Class 12 Physics Notes Chapter 6 (Electromagnetic Induction) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 6: Electromagnetic Induction. This is a crucial chapter, not just for your board exams but also for various competitive government exams where Physics is a component. We'll be focusing on the core concepts, often tested through conceptual questions similar to those found in your NCERT Exemplar book.

Chapter 6: Electromagnetic Induction - Detailed Notes

1. Magnetic Flux (ΦB)

• Definition: Magnetic flux through any surface placed in a magnetic field is the measure of the total number of magnetic field lines passing normally through that surface.
• Mathematical Representation: For a uniform magnetic field B passing through a planar area A, the flux is given by the scalar product:
  ΦB = B ⋅ A = BA cos θ
  where θ is the angle between the magnetic field vector B and the area vector A (which is perpendicular to the surface).
• Scalar Quantity: Magnetic flux is a scalar quantity.
• SI Unit: Weber (Wb). 1 Wb = 1 Tesla-meter² (T m²).
• CGS Unit: Maxwell (Mx). 1 Wb = 10⁸ Mx.
• Key Idea: Flux changes if B changes, A changes, or the angle θ changes. This change is the root cause of induced EMF.

2. Faraday's Law of Electromagnetic Induction

• Statement: Whenever the magnetic flux linked with a closed circuit changes, an electromotive force (EMF), and hence a current (if the circuit is closed), is induced in it. The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linked with the circuit.
• Mathematical Representation:
  ε = - dΦB / dt
  For a coil with N turns:
  ε = - N (dΦB / dt)
• Induced Current: If the resistance of the closed circuit is R, the induced current is:
  I = ε / R = (-1/R) * (dΦB / dt)
• Induced Charge: The total charge induced in a time interval Δt during which flux changes by ΔΦB is:
  q = ∫ I dt = ∫ (ε/R) dt = (-1/R) ∫ (dΦB/dt) dt = - ΔΦB / R
  Note: Induced charge depends on the net change in flux and resistance, but not on the time taken for the change.

3. Lenz's Law

• Statement: The direction of the induced EMF or induced current is always such that it opposes the change in magnetic flux that produced it.
• Explanation: This law determines the direction of the induced current (and hence the polarity of the induced EMF).
  • If flux is increasing, the induced current creates a magnetic field opposing the increase (i.e., opposite to the original field).
  • If flux is decreasing, the induced current creates a magnetic field supporting the decrease (i.e., in the same direction as the original field).
• Conservation of Energy: Lenz's law is a direct consequence of the law of conservation of energy. Work has to be done against the opposing force (predicted by Lenz's law) to cause the change in flux, and this mechanical work gets converted into electrical energy (the induced EMF/current).

4. Motional Electromotive Force (Motional EMF)

• Definition: The EMF induced across the ends of a conductor due to its motion in a magnetic field.
• Straight Conductor: Consider a straight conductor of length 'l' moving with velocity 'v' perpendicular to its length and perpendicular to a uniform magnetic field 'B'.
  ε = Blv
  Derivation: The magnetic Lorentz force on free electrons (Fm = q(v × B)) causes them to accumulate at one end, creating an electric field E inside the conductor. In equilibrium, Fe = Fm => qE = qvB => E = vB. The potential difference (EMF) is ε = El = Blv.
• General Case: If v makes an angle θ with B, ε = Blv sin θ. If the velocity is not perpendicular to the length, only the component perpendicular to both B and l contributes.
• Rotating Rod: A conducting rod of length 'l' rotating with angular velocity 'ω' about one end in a uniform magnetic field 'B' perpendicular to the plane of rotation.
  ε = ½ Bωl²
• Arbitrary Motion/Field: The most general expression involves integrating the motional electric field (v × B) along the length of the conductor:
  ε = ∫ (v × B) ⋅ dl

