Electromagnetic Induction & Alternating Current
Complete Class 12 Physics Notes for Bihar Board & NEET Preparation
Section A: Electromagnetic Induction (EMI)
Key Concept: Electromagnetic induction is the phenomenon of inducing an electromotive force (emf) or current in a closed circuit due to a changing magnetic field.
1. Introduction
Electromagnetic induction was discovered by Michael Faraday in 1831 and mathematically developed by James Clerk Maxwell. This fundamental principle underlies the operation of generators, transformers, and many electrical devices.
2. Faraday's Experiments
Faraday performed several experiments demonstrating that electric current can be produced by changing magnetic flux:
Experiment 1: Motion of Magnet and Coil
When a magnet is moved towards a stationary coil connected to a galvanometer, the galvanometer needle deflects. When moved away, it deflects in the opposite direction. No deflection occurs when the magnet is stationary.
Conclusion: Current is induced only when there's a change in magnetic flux.
Experiment 2: Two Coils (Primary & Secondary)
A primary coil connected to a battery and a secondary coil connected to a galvanometer. When the key is closed or opened in the primary circuit, momentary deflection occurs in the galvanometer.
Conclusion: Emf is induced in the secondary coil due to changing current in the primary coil → changing magnetic field → changing flux → induced emf.
3. Faraday's Laws of Electromagnetic Induction
First Law: Whenever magnetic flux changes, an emf is induced.
Second Law: e = -dΦB/dt
Where e = induced emf, ΦB = magnetic flux = B·A·cosθ
4. Lenz's Law – Direction of Induced Current
Proposed by Heinrich Lenz (1834):
Statement: The direction of induced current is such that it opposes the cause which produces it.
The negative sign in Faraday's law represents Lenz's Law.
Example:
When a magnet moves towards a coil, the approaching north pole increases magnetic flux through the coil. The induced current flows in such a direction that it repels the approaching pole (to oppose increase in flux).
5. Magnetic Flux
ΦB = B A cosθ
Where B = magnetic field strength, A = area of loop, θ = angle between B and normal to the surface.
6. Induced emf and Current
When circuit is closed:
I = e/R = -(1/R)(dΦB/dt)
When circuit is open: Only emf is induced, no current flows.
7. Types of Induced emf
1. Motional emf: When a conductor moves in a magnetic field.
e = B l v sinθ
Where l = length, v = velocity.
2. Induced emf due to changing magnetic field: When magnetic flux changes due to variation in magnetic field strength or coil orientation.
8. Eddy Currents
When a metal plate moves through a magnetic field, circulating currents called eddy currents are induced within it.
Effects:
- Heating effect (used in induction furnace)
- Magnetic damping (used in speedometers)
- Energy loss in transformers
Applications:
- Braking in trains: Magnetic braking using eddy currents
- Electric meters: Damping motion of needle
- Induction stove: Eddy currents heat metal utensils
9. Self Induction
When the current in a coil changes, the magnetic flux linked with the same coil also changes, inducing emf in the same coil.
e = -L(di/dt)
Where L = self-inductance of the coil.
Unit: Henry (H) - When a current change of 1 A/s induces an emf of 1 V.
Expression for L: For a solenoid of length l, area A, and N turns:
L = μ₀N²A/l
10. Mutual Induction
When current in one coil changes, emf is induced in another nearby coil due to magnetic coupling.
e₂ = -M(di₁/dt)
Where M = mutual inductance.
Expression for M:
M = N₂Φ₂₁/I₁
11. Energy Stored in an Inductor
U = ½ L I²
This energy is stored in the magnetic field.
12. L-R Circuit (Growth and Decay of Current)
When a switch is closed in an LR circuit:
i = I(1 - e^(-Rt/L))
When switch is opened:
i = I e^(-Rt/L)
Where τ = L/R is the time constant.
