Electromagnetic Induction

1. Magnetic Flux & Faraday's Laws

Magnetic Flux (Φ_B): Number of magnetic field lines passing normally through a surface.

Φ_B = B ⋅ A = BA cosθ
SI Unit: Weber (Wb) or Tesla-meter² (Tm²).

Faraday's Laws of Induction

First Law: Whenever magnetic flux linked with a coil changes, an induced EMF is produced.
Second Law: The magnitude of induced EMF is equal to the rate of change of magnetic flux.

ε = - N (dΦ_B / dt)

(Negative sign indicates direction - Lenz's Law)

2. Lenz's Law

The direction of induced EMF (or current) is such that it opposes the cause which produces it.

It is based on Conservation of Energy.
Mechanical work done in moving magnet against repulsive/attractive force is converted into Electrical Energy.

N-pole approaching coil → Face becomes N-pole (Repulsion).
N-pole receding from coil → Face becomes S-pole (Attraction).

(Asked in NEET 2019, 2022)

3. Motional Electromotive Force

EMF induced in a conductor moving in a magnetic field.

ε = Blv

(B, l, v are mutually perpendicular)

Rotational EMF

Rod of length l rotating with angular velocity ω:

ε = ½ B ω l² = B A f

(Asked in NEET 2018, 2021)

4. Eddy Currents

Circulating currents induced in bulk pieces of conductors when magnetic flux changes.

Disadvantages: Heat production (Energy loss).
Minimization: Using Laminated cores.
Applications: Magnetic Braking in trains, Induction Furnace, Speedometer.

(Asked in NEET 2019)

5. Inductance

Self Induction

Production of induced EMF in a coil when current in itself changes.

Φ = LI → ε = -L (dI/dt)
Solenoid: L = μ₀ n² Al
Energy Stored: U = ½ L I²

Mutual Induction

EMF in secondary coil when current in primary coil changes.

Φ_s = MI_p → ε_s = -M (dI_p/dt)
Co-axial Solenoids: M = μ₀ n₁ n₂ Al
Coupling Coefficient: k = M / √(L₁L₂)

(Asked in NEET 2016, 2017, 2023)

6. AC Generator

Converts Mechanical Energy into Electrical Energy (Sine Wave).

Principle: Electromagnetic Induction.
Flux Φ = NBA cos(ωt)
EMF ε = -dΦ/dt = NBAω sin(ωt)
Max EMF ε₀ = NBAω

Numericals: Electromagnetic Induction

Q1. A coil of 100 turns and area 0.1 m² is placed perpendicular to a magnetic field of 0.2 T. The field acts for 0.1 s and then vanishes. Calculate induced EMF.

Solution:
Initial Flux Φ₁ = NBA = 100 × 0.2 × 0.1 = 2 Wb.
Final Flux Φ₂ = 0.
Rate of change = (Φ₂ - Φ₁) / t = (0 - 2) / 0.1 = -20 Wb/s.
|ε| = 20 V.

Q2. A jet plane with wing span 25 m is flying at 1800 km/h. Earth's vertical field is 5 × 10^-4 T. Find voltage between wing tips.

Solution:
v = 1800 × (5/18) = 500 m/s.
l = 25 m, B = 5 × 10^-4 T.
ε = B l v
ε = (5 × 10^-4) × 25 × 500
ε = 125 × 500 × 10^-4
ε = 62500 × 10^-4 = 6.25 V.

Q3. An inductor of inductance 50 mH carries a current of 2 A. Calculate energy stored in it.

Solution:
U = ½ L I²
U = 0.5 × (50 × 10^-3) × 2²
U = 0.5 × 0.05 × 4
U = 0.1 Joules.

Q4. Current in a coil falls from 5 A to 0 A in 0.1 s. If induced EMF is 200 V, find self impedance (inductance) of the coil.

Solution:
ε = -L (dI/dt)
200 = -L (0 - 5) / 0.1
200 = L (50)
L = 200 / 50 = 4 H.

Q5. A 1 m metal rod rotates with frequency 50 Hz with one end hinged at center and other at rim of a circular ring in B-field 1 T. Find EMF.

Solution:
ε = B A f = B (πl²) f
ε = 1 × 3.14 × (1)² × 50
ε = 3.14 × 50 = 157 V.

Q6. Two concentric circular coils have radii r and R (R >> r). Find mutual inductance.

Solution:
Assume current I in outer coil R.
Magnetic field at center B = μ₀I / 2R.
Flux through inner coil Φ = B × πr².
Φ = (μ₀I / 2R) πr².
Since Φ = MI, M = μ₀ π r² / 2R.