5. Eddy Currents (Foucault Currents)

• Definition: Induced circulating currents produced in the bulk piece of a conductor when the magnetic flux linked with it changes. They flow in closed loops within the conductor.
• Nature: Their direction is given by Lenz's law, opposing the change in flux.
• Effects:
  • Heating: Due to the resistance of the conductor (I²R loss), eddy currents produce heat (undesirable in transformers, motor cores).
  • Damping: They produce a drag force opposing the motion that causes them (electromagnetic damping).
• Minimization: To reduce eddy currents in devices like transformer cores and armatures, the core is laminated (made of thin sheets insulated from each other). This increases the resistance to the path of eddy currents.
• Applications:
  • Magnetic Braking in trains
  • Electromagnetic Damping (in galvanometer coils)
  • Induction Furnace (to melt metals)
  • Induction Cooktops
  • Speedometers

6. Inductance

• Definition: The property of an electrical circuit by virtue of which it opposes any change in the current flowing through it. It's the electrical analogue of inertia.
• Types:
  • Self-Inductance (L): The property of a coil by virtue of which it opposes a change in the current flowing through itself.
    • Magnetic flux linked with the coil is proportional to the current: ΦB ∝ I => ΦB = LI
    • Induced EMF (Back EMF): ε = - dΦB / dt = - L (dI/dt)
    • SI Unit: Henry (H). 1 H = 1 Wb/A = 1 V s / A.
    • L depends on the geometry (size, shape, number of turns) of the coil and the permeability of the medium inside.
    • Example: Self-inductance of a long solenoid: L = μ₀n²Al = μ₀N²A/l (where n=N/l is turns per unit length, N is total turns, A is area, l is length).
  • Mutual Inductance (M): The property of a pair of coils by virtue of which one coil opposes any change in the current flowing through the other coil by inducing an EMF in itself.
    • Flux linked with coil 2 due to current I₁ in coil 1: Φ₂₁ = M₂₁I₁
    • EMF induced in coil 2: ε₂ = - dΦ₂₁ / dt = - M₂₁ (dI₁/dt)
    • Similarly, Φ₁₂ = M₁₂I₂ and ε₁ = - M₁₂ (dI₂/dt)
    • Reciprocity Theorem: M₁₂ = M₂₁ = M
    • SI Unit: Henry (H).
    • M depends on the geometry of both coils, their relative separation, orientation, and the permeability of the medium.
    • Example: Mutual inductance of two long coaxial solenoids.

7. Energy Stored in an Inductor

• Work has to be done by the external source against the back EMF to establish a current in an inductor. This work is stored as magnetic potential energy.
• Energy Stored: U = ½ LI²
• Magnetic Energy Density (uB): Energy stored per unit volume in a magnetic field.
  uB = U / Volume = (½ LI²) / (Al) = ½ (μ₀n²Al) I² / (Al) = ½ μ₀n²I²
  Since B = μ₀nI for a solenoid, I = B / (μ₀n). Substituting this:
  uB = ½ μ₀n² [B / (μ₀n)]² = ½ μ₀n² B² / (μ₀²n²) = B² / (2μ₀)
  This is a general result for energy density in a magnetic field.

8. AC Generator (Dynamo)

• Principle: Electromagnetic Induction. When a coil rotates in a uniform magnetic field, the magnetic flux linked with it changes continuously, inducing an alternating EMF.
• Components:
  • Armature Coil (rotating coil with N turns, area A)
  • Field Magnets (produce uniform magnetic field B)
  • Slip Rings (maintain continuous contact between rotating coil and external circuit)
  • Brushes (carbon brushes press against slip rings to conduct current)
• Working: As the coil rotates with angular velocity ω, the angle θ between B and A changes as θ = ωt (assuming θ=0 at t=0).
  ΦB = NBA cos θ = NBA cos(ωt)
• Induced EMF:
  ε = - dΦB / dt = - d/dt [NBA cos(ωt)] = -NBA [-sin(ωt) * ω]
  ε = NBAω sin(ωt)
  ε = ε₀ sin(ωt)
  where ε₀ = NBAω is the peak value (amplitude) of the induced EMF.
• Output: The induced EMF and current are sinusoidal (alternating).

