Electromagnetic Induction Class 12 – Faraday’s Laws, NCERT Solutions & CBSE PYQs

Here are some important pointers regarding electromagnetic induction, drawing on the provided sources:

• Inter-relatedness of Electricity and Magnetism: Historically, electricity and magnetism were seen as separate phenomena. However, experiments in the early 19th century by Oersted and Ampere demonstrated that moving electric charges produce magnetic fields. This led to the converse question: Can moving magnets produce electric currents? Michael Faraday in England and Joseph Henry in USA conclusively showed around 1830 that electric currents are induced in closed coils when subjected to changing magnetic fields.
• Electromagnetic Induction Defined: This phenomenon, where electric current is generated by varying magnetic fields, is appropriately called electromagnetic induction. Its discovery is not just of theoretical interest but has immense practical utility, directly leading to the development of modern-day generators and transformers, which underpin today's civilization.
• Faraday and Henry's Key Experiments:
  • Experiment 6.1 (Magnet and Coil): When a bar magnet is moved towards or away from a coil, an electric current is induced in the coil, indicated by a galvanometer deflection. The deflection lasts only as long as the magnet is in motion. The direction of current reverses when the magnet's motion direction or its pole (North/South) is reversed. The induced current is larger with faster motion. This showed that relative motion between the magnet and the coil is responsible for current induction.
  • Experiment 6.2 (Two Coils): Replacing the magnet with a current-carrying coil (C2) also induces current in a nearby coil (C1) when C2 is moved towards or away from C1. Again, it is the relative motion between the coils that induces the current.
  • Experiment 6.3 (Stationary Coils with Changing Current): Faraday further demonstrated that relative motion isn't an absolute requirement. If two stationary coils are used, with one (C2) connected to a battery via a tapping key and the other (C1) to a galvanometer, a momentary deflection occurs in C1 when the key is pressed (current in C2 rises) or released (current in C2 falls). No deflection is observed if the key is held pressed continuously (current in C2 is constant). Inserting an iron rod dramatically increases the deflection.
• Underlying Principle: Change in Magnetic Flux: Faraday's great insight was that an electromotive force (emf) is induced in a coil when magnetic flux through the coil changes with time. The observations from the experiments confirm this:
  • In Experiments 6.1 and 6.2, the motion changes the magnetic flux linked with coil C1, inducing emf.
  • In Experiment 6.3, pressing or releasing the key causes the current in C2 (and thus its magnetic field) to change, which in turn changes the magnetic flux through the neighboring coil C1, producing an induced emf.
• Magnetic Flux (ΦB):
  • It is defined similarly to electric flux.
  • For a plane area A in a uniform magnetic field B, ΦB = B ⋅ A = BA cos θ, where θ is the angle between the magnetic field B and the area vector A.
  • It is a scalar quantity.
  • The SI unit of magnetic flux is weber (Wb) or tesla meter squared (T m²).
• Faraday's Law of Electromagnetic Induction:
  • The magnitude of the induced emf (ε) in a circuit is equal to the time rate of change of magnetic flux (ΦB) through the circuit.
  • Mathematically: ε = -dΦB/dt.
  • For a coil with N turns, where the flux through each turn is the same: ε = -N dΦB/dt.
  • The negative sign is significant and relates to Lenz's Law.
  • Induced emf can be increased by increasing the number of turns (N).
  • Magnetic flux can be varied by changing the magnetic field (B), the area of the coil (A), or the angle (θ) between the field and the area.
