📝 Chapter Notes & Revision

Alternating Current

🏫 MP BoardClass 12Physics

📐 Formula & Cheat Sheet (English)

Class 12 Physics Quick Revision Notes & Formula Sheet

Chapter 7: Alternating Current (प्रत्यावर्ती धारा)

1. Basic Definitions & Equations

  • Alternating Current (AC): An electric current whose magnitude changes continuously with time and direction reverses periodically.
  • Instantaneous Voltage & Current:
    • E = E₀ sin(ωt)
    • I = I₀ sin(ωt + ϕ) (where E₀, I₀ are peak values, ω is angular frequency, and ϕ is phase difference)
  • Time Period (T): Time taken to complete one full cycle. T = 2π / ω
  • Frequency (f or ν): Number of cycles completed per second.
    • f = 1 / T = ω / (2π)
    • Standard AC supply frequency in India = 50 Hz (Angular frequency ω = 100π rad/s).

2. Peak, Average, and RMS Values

QuantityRelation with Peak Value (I₀ or E₀)Value / Formula
Peak ValueMaximum value of AC in a cycleI₀, E₀
Mean / Average Value (Over half cycle)I_avg = (2 / π) × I₀I_avg ≈ 0.637 I₀
Mean / Average Value (Over full cycle)I_avg(full cycle) = 00
RMS / Effective Value (Root Mean Square)I_rms = I₀ / √2I_rms ≈ 0.707 I₀
RMS VoltageE_rms = E₀ / √2E_rms ≈ 0.707 E₀

Note: AC measuring instruments (like AC voltmeter/ammeter) work on the heating effect of current and always measure RMS values, not peak values.

3. AC Circuits with Single Elements

A. Purely Resistive Circuit (Pure R)

  • Voltage: E = E₀ sin(ωt)
  • Current: I = I₀ sin(ωt)
  • Phase Difference (ϕ): 0 (Voltage and Current are in phase).
  • Opposition: Resistance R

B. Purely Inductive Circuit (Pure L)

  • Voltage: E = E₀ sin(ωt)
  • Current: I = I₀ sin(ωt - π/2)
  • Phase Difference (ϕ): Voltage leads current by 90° (π/2 rad).
  • Inductive Reactance (X_L):
    • X_L = ωL = 2πfL (Unit: Ohm Ω)
    • For Direct Current (DC): f = 0 $\rightarrow$ X_L = 0 (Inductor acts as a simple conductor to DC).

C. Purely Capacitive Circuit (Pure C)

  • Voltage: E = E₀ sin(ωt)
  • Current: I = I₀ sin(ωt + π/2)
  • Phase Difference (ϕ): Current leads voltage by 90° (π/2 rad).
  • Capacitive Reactance (X_C):
    • X_C = 1 / (ωC) = 1 / (2πfC) (Unit: Ohm Ω)
    • For Direct Current (DC): f = 0 $\rightarrow$ X_C = ∞ (Capacitor completely blocks DC).

4. Series LCR Circuit

An AC circuit containing Inductor ($L$), Capacitor ($C$), and Resistor ($R$) connected in series.

  • Total Impedance (Z): The net effective opposition offered by L, C, and R to AC flow.
    • Z = √[ R² + (X_L - X_C)² ]
    • Z = √[ R² + (ωL - 1 / (ωC))² ]
  • Resultant Voltage: E_rms = √[ V_R² + (V_L - V_C)² ]
  • Phase Angle (ϕ):
    • tan ϕ = (X_L - X_C) / R = (V_L - V_C) / V_R
    • If X_L > X_C: Circuit is Inductive (Voltage leads current).
    • If X_C > X_L: Circuit is Capacitive (Current leads voltage).
    • If X_L = X_C: Circuit is Purely Resistive (Resonance condition).

5. Resonance in Series LCR Circuit

Resonance occurs when inductive reactance equals capacitive reactance (X_L = X_C).