Summary of Section A
| Concept | Formula | Unit |
|---|---|---|
| Induced emf | e = -dΦ/dt | Volt |
| Magnetic flux | Φ = B A cosθ | Weber |
| Motional emf | e = B l v sinθ | Volt |
| Self Induction | e = -L di/dt | Henry |
| Mutual Induction | e = -M di/dt | Henry |
| Energy stored | U = ½ L I² | Joule |
| Time constant | τ = L/R | Second |
Section B: Alternating Current (AC)
Key Concept: Alternating Current (AC) is an electric current whose magnitude and direction change periodically with time.
1. Introduction to Alternating Current
Unlike Direct Current (DC), which flows in one direction only, AC changes its direction continuously.
The instantaneous value of AC at any time t is given by:
i = i₀ sin(ωt) or i = i₀ cos(ωt)
Where i₀ = maximum or peak current, ω = angular frequency (rad/s), t = time.
The frequency (f) and time period (T) are related as:
ω = 2πf = 2π/T
2. Generation of Alternating EMF
An alternating emf is produced when a coil rotates in a uniform magnetic field:
e = e₀ sin(ωt)
Where e₀ = N B A ω (N = number of turns, B = magnetic field, A = area)
3. Characteristics of AC
| Term | Symbol | Meaning |
|---|---|---|
| Peak value | I₀, V₀ | Maximum current or voltage |
| Instantaneous value | I, V | Value at any instant |
| Average value | Iavg = 2I₀/π | Mean over half cycle |
| RMS value | Irms = I₀/√2 | Effective current producing same heating as DC |
4. RMS and Average Values
RMS Value: Root Mean Square value gives the effective current.
Irms = I₀/√2, Vrms = V₀/√2
Average Value:
Iavg = 2I₀/π, Vavg = 2V₀/π
5. Phase and Phase Difference
When two alternating quantities have the same frequency but reach their maximum and zero values at different times, they have a phase difference:
φ = ωt₂ - ωt₁
6. AC Through Pure Resistor
Let V = V₀ sin(ωt). According to Ohm's law:
I = V/R = (V₀/R) sin(ωt)
In a pure resistor, current and voltage are in phase.
Power in resistor:
P = VI = V₀I₀ sin²(ωt)
Average power: Pavg = ½ V₀I₀ = VrmsIrms
7. AC Through Pure Inductor
If AC voltage is applied to an inductor L:
V = L dI/dt
I = I₀ sin(ωt - π/2)
Current lags voltage by 90°.
Inductive Reactance:
XL = ωL = 2πfL
Opposes the flow of AC.
8. AC Through Pure Capacitor
For a capacitor of capacitance C:
I = I₀ sin(ωt + π/2)
Current leads voltage by 90°.
Capacitive Reactance:
XC = 1/(ωC)
9. L-C-R Series Circuit (AC)
In an LCR series circuit, all three elements are connected in series:
V = √[VR² + (VL - VC)²]
Impedance: Z = √[R² + (XL - XC)²]
Phase angle: tanφ = (XL - XC)/R
Current: I = V/Z
10. Resonance in AC Circuit
Condition: When XL = XC, impedance is minimum and current is maximum.
Z = R
Resonant frequency: fr = 1/(2π√LC)
Applications: Radio and TV tuning circuits, filters and oscillators.
11. Power in AC Circuit
P = V I cosφ
Where φ = phase difference, cosφ = Power Factor
12. Power Factor
Power Factor = cosφ = R/Z
If cosφ = 1 → 100% efficiency. If cosφ < 1 → Power loss due to reactance.
13. Quality Factor (Q)
Q = (1/R) √(L/C)
A high Q means sharper resonance.
Summary of Section B
| Quantity | Formula | Remarks |
|---|---|---|
| RMS value | I₀/√2 | Effective current |
| Average value | 2I₀/π | For half cycle |
| Inductive reactance | XL = 2πfL | Increases with f |
| Capacitive reactance | XC = 1/(2πfC) | Decreases with f |
| Impedance | Z = √[R² + (XL - XC)²] | Total opposition |
| Resonant frequency | fr = 1/(2π√LC) | Maximum current |
| Power | P = V I cosφ | True power |
Section C: Applications & Advanced Concepts
Key Concept: This section covers practical applications of electromagnetic induction and AC principles, including transformers, power transmission, and LC oscillations.
1. LC Oscillations
An LC circuit consists of an inductor (L) and a capacitor (C) connected together. When charged, it can produce electrical oscillations.
Equation of LC Oscillations:
d²q/dt² + (1/LC)q = 0
Angular frequency of oscillation:
ω = 1/√LC
Frequency of oscillation:
f = 1/(2π√LC)
Total energy in LC circuit:
E = ½ L I₀² = ½ (q₀²/C)
This energy oscillates between the electric field of the capacitor and the magnetic field of the inductor.
2. Transformers
Transformers are devices that transfer electrical energy between circuits through electromagnetic induction, changing AC voltage levels.
Working Principle: Mutual induction between primary and secondary coils.
Transformer Equation:
Vs/Vp = Ns/Np = Ip/Is
Where V = voltage, N = number of turns, I = current, subscripts s = secondary, p = primary.
Types of Transformers:
- Step-up transformer: Increases voltage (Ns > Np)
- Step-down transformer: Decreases voltage (Ns < Np)
Efficiency of transformer:
η = (Output power / Input power) × 100%
Energy losses in transformers:
- Copper losses (I²R losses in windings)
- Iron losses (eddy currents and hysteresis)
- Flux leakage
3. Power Transmission
Electrical power is transmitted over long distances at high voltages to reduce energy loss.
Power loss in transmission lines:
Ploss = I²R
Where I = current, R = resistance of transmission lines.
Since power P = VI, for constant power, higher voltage means lower current, which reduces I²R losses.
Typical power transmission system:
- Generation at power plant (11-25 kV)
- Step-up to transmission voltage (132-765 kV)
- Long-distance transmission
- Step-down to sub-transmission voltage (33-132 kV)
- Further step-down for distribution (11-33 kV)
- Final step-down for consumers (220V/440V)
4. NEET-Focused Numerical Problems
Problem 1: Faraday's Law
A coil of 100 turns and area 0.1 m² is placed in a magnetic field of 0.5 T. If the coil is rotated from a position perpendicular to the field to parallel in 0.2 seconds, calculate the average emf induced.
Problem 2: LCR Circuit
An LCR series circuit with R = 10Ω, L = 0.1 H, and C = 10μF is connected to a 200V, 50Hz AC source. Calculate the impedance and current in the circuit.
Problem 3: Transformer
A step-down transformer has 500 turns in primary and 50 turns in secondary. If the primary voltage is 220V, find the secondary voltage. If the secondary current is 5A, find the primary current (assuming 100% efficiency).
5. Important Derivations for NEET
Derivation 1: Expression for induced emf in a rotating coil
Derivation 2: Impedance of LCR series circuit
6. Quick Revision Notes
Key Points to Remember:
- Faraday's Law: e = -dΦ/dt
- Lenz's Law: Direction of induced current opposes the change causing it
- Self inductance: e = -L di/dt
- Mutual inductance: e = -M di/dt
- RMS value = Peak value/√2
- In pure inductor: Current lags voltage by 90°
- In pure capacitor: Current leads voltage by 90°
- Resonance in LCR circuit when XL = XC
- Power in AC circuit: P = VIcosφ
- Transformer: Vs/Vp = Ns/Np = Ip/Is
7. Common Mistakes to Avoid in Exams
- Forgetting the negative sign in Faraday's law (Lenz's law)
- Confusing RMS values with average values
- Mixing up when current leads or lags voltage in capacitive/inductive circuits
- Not converting units properly (e.g., μF to F, mH to H)
- Forgetting to consider phase difference in AC power calculations
- Confusing self-inductance with mutual inductance
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