Class 12 Physics Quick Revision Notes & Formula Sheet

Chapter 7: Alternating Current (प्रत्यावर्ती धारा)

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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).

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2. Peak, Average, and RMS Values

Quantity	Relation with Peak Value (I₀ or E₀)	Value / Formula
Peak Value	Maximum value of AC in a cycle	I₀, E₀
Mean / Average Value (Over half cycle)	I_avg = (2 / π) × I₀	I_avg ≈ 0.637 I₀
Mean / Average Value (Over full cycle)	I_avg(full cycle) = 0	0
RMS / Effective Value (Root Mean Square)	I_rms = I₀ / √2	I_rms ≈ 0.707 I₀
RMS Voltage	E_rms = E₀ / √2	E_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.

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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).

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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).

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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.

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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

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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

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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.

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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)

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💡 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 $\cos \phi$.