frequency.
    • Im = Vm / XC
    • Phase: Current leads the voltage by π/2 (or 90°).
    • Phasor Diagram: I phasor is π/2 ahead of the V phasor.
    • Power Dissipation: Average power consumed by a pure capacitor over a full cycle is zero.

5. Series LCR Circuit

  • An AC voltage v = Vm sin(ωt) is applied across a series combination of R, L, and C.
  • Let VR, VL, VC be the voltage drops across R, L, C respectively.
  • Using phasors:
    • VR is in phase with current i.
    • VL leads i by π/2.
    • VC lags i by π/2.
  • The net voltage v is the phasor sum of VR, VL, and VC.
  • Impedance (Z): The total effective opposition to AC flow in the LCR circuit.
    • Z = √[ R² + (XL - XC)² ] = √[ R² + (ωL - 1/ωC)² ] (Unit: Ohm)
  • Peak Current: Im = Vm / Z
  • RMS Current: Irms = Vrms / Z
  • Phase Difference (φ): The angle by which the total voltage v leads or lags the current i.
    • tan φ = (XL - XC) / R = (ωL - 1/ωC) / R
    • If XL > XC (Inductive): Circuit is predominantly inductive, φ is positive (Voltage leads current).
    • If XC > XL (Capacitive): Circuit is predominantly capacitive, φ is negative (Current leads voltage).
    • If XL = XC (Resistive): Circuit is purely resistive at resonance, φ = 0.

6. Resonance in Series LCR Circuit

  • Condition: The frequency at which the current amplitude Im becomes maximum (or impedance Z becomes minimum). This happens when XL = XC.
  • Resonant Angular Frequency (ω₀):
    • ω₀L = 1 / (ω₀C)
    • ω₀² = 1 / (LC)
    • ω₀ = 1 / √(LC)
  • Resonant Frequency (f₀): f₀ = ω₀ / (2π) = 1 / (2π√(LC))
  • At Resonance:
    • XL = XC
    • Impedance Z = R (Minimum value).
    • Current Im = Vm / R (Maximum value).
    • Phase difference φ = 0 (Voltage and current are in phase).
    • The circuit behaves like a purely resistive circuit.
  • Quality Factor (Q-factor): Measures the sharpness of resonance. It's the ratio of resonant frequency to the bandwidth (range of frequencies over which power is >= half the maximum power).
    • Q = ω₀ / (Δω) = (ω₀L) / R = 1 / (ω₀CR)
    • Higher Q means sharper resonance (more selective circuit).

7. Power in AC Circuits

  • Instantaneous Power: p = v * i = [Vm sin(ωt)] * [Im sin(ωt + φ)]
  • Average Power (Pavg): The average power dissipated per cycle.
    • Pavg = Vrms * Irms * cos φ
    • Pavg = Irms² * Z * cos φ
    • Since cos φ = R/Z, we also get Pavg = Irms² * R. This shows power is dissipated only across the resistor.
  • Power Factor (cos φ):
    • cos φ = R / Z = R / √[ R² + (XL - XC)² ]
    • It represents the fraction of apparent power (Vrms * Irms) that is used for actual work (true power).
    • Value ranges from 0 to 1.
    • Pure R circuit: φ = 0, cos φ = 1 (Max power).
    • Pure L or C circuit: φ = ±π/2, cos φ = 0 (Zero average power).
    • LCR circuit at resonance: φ = 0, cos φ = 1 (Max power for given Vrms, Irms).
  • Wattless Current: The component of current that consumes no average power (Irms sin φ). This component is maximum when cos φ = 0 (pure L or C).

8. LC Oscillations

  • When a charged capacitor is connected across an inductor, the charge (and energy) oscillates between the capacitor (as electric field energy UE = q²/2C) and the inductor (as magnetic field energy UB = ½ LI²).
  • These are undamped oscillations (assuming zero resistance).
  • The natural frequency of oscillation is ω₀ = 1 / √(LC).
  • Analogous to mechanical Simple Harmonic Motion (SHM).