• Lenz's Law and Conservation of Energy:
  • Lenz's Law states that the polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.
  • The negative sign in Faraday's law represents this opposing effect.
  • This law is a direct consequence of the law of conservation of energy. If the induced current were to aid the change in flux, it would lead to a perpetual motion machine, which violates energy conservation. Instead, work must be done against the opposing force (e.g., repulsive force when moving a magnet towards a coil), and this energy is converted into electrical energy, often dissipated as Joule heating.
• Motional Electromotive Force (emf):
  • This is the emf induced when a conductor moves in a uniform and time-independent magnetic field.
  • For a straight conductor of length l moving with velocity v perpendicular to a magnetic field B, the induced motional emf is ε = Blv.
  • This can be explained by the Lorentz force acting on the free charge carriers within the moving conductor.
• Inductance:
  • Inductance is a property of a coil or circuit that relates the magnetic flux linkage to the current flowing through it.
  • Flux linkage (NΦB) is proportional to the current (I): NΦB ∝ I.
  • The constant of proportionality is called inductance (L).
  • Inductance depends on the geometry of the coil and the intrinsic material properties (like permeability of the core).
  • It is a scalar quantity.
  • The SI unit of inductance is henry (H), named in honor of Joseph Henry.
• Mutual Inductance (M):
  • This occurs when a changing current in one coil (coil 2) induces an emf in a nearby coil (coil 1).
  • The flux linkage in coil 1 due to current in coil 2 is N₁Φ₁ = M₁₂I₂.
  • The induced emf in coil 1 is ε₁ = -M dI₂/dt.
  • A general equality exists: M₁₂ = M₂₁ = M. This means the mutual inductance between two coils is the same regardless of which coil is carrying the current and inducing flux in the other.
• Self-Inductance (L):
  • This phenomenon occurs when emf is induced in a single isolated coil due to a change of flux through the coil caused by varying the current through the same coil.
  • The flux linkage in the coil is NΦB = LI.
  • The self-induced emf (also called back emf) is given by ε = -L dI/dt.
  • The self-induced emf always opposes any change (increase or decrease) of current in the coil.
  • Self-inductance plays the role of inertia in an electromagnetic circuit; it is the electromagnetic analogue of mass in mechanics, opposing the growth and decay of current.
  • The energy required to establish a current I in an inductor is stored as magnetic potential energy: W = ½ LI².
  • The magnetic energy density (energy per unit volume) in a magnetic field B is uB = B² / (2μ₀). This is analogous to the electrostatic energy density in an electric field.
• AC Generator:
  • An exceptionally important application of electromagnetic induction is the generation of alternating currents (ac).
  • An AC generator converts mechanical energy into electrical energy.
  • The basic principle involves a coil (armature) mounted on a rotor shaft, mechanically rotated in a uniform magnetic field.
  • The rotation of the coil causes the magnetic flux through it to change, inducing an emf.
  • If a coil of N turns and area A is rotated at an angular speed ω (or frequency n revolutions per second) in a uniform magnetic field B, the instantaneous induced emf is ε = NBAω sin ωt or ε = NBA(2πn) sin(2πnt).
  • The maximum value of the emf is ε₀ = NBAω.
  • The sign (polarity) of the emf changes with time, leading to an alternating current.
  • In large-scale commercial generators, mechanical energy is provided by sources like falling water (hydro-electric), steam from heated water (thermal), or nuclear fuel (nuclear power generators).
  • Modern generators can produce very high electric power outputs, for example, 500 MW.

📘 Chapter 6: Electromagnetic Induction – Class 12 CBSE

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🔑 Key Concepts & Derivations

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1. Faraday’s Laws of Electromagnetic Induction

• First Law: Whenever magnetic flux linked with a circuit changes, an emf is induced.

• Second Law: Magnitude of emf:

  $$\varepsilon = -\frac{d\Phi_B}{dt}$$

  where $\Phi_B = \vec{B} \cdot \vec{A}$

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2. Lenz’s Law

• The direction of induced current is such that it opposes the cause producing it.

• Expressed in the negative sign in Faraday’s law:

  $$\varepsilon = -\frac{d\Phi}{dt}$$

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3. Motional Electromotive Force (emf)

• emf induced in a moving conductor:

  $$\varepsilon = B l v \sin \theta$$

  For perpendicular motion: $\varepsilon = B l v$

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4. Eddy Currents

• Circulating currents induced in solid conductors due to changing magnetic flux.

• Applications: Induction furnace, speedometers, magnetic brakes.

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5. Self Induction

• emf induced in a coil due to change in its own current.

• Self-induced emf:

  $$\varepsilon = -L \frac{dI}{dt}$$

• $L$: self-inductance, SI unit: Henry (H)

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6. Mutual Induction

• emf induced in one coil due to change in current in a nearby coil.

• Mutually induced emf:

  $$\varepsilon = -M \frac{dI}{dt}$$

• $M$: mutual inductance

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7. Energy Stored in Inductor

$$U = \frac{1}{2} L I^2$$

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8. AC Generator (Alternator)

• Converts mechanical energy into electrical energy using electromagnetic induction.

• emf produced:

  $$\varepsilon = NBA \omega \sin(\omega t)$$

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📐 Important Formulas Recap

Concept	Formula
Faraday’s Law	$\varepsilon = -\frac{d\Phi}{dt}$
Motional emf	$\varepsilon = B l v$
Self-induced emf	$\varepsilon = -L \frac{dI}{dt}$
Mutual-induced emf	$\varepsilon = -M \frac{dI}{dt}$
Magnetic flux	$\Phi = B A \cos\theta$
Inductor Energy	$U = \frac{1}{2} L I^2$

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📗 NCERT Exercise Questions with Solutions

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Q1. A magnetic field of 100 G (1 G = 10⁻⁴ T) is perpendicular to a coil of 100 turns and radius 10 cm. The field is reduced to zero in 0.01 s. Find induced emf.

Answer:

$$\Phi = B \cdot A = 100 \times 10^{-4} \cdot \pi (0.1)^2 = \pi \times 10^{-3} \, Wb$$

Change in flux $\Delta \Phi = \pi \times 10^{-3}$

$$\varepsilon = \frac{N \Delta \Phi}{\Delta t} = \frac{100 \cdot \pi \cdot 10^{-3}}{0.01} = 10\pi \, V$$

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Q2. Derive expression for emf induced in a straight conductor moving in a magnetic field.

Answer:
From Lorentz force:
$F = qvB \Rightarrow \varepsilon = B l v$
where:

• $l$ = length of conductor

• $v$ = velocity perpendicular to field

• $B$ = magnetic field strength

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Q3. Define self-inductance and derive its expression.

Answer:

Self-inductance is the property of a coil by virtue of which it opposes any change in current flowing through it.

$$\varepsilon = -L \frac{dI}{dt}, \quad L = \frac{N \Phi}{I}$$

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Q4. What is eddy current? State any two applications.

Answer:
Eddy currents are loops of induced currents formed in conductors when exposed to changing magnetic fields.

Applications:

• Magnetic brakes in trains

• Induction furnace

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Q5. Explain the working of an AC generator with diagram.

Answer:

• A coil rotates in a uniform magnetic field.

• Flux linkage changes → emf is induced.

• Emf:

  $$\varepsilon = NBA\omega \sin(\omega t)$$

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📋 CBSE Previous Year Questions (PYQs) with Best Answers

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🔹 Q (2023)

State Lenz’s law. Show how it obeys law of conservation of energy.

Answer:
Lenz’s law: Induced current opposes the cause that produces it.

Energy Conservation: The opposing current requires work to be done, thus energy is conserved by converting mechanical → electrical energy.

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🔹 Q (2022)

A rod of length 0.5 m moves with speed 2 m/s in perpendicular magnetic field of 0.3 T. Find emf.

Answer:

$$\varepsilon = B l v = 0.3 \times 0.5 \times 2 = 0.3 \, V$$

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🔹 Q (2021)

Define mutual inductance and give its SI unit.

Answer:
Mutual inductance: The property of a pair of coils where change in current in one induces emf in the other.
Unit: Henry (H)

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🔹 Q (2020)

Describe principle and working of an AC generator.

Answer:

• Based on electromagnetic induction.

• Rotating coil cuts magnetic flux → emf induced:

  $$\varepsilon = NBA \omega \sin(\omega t)$$

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