  • Resonance Condition: ωL = 1 / (ωC)
  • Resonant Angular Frequency (ω_r): ω_r = 1 / √(LC)
  • Resonant Frequency (f_r): f_r = 1 / (2π √(LC))
  • Key Characteristics at Resonance:
    1. Impedance is minimum: Z_min = R
    2. Current is maximum: I_max = E_rms / R
    3. Circuit is purely resistive (ϕ = 0°).
    4. Power factor is maximum (cos ϕ = 1).

Sharpness of Resonance & Quality Factor (Q-Factor)

  • Q = (1 / R) × √(L / C)
  • Q = (ω_r × L) / R = 1 / (ω_r × C × R)
  • High Q factor means sharper resonance and better selectivity.

6. Power in AC Circuits

  • Instantaneous Power: P = E × I
  • Average Power ($P_{avg}$):
    • P_avg = E_rms × I_rms × cos ϕ
  • Power Factor ($\cos \phi$):
    • cos ϕ = R / Z = True Power / Apparent Power
    • For Pure R: ϕ = 0° $\rightarrow$ cos ϕ = 1 (Maximum power dissipation)
    • For Pure L or C: ϕ = 90° $\rightarrow$ cos ϕ = 0 (Zero power dissipation)
    • For LCR at Resonance: cos ϕ = 1

Wattless Current (शक्तिहीन धारा)

  • When current flows through a circuit containing only pure Inductor ($L$) or pure Capacitor ($C$), phase difference ϕ = 90°.
  • Average Power consumed P_avg = 0. This current is called Wattless Current.
  • Wattless Component of Current: I_wattless = I_rms sin ϕ

7. LC Oscillations

  • When a charged capacitor ($C$) is connected to an inductor ($L$), electrical energy stored in $C$ oscillates back and forth into magnetic energy in $L$.
  • Frequency of Oscillations: f = 1 / (2π √(LC))
  • Total Energy conserved: E = (1/2) (q² / C) + (1/2) L I² = Constant

8. Transformer (ट्रांसफॉर्मर)

A device used to step-up or step-down alternating voltage based on the principle of Mutual Induction (अन्योन्य प्रेरण).

  • Transformation Ratio (K):

    • K = N_s / N_p = E_s / E_p = I_p / I_s (where p = primary coil, s = secondary coil)

  • Types of Transformers:

    1. Step-Up Transformer:
      • N_s > N_p and K > 1
      • Increases Voltage (E_s > E_p), decreases current (I_s < I_p).
    2. Step-Down Transformer:
      • N_s < N_p and K < 1
      • Decreases Voltage (E_s < E_p), increases current (I_s > I_p).

  • Efficiency of Transformer ($\eta$):

    • η = (Output Power / Input Power) × 100%
    • η = (E_s × I_s) / (E_p × I_p) × 100%

Major Energy Losses in Transformers & Minimization Methods:

  1. Copper Loss ($I²R$ loss): Heat loss in copper windings $\rightarrow$ Minimized by using thick wires.
  2. Eddy Current Loss (भंवर धारा हानि): Heating in iron core $\rightarrow$ Minimized using a laminated iron core.
  3. Hysteresis Loss (शिथिल्य हानि): Repeated magnetization/demagnetization $\rightarrow$ Minimized by using a soft iron core.
  4. Flux Leakage: Magnetic flux lost to air $\rightarrow$ Minimized by winding primary and secondary coils over one another.

9. AC Generator (Dynamo)

  • Principle: Electromagnetic Induction (Faraday's Laws). Converts mechanical energy into electrical energy.
  • Induced EMF Equation:
    • e = E₀ sin(ωt)
    • Peak EMF: E₀ = N B A ω (where $N$ = number of turns, $A$ = area, $B$ = magnetic field, $ω$ = angular speed)

💡 MP Board Exam Important Tips:

  1. Derivations to Focus:
    • Expression for RMS current ($I_{rms} = I_0 / \sqrt{2}$).
    • Impedance and Phase for Series LCR circuit.
    • Resonant frequency derivation.
    • Transformer principle, working, and efficiency.
  2. Definitions Often Asked: Wattless current, Quality Factor, Power factor, Reactance vs Impedance.
  3. Numerical Topics: Calculating $I_{rms}$, $X_L$, $X_C$, $Z$, resonant frequency $f_r$, and power factor $\cos \phi$.