Multiple Choice Questions (MCQs)

1. The magnetic flux linked with a coil (in Wb) is given by the equation ΦB = 5t² + 3t + 16. The magnitude of induced EMF in the coil at the fourth second will be:
   (a) 10 V
   (b) 43 V
   (c) 108 V
   (d) 16 V

2. Lenz's law is a consequence of the law of conservation of:
   (a) Charge
   (b) Mass
   (c) Momentum
   (d) Energy

3. A conducting circular loop is placed in a uniform magnetic field B with its plane perpendicular to the field. The radius of the loop starts shrinking at a constant rate 'α'. The induced EMF in the loop at the instant when the radius is 'r' is:
   (a) πr²αB
   (b) 2πrαB
   (c) πrαB
   (d) 2r²αB

4. The self-inductance L of a solenoid of length l and area of cross-section A, with a fixed number of turns N, increases as:
   (a) l and A increase
   (b) l decreases and A increases
   (c) l increases and A decreases
   (d) l and A decrease

5. Eddy currents are produced when:
   (a) A metal is kept in varying magnetic field
   (b) A metal is kept in steady magnetic field
   (c) A circular coil is placed in a magnetic field
   (d) Current is passed through a circular coil

6. A horizontal straight wire 10 m long extending from east to west is falling with a speed of 5.0 m s⁻¹, at right angles to the horizontal component of the earth’s magnetic field, 0.30 × 10⁻⁴ Wb m⁻². The instantaneous value of the EMF induced in the wire is:
   (a) 1.5 × 10⁻³ V
   (b) 1.5 × 10⁻⁴ V
   (c) 2.5 × 10⁻³ V
   (d) 2.5 × 10⁻⁴ V

7. Two coils have a mutual inductance of 0.005 H. The current changes in the first coil according to the equation I = I₀ sin(ωt), where I₀ = 10 A and ω = 100π rad/s. The maximum value of EMF induced in the second coil is:
   (a) 2π V
   (b) 5π V
   (c) π V
   (d) 4π V

8. The energy stored in a 50 mH inductor carrying a current of 4 A is:
   (a) 0.4 J
   (b) 0.1 J
   (c) 0.04 J
   (d) 4.0 J

9. In an AC generator, the peak value of the induced EMF depends on:
   (a) Number of turns (N) only
   (b) Area of coil (A) and Magnetic field (B) only
   (c) Angular speed (ω) only
   (d) N, B, A, and ω

10. When the current in a coil changes from 5 A to 2 A in 0.1 s, an average voltage of 50 V is produced. The self-inductance of the coil is:
    (a) 1.67 H
    (b) 6 H
    (c) 3 H
    (d) 0.67 H

Answers to MCQs:

1. (b) [ε = |dΦB/dt| = |10t + 3|. At t=4s, ε = 10(4) + 3 = 43 V]
2. (d)
3. (b) [ΦB = BA = B(πr²). |ε| = |dΦB/dt| = Bπ |d(r²)/dt| = Bπ |2r (dr/dt)|. Since radius shrinks, dr/dt = -α. |ε| = Bπ(2rα) = 2πrαB]
4. (b) [L = μ₀N²A/l. For fixed N, L increases if A increases and l decreases]
5. (a)
6. (a) [ε = Blv = (0.30 × 10⁻⁴ T) × (10 m) × (5.0 m/s) = 1.5 × 10⁻³ V]
7. (b) [ε₂ = -M (dI₁/dt). I₁ = 10 sin(100πt). dI₁/dt = 10 × 100π cos(100πt) = 1000π cos(100πt). Max value of dI₁/dt is 1000π. Max |ε₂| = M × (Max |dI₁/dt|) = 0.005 H × 1000π A/s = 5π V]
8. (a) [U = ½ LI² = ½ (50 × 10⁻³ H) × (4 A)² = ½ × 50 × 10⁻³ × 16 = 400 × 10⁻³ J = 0.4 J]
9. (d) [ε₀ = NBAω]
10. (a) [|ε| = L |ΔI/Δt|. 50 V = L |(2 A - 5 A) / 0.1 s| = L |-3 A / 0.1 s| = L (30 A/s). L = 50 V / 30 A/s = 5/3 H ≈ 1.67 H]

Make sure you understand the underlying principles behind each concept and formula. Pay special attention to Lenz's Law applications and the factors affecting inductance. Good luck with your preparation!