Q7. A coil of resistance 10 Ω has flux change from 10 Wb to 2 Wb. Calculate charge flowing through it.

Solution:
Δq = ΔΦ / R
ΔΦ = 10 - 2 = 8 Wb.
Δq = 8 / 10 = 0.8 C.

Q8. A rectangular coil of 20 turns and area 200 cm² rotates at 3000 rpm in B=0.5 T. Find max EMF.

Solution:
ε₀ = NBAω
ω = 2πf = 2π(3000/60) = 100π rad/s.
A = 200 cm² = 200 × 10^-4 m².
ε₀ = 20 × 0.5 × (2×10^-2) × 100π
ε₀ = 10 × 0.02 × 314
ε₀ = 0.2 × 314 = 62.8 V.

Q9. A 0.5 m long conductor moves at 20 m/s perpendicular to B = 2 T. If it is part of a circuit of 5 Ω, find force required to move it.

Solution:
Induced EMF ε = Blv = 2 × 0.5 × 20 = 20 V.
Current I = ε/R = 20/5 = 4 A.
Magnetic Force F = BIl = 2 × 4 × 0.5.
F = 4 N.

Q10. A solenoid has 1000 turns wound on a 50 cm long core of radius 2 cm. Calculate self-inductance.

Solution:
L = μ₀ n² Al
n = N/l = 1000 / 0.5 = 2000 turns/m.
A = πr² = π(0.02)² = 4π × 10^-4 m².
L = (4π × 10^-7) × (2000)² × (4π × 10^-4) × 0.5
L = 4π × 4 × 10⁶ × 2π × 10^-4 × 10^-7
L = 32 π² × 10^-5 ≈ 320 × 10^-5 = 3.2 mH.

Important Formulae

1. Faraday's Law & Flux

ε = - N (dΦ / dt)

Φ = B ⋅ A = BA cosθ
Current I = |ε| / R
Charge Δq = ΔΦ / R

2. Motional EMF

Translational:

ε = B l v

Rotational (Rod):

ε = ½ B ω l²

Resulting Force F = I l B = (B²l²v)/R
Power P = F v = (B²l²v²)/R

3. Inductance

Self Inductance:

ε = - L (dI/dt)

Mutual Inductance:

ε_s = - M (dI_p/dt)

L (Solenoid) = μ₀ n² V
Energy U = ½ L I²

4. Combination of Inductors

Series:

L_eq = L₁ + L₂ ± 2M

Parallel (M=0):

1/L_eq = 1/L₁ + 1/L₂

20 NEET Golden Facts

1. Induced Charge: Depends only on change in flux (ΔΦ) and resistance, NOT on time rate. q = ΔΦ/R.
2. Lenz's Law: It is a consequence of Law of Conservation of Energy.
3. Inertia of Electricity: Self-inductance is called electrical inertia because it opposes change in current.
4. Sparking: Occurs during 'breaking' of circuit (switch off) due to large induced EMF in inductor.
5. Doubling Turns: If turns N are doubled in same length, L becomes 4 times (since L ∝ N²).
6. Dead Beat Galvanometer: Coil wound on metallic frame to produce eddy currents which damp oscillations quickly.
7. Metal Detector: Works on principle of change in inductance and resonance in AC circuit.
8. Free Fall Magnet: Acceleration of magnet falling through a coil is less than g (due to opposing force).
9. Cut Ring: If the ring has a cut (open circuit), EMF is induced but no Current flows (No opposition to magnet).
10. DC Block: Inductor allows DC to flow easily (f=0, X_L=0) but blocks AC.
11. Plane Flies: Potential difference develops between wing tips of aircraft flying in Earth's B-field (Vertical component).
12. AC Generator: Slip rings produce AC. Split rings (Commutator) produce DC.
13. Non-Inductive: Resistance coils are double wound to make net inductance zero.
14. Flux Max: Flux is max when θ=0. EMF is max when θ=90 (since EMF ∝ sinθ).
15. Iron Core: Putting iron core inside solenoid increases L drastically (by factor μ_r).
16. Dimensions: [L/R] has dimensions of Time. [L] = [ML²T^-2A^-2].
17. Disc Rotation: Metal disc rotating in B-field cuts flux lines → EMF induced between Center and Rim.
18. 1 Henry: Current changing at 1 A/s inducing 1 V EMF.
19. Coupling: Tight coupling (k=1) means 100% flux linkage. Loose coupling (k<1).
20. Fleming's Right Hand: Used to find direction of Induced Current. (Left hand is for Force).

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