9. Transformers

  • Principle: Mutual Induction. An AC voltage applied to the primary coil induces a varying magnetic flux, which links with the secondary coil and induces an AC voltage across it.
  • Ideal Transformer (No energy loss):
    • Voltage Ratio: Vs / Vp = Ns / Np
    • Current Ratio: Is / Ip = Np / Ns
    • Turns Ratio: k = Ns / Np
    • Vs / Vp = Ns / Np = Ip / Is = k
    • Input Power = Output Power (Vp * Ip = Vs * Is)
  • Types:
    • Step-up Transformer: Ns > Np => Vs > Vp and Is < Ip. Increases voltage, decreases current.
    • Step-down Transformer: Ns < Np => Vs < Vp and Is > Ip. Decreases voltage, increases current.
  • Efficiency (η): η = (Output Power) / (Input Power) = (Vs * Is) / (Vp * Ip)
    • For an ideal transformer, η = 1 (or 100%).
    • Practical transformers have η < 1 due to losses.
  • Energy Losses: Flux leakage, resistance of windings (copper loss), eddy currents (iron loss), hysteresis loss. Lamination of the core minimizes eddy currents.
  • Use: Widely used in power transmission (step-up at generation, step-down at distribution), voltage adaptation in devices.

Practice MCQs

Here are 10 multiple-choice questions to test your understanding:

  1. The RMS value of an AC voltage given by v = 200√2 sin(100πt) V is:
    (a) 200√2 V
    (b) 100 V
    (c) 200 V
    (d) 100√2 V

  2. In a pure inductive circuit, the current:
    (a) Lags behind the voltage by π/2
    (b) Leads the voltage by π/2
    (c) Is in phase with the voltage
    (d) Lags behind the voltage by π

  3. The opposition offered by a capacitor to the flow of alternating current is called:
    (a) Inductive Reactance
    (b) Impedance
    (c) Resistance
    (d) Capacitive Reactance

  4. At resonance frequency in a series LCR circuit, the impedance is:
    (a) Maximum
    (b) Equal to R
    (c) Equal to XL
    (d) Equal to XC

  5. The power factor of a pure resistive AC circuit is:
    (a) 0
    (b) 1
    (c) 0.5
    (d) 1/√2

  6. A transformer works on the principle of:
    (a) Self Induction
    (b) Mutual Induction
    (c) Electromagnetic Damping
    (d) Eddy Currents

  7. In a step-up transformer:
    (a) Ns < Np
    (b) Vs < Vp
    (c) Is > Ip
    (d) Ns > Np

  8. The quality factor (Q) of a series LCR circuit with L=2.0 H, C=32 µF, and R=10 Ω is:
    (a) 25
    (b) 50
    (c) 10
    (d) 20

  9. Wattless current flows in an AC circuit when the circuit is:
    (a) Purely Resistive
    (b) Containing R and L only
    (c) Containing R and C only
    (d) Purely Inductive or Purely Capacitive

  10. An AC source is connected to a resistor R. If the frequency of the source is increased, the power dissipated in the resistor will:
    (a) Increase
    (b) Decrease
    (c) Remain the same
    (d) Become zero

Answers to MCQs:

  1. (c) [Vrms = Vm/√2 = (200√2)/√2 = 200 V]
  2. (a) [Standard property of inductor in AC]
  3. (d) [Definition]
  4. (b) [At resonance XL=XC, so Z = √(R² + (XL-XC)²) = R]
  5. (b) [For pure R, φ=0, cos φ = 1]
  6. (b) [Fundamental principle]
  7. (d) [Definition of step-up transformer]
  8. (a) [ω₀ = 1/√(LC) = 1/√(2 * 32 * 10⁻⁶) = 1/√(64 * 10⁻⁶) = 1/(8 * 10⁻³) = 125 rad/s. Q = ω₀L/R = (125 * 2) / 10 = 250 / 10 = 25]
  9. (d) [Wattless means cos φ = 0, which occurs when φ = ±π/2, i.e., pure L or C]
  10. (c) [Power in resistor P = Vrms²/R. Vrms and R are independent of frequency]

Remember to thoroughly revise these concepts and practice numerical problems, especially those involving LCR circuits and transformers. Good luck with your preparation!

Read more