Electromagnetic Induction

6.1 INTRODUCTION

**Electromagnetic Induction** is the phenomenon in which electric currents are generated in conductors by varying magnetic fields. This chapter bridges the gap between electricity and magnetism, establishing that moving electric charges produce magnetic fields and changing magnetic fields produce electric currents.

**Historical Context:**

Early 19th century: Oersted, Ampere discovered that electric current deflects magnetic compass needles — electricity and magnetism are related phenomena

1830: Michael Faraday (England) and Joseph Henry (USA) demonstrated conclusively that changing magnetic fields induce electric currents in closed loops

**Practical Importance:** All modern electrical generators, transformers, and alternating current systems depend on electromagnetic induction. Today's technological civilization is built on this discovery.

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6.2 THE EXPERIMENTS OF FARADAY AND HENRY

Experiment 6.1: Magnet and Coil Relative Motion

**Setup:** A bar magnet moved toward/away from a stationary coil connected to a galvanometer.

**Key Observations:**

**Deflection occurs only during motion** — when magnet is stationary, no deflection (no current)

**Direction reverses** when magnet moves in opposite direction

**Magnitude increases** with faster motion (higher rate of change of flux)

**Effect is mutual:** Same results if magnet is held fixed and coil moves

**Conclusion:** It is **relative motion** between magnet and coil that induces current, not absolute motion

**Physical Principle:** The relative motion changes the magnetic flux through the coil, inducing an emf.

Experiment 6.2: Two Coils — Moving Current-Carrying Coil

**Setup:** A coil C₂ (connected to battery) is moved toward/away from stationary coil C₁ (connected to galvanometer).

**Key Observations:**

Current is induced in C₁ when C₂ moves

Direction of induced current reverses when direction of motion reverses

Effect disappears when C₂ is stationary

Larger deflection when motion is faster

**Conclusion:** Again, **relative motion** between two coils causes induction

**Significance:** No permanent magnet needed; time-varying magnetic field from a moving current-carrying coil can induce current in another coil.

Experiment 6.3: Stationary Coils with Changing Current

**Setup:** Two coils C₁ and C₂ held **stationary**. C₂ connected to battery through tapping key K; C₁ connected to galvanometer.

**Key Observations:**

**Momentary deflection when key is pressed** — current in C₂ rises from zero to maximum

**No deflection while key is held pressed** — current in C₂ is constant, no change in flux

**Momentary deflection in opposite direction when key is released** — current in C₂ falls from maximum to zero

**Deflection increases dramatically** when iron core is inserted along coil axis (increases magnetic field and flux change)

**Critical Insight:** **Relative motion is NOT required.** What matters is **change in magnetic flux through the circuit**, whether due to motion or change in current (and hence magnetic field).

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6.3 MAGNETIC FLUX

**Definition:** Magnetic flux (Φ_B) is the measure of the total magnetic field passing through a surface. It quantifies how much magnetic field "pierces" through an area.

Magnetic Flux Through a Uniform Field

For a plane surface of area **A** in uniform magnetic field **B**:

**Φ_B = B · A = BA cos θ** [Equation 6.1]

Where:

**B** = magnitude of magnetic field (Tesla, T)

**A** = area of surface (m²)

**θ** = angle between magnetic field vector **B** and area vector **A** (normal to surface)

**Φ_B** = magnetic flux (Weber, Wb) or (T·m²)

**Key Points:**

Flux is **maximum** when θ = 0° (field perpendicular to surface): Φ_B = BA

Flux is **zero** when θ = 90° (field parallel to surface): Φ_B = 0

Flux can be **negative** (field pointing inward) or **positive** (field pointing outward)

Magnetic flux is a **scalar quantity** (has magnitude and sign, not direction)

Magnetic Flux Through Non-Uniform Fields or Curved Surfaces

For surfaces where field is non-uniform or surface is curved:

**Φ_B = Σ B_i · dA_i = ∫ B · dA** [Equation 6.2]

Where:

**B_i** = magnetic field at area element i

**dA_i** = small area element (vector, normal to surface)

Summation is over all area elements

**SI Unit:**

**1 Weber (Wb) = 1 T·m²**

1 Wb = 10⁸ Maxwell (older CGS unit)

**Example:** A circular loop of radius 0.1 m perpendicular to a magnetic field of 0.5 T has flux:

Φ_B = BA = 0.5 × π(0.1)² = 0.005π Wb ≈ 0.0157 Wb

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6.4 FARADAY'S LAW OF ELECTROMAGNETIC INDUCTION

**Faraday's Great Insight:** After analyzing all his experiments, Faraday discovered that an **emf (electromotive force) is induced in a circuit when the magnetic flux through it changes with time**. The magnitude of induced emf depends on **how fast** the flux changes.

Faraday's Law — Mathematical Statement

**ε = -dΦ_B/dt** [Equation 6.3]

**For N-turn coil:**

**ε = -N(dΦ_B/dt)** [Equation 6.4]

Where:

**ε** = induced emf (Volt, V)

**dΦ_B/dt** = rate of change of magnetic flux (Wb/s or V)

**N** = number of turns in coil

**Negative sign** = represents direction of induced emf (explained in Lenz's Law section)

Physical Interpretation of Faraday's Law

1. **Magnitude of induced emf is proportional to:**

**Rate of change of magnetic flux** (dΦ_B/dt) — faster change → larger emf

**Number of turns N** — more turns → larger total emf

2. **The induced emf opposes the change** that produces it (negative sign)

3. **Flux can change due to:**

**Change in magnetic field B** (e.g., magnet approaching coil)

**Change in area A** (e.g., expanding/contracting loop or moving conductor)

**Change in angle θ** (e.g., rotating coil in magnetic field)

**Any combination** of above

Derivation of Faraday's Law (for moving conductor in uniform field)

Consider rectangular loop PQRS with one side PQ of length l moving with velocity v perpendicular to uniform magnetic field B (perpendicular to loop plane):

**Φ_B = B · A = B · l · x**

where x = variable distance from reference

**dΦ_B/dt = Bl(dx/dt) = Blv**

**ε = -dΦ_B/dt = -Blv** (magnitude: **ε = Blv**)

This is the motional emf formula for a straight conductor moving in a magnetic field.

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6.5 LENZ'S LAW AND CONSERVATION OF ENERGY

Lenz's Law — Statement

**The polarity (direction) of the induced emf is such that the induced current opposes the change in magnetic flux that produced it.**

The **negative sign** in Faraday's law (Equation 6.3) mathematically represents Lenz's law.

Physical Explanation with Examples

**Example 1: Magnet Approaching Coil**

**Situation:** North pole of magnet pushed toward coil

**Flux change:** Magnetic flux through coil **increases**

**Induced current direction:** Flows to create magnetic field opposing this increase

**Induced field:** Must have North pole facing approaching North pole of magnet (repulsive)

**Result:** Induced current flows **counter-clockwise** (from observer's view on magnet side)

**Example 2: Magnet Receding from Coil**

**Situation:** North pole of magnet pulled away from coil

**Flux change:** Magnetic flux through coil **decreases**

**Induced current direction:** Flows to create magnetic field opposing this decrease

**Induced field:** Must have South pole facing receding North pole of magnet (attractive)

**Result:** Induced current flows **clockwise** (from observer's view on magnet side)

Connection to Conservation of Energy

**Why Lenz's law must be true:**

If induced current could **aid** rather than oppose flux change, then:

1. Magnet approaching coil would be attracted faster and faster

2. Its kinetic energy and velocity would increase continuously

3. No external work needed — perpetual motion machine possible

4. **Violates Law of Conservation of Energy**

**Energy balance in reality:**

When magnet approaches or recedes from coil, person doing work against magnetic force

**Work done by person = Heat dissipated** as Joule heat in coil resistance

Energy is conserved; perpetual motion is impossible

**For moving loops in magnetic field:**

Mechanical work done in moving loop = Electrical energy dissipated in resistance

**Power dissipated: P = ε²/R = (Blv)²/R**

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6.6 MOTIONAL ELECTROMOTIVE FORCE

Definition and Physical Principle

**Motional emf** is the emf induced in a straight conductor moving through a uniform magnetic field due to the change in magnetic flux through the circuit containing the conductor.

Derivation for Straight Conductor

**Setup:** Rectangular conductor PQRS with movable arm PQ of length l, moving with constant velocity v perpendicular to uniform magnetic field B (perpendicular to plane of loop).

**Step 1 — Express magnetic flux:**

If distance RQ = x (changing with time),

**Φ_B = B · A = B · l · x**

**Step 2 — Find rate of change of flux:**

**dΦ_B/dt = Bl(dx/dt) = Bl · v**

(since dx/dt = velocity v)

**Step 3 — Apply Faraday's Law:**

**ε = -dΦ_B/dt = -Blv**

**Magnitude of motional emf:**

**ε = Blv** [Equation 6.5]

Where:

**B** = magnetic field strength (T)

**l** = length of conductor perpendicular to both B and v (m)

**v** = velocity of conductor (m/s)

**ε** = induced emf (V)

Physical Interpretation

1. **Lorentz force on charge carriers:** When conductor moves through magnetic field, free electrons in conductor experience Lorentz force **F = q(v × B)**

2. **Charge separation:** Electrons accumulate at one end (say Q), leaving positive charges at other end (P)

3. **Potential difference develops:** This charge separation creates an emf between ends P and Q

4. **Direction (Right-hand rule):** If **v** points left and **B** points out of page, then **v × B** points upward, so electrons accumulate at top → bottom end becomes positive

Induced Current in Motional Emf

If conductor PQ moves in closed circuit of total resistance R:

**Current: I = ε/R = Blv/R**

**Direction:** Using Lenz's law — current direction opposes flux increase/decrease:

**If loop area increases:** Current direction creates field opposing applied field

**If loop area decreases:** Current direction creates field reinforcing applied field

Practical Examples

**Example 1: Rod Moving in Uniform Field**

Rod PQ, length 0.2 m

Magnetic field B = 0.5 T (perpendicular to plane)

Velocity v = 2 m/s

**ε = Blv = 0.5 × 0.2 × 2 = 0.2 V**

**Example 2: Generator Principle**

When conductor rotates in magnetic field (like turbine blade), motional emf is generated

This is the principle of AC generators and dynamos

Multiple conductors (coil) increase total emf: **ε_total = NBlv**

Key Distinctions

| Aspect | Motional EMF | Induced EMF (General) |

|--------|-------------|----------------------|

| **Cause** | Moving conductor in static B field | Changing magnetic flux (any reason) |

| **Formula** | ε = Blv | ε = -dΦ_B/dt |

| **Mechanism** | Lorentz force on charge carriers | Faraday's law |

| **Examples** | Moving rod, rotating coil, vehicle speedometer | Transformer, changing current coil |

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6.4 (Extended) NUMERICAL APPLICATIONS OF FARADAY'S LAW

Worked Example 6.2: Square Loop in Changing Magnetic Field

**Problem:** Square loop of side 10 cm and resistance 0.5 Ω placed vertically in east-west plane. Uniform magnetic field 0.10 T set up in northeast direction (45° to plane). Field decreases to zero in 0.70 s at steady rate. Find induced emf and current.

**Solution:**

**Step 1 — Find initial magnetic flux:**

Area: A = (10 × 10⁻²)² = 10⁻² m²

Angle between B and area normal: θ = 45°

**Φ_B(initial) = BA cos θ = 0.10 × 10⁻² × cos 45° = 10⁻³ × (1/√2) = 0.707 × 10⁻³ Wb**

**Step 2 — Find final flux:**

**Φ_B(final) = 0** (field becomes zero)

**Step 3 — Calculate rate of change:**

**ΔΦ_B = 0 - 0.707 × 10⁻³ = -0.707 × 10⁻³ Wb**

**Δt = 0.70 s**

**dΦ_B/dt = -0.707 × 10⁻³ / 0.70 = -1.01 × 10⁻³ Wb/s**

**Step 4 — Apply Faraday's Law (N=1):**

**ε = |dΦ_B/dt| = 1.0 × 10⁻³ V = 1.0 mV**

**Step 5 — Calculate current:**

**I = ε/R = (1.0 × 10⁻³) / 0.5 = 2.0 × 10⁻³ A = 2.0 mA**

**Note:** Earth's magnetic field also present but doesn't contribute because it's constant (steady field produces no emf).

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Worked Example 6.3: Rotating Coil in Earth's Magnetic Field

**Problem:** Circular coil, radius 10 cm, 500 turns, resistance 2 Ω placed with plane perpendicular to horizontal component of earth's field. Rotated 180° about vertical diameter in 0.25 s. Earth's field: B_h = 3.0 × 10⁻⁵ T. Find induced emf and current.

**Solution:**

**Step 1 — Find initial flux (θ = 0°):**

Area: A = π × (10 × 10⁻²)² = π × 10⁻² m²

**Φ_B(initial) = B_h × A × cos 0° = 3.0 × 10⁻⁵ × π × 10⁻² × 1**

**Φ_B(initial) = 3π × 10⁻⁷ Wb**

**Step 2 — Find final flux after 180° rotation (θ = 180°):**

**Φ_B(final) = B_h × A × cos 180° = 3.0 × 10⁻⁵ × π × 10⁻² × (-1)**

**Φ_B(final) = -3π × 10⁻⁷ Wb**

**Step 3 — Calculate total flux change:**

**ΔΦ_B = Φ_B(final) - Φ_B(initial) = -3π × 10⁻⁷ - 3π × 10⁻⁷ = -6π × 10⁻⁷ Wb**

**Step 4 — Apply Faraday's Law for N turns:**

**ε = N × |ΔΦ_B/Δt| = 500 × (6π × 10⁻⁷) / 0.25**

**ε = 500 × 24π × 10⁻⁷ = 12,000π × 10⁻⁷ V**

**ε = 3.77 × 10⁻³ V ≈ 3.8 mV**

**Step 5 — Calculate current:**

**I = ε/R = 3.77 × 10⁻³ / 2 = 1.885 × 10⁻³ A ≈ 1.9 mA**

**Important Note:** These are **estimated average values**. Instantaneous emf and current vary during rotation depending on rotational speed at each instant. Values are averages over 0.25 s time interval.

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Worked Example 6.4: Direction of Induced Current Using Lenz's Law

**Problem:** Three planar loops (rectangular, triangular, irregular) moving out of/into uniform magnetic field region perpendicular to loop plane (field pointing away from reader). Determine induced current direction.

**Solutions Using Lenz's Law:**

**(i) Rectangular loop entering field region:**

**Flux change:** Magnetic flux **increases** as more area enters field

**Opposition needed:** Induced current creates field opposing this increase (into the page)

**Current direction:** By right-hand rule, **flows counter-clockwise** (path: b→c→d→a→b)

**(ii) Triangular loop exiting field region:**

**Flux change:** Magnetic flux **decreases** as loop leaves field

**Opposition needed:** Induced current creates field aiding field (out of page) to oppose decrease

**Current direction:** By right-hand rule, **flows clockwise** (path: b→a→c→b)

**(iii) Irregular loop exiting field region:**

**Flux change:** Magnetic flux **decreases** as loop moves out

**Opposition needed:** Induced field should point out of page (same as original field)

**Current direction:** **Flows counter-clockwise** (path: c→d→a→b→c)

**Key Principle:** Use Lenz's law in three steps:

1. Determine if flux is increasing or decreasing

2. Determine field direction needed to oppose this change

3. Use right-hand rule: curl fingers in current direction, thumb points in field direction

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Worked Example 6.5: Analysis of Various Physical Situations

**Problem (a):** Stationary closed loop in fixed magnetic field of strong permanent magnets. Can current be generated?

**Answer:** **No.** However strong the magnets:

Flux through loop is constant (time-independent)

dΦ_B/dt = 0

ε = -dΦ_B/dt = 0

No current generated

**Key principle:** Only **changing flux** induces emf; static fields do not

**Problem (b):** Closed loop moving normal to constant electric field between capacitor plates. Electric field normal to loop plane.

**Cases:**

**(i) Loop entirely inside capacitor:** Electric field is uniform and constant throughout loop region

**No change in electric flux** through loop

**No emf induced** (electromagnetic induction depends on magnetic flux, not electric flux)

**(ii) Loop partially outside capacitor:** Loop partially in field, partially in zero-field region

**Change in electric flux** is occurring

**Still no emf induced** (only time-varying **magnetic** fields induce emf in conductors, not electric fields)

**Answer:** **No current in either case.**

**Problem (c):** Rectangular loop vs circular loop both moving out of uniform magnetic field region at constant velocity v. Which has constant induced emf?

**Analysis:**

**Rectangular Loop:**

As rectangle exits field, straight edge always cuts field at constant rate

Area changing uniformly: dA/dt = l × v = constant (where l = width)

**dΦ_B/dt = B × dA/dt = constant**

**ε = Blv = constant**

**Answer: Rectangular loop has constant emf**

**Circular Loop:**

As circle exits, curved boundary means area changes non-uniformly

At side edges: dA/dt is maximum; at top/bottom: dA/dt is less

**dA/dt is not constant**

**dΦ_B/dt varies** during exit

**ε is not constant**

**Conclusion:** Rectangular loop experiences constant emf during uniform exit; circular loop does not.

**Problem (d):** Rotating rod between capacitor plates (rod and plates both rotating at angular velocity ω). Predict capacitor polarity.

**Setup from Figure 6.9:**

Rotating rod between capacitor plates oriented radially

Rod rotates with plates, both at angular velocity ω

Magnetic field present during rotation (rotating charged rod creates field)

**Solution:**

As rod rotates, free electrons experience centrifugal force (in rotating frame) and magnetic Lorentz force

Electrons accumulate at one end (e.g., outer end at larger radius)

Outer end becomes **negative**, inner end becomes **positive**

If plate A is connected to inner end and plate B to outer end:

**Plate A polarity: Positive** (relative to plate B)

**Plate B polarity: Negative** (relative to plate A)

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EXAM-IMPORTANT SUMMARY TABLE

| Concept | Formula | Key Points |

|---------|---------|-----------|

| **Magnetic Flux** | Φ_B = BA cos θ | Maximum when θ=0°; scalar quantity; unit = Wb = T·m² |

| **Faraday's Law (1 turn)** | ε = \|dΦ_B/dt\| | Magnitude only; direction from Lenz's law |

| **Faraday's Law (N turns)** | ε = N\|dΦ_B/dt\| | N-turn coil increases emf by factor N |

| **Motional EMF** | ε = Blv | For straight conductor length l, perpendicular to B and v |

| **Induced Current** | I = ε/R | Requires complete circuit; opposes flux change per Lenz |

| **Lenz's Law** | Direction opposes Δ flux | Conservative: prevents perpetual motion; conserves energy |

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IMPORTANT FORMULAS FOR QUICK REFERENCE

**Magnetic Flux (uniform field):** Φ_B = BA cos θ

**Magnetic Flux (non-uniform):** Φ_B = ∫B·dA

**Faraday's Law:** ε = -N(dΦ_B/dt)

**Motional EMF:** ε = Blv

**Induced Current:** I = ε/R = N|dΦ_B/dt|/R

**Power dissipated:** P = I²R = ε²/R = (Blv)²/R

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BOARD EXAM PREPARATION CHECKLIST

✓ Understand all three Faraday-Henry experiments and what each demonstrates

✓ Define magnetic flux and distinguish from electric flux

✓ State Faraday's law mathematically and explain negative sign using Lenz's law

✓ Apply Lenz's law using right-hand rule for current direction (at least 5 different scenarios)

✓ Derive motional emf (ε = Blv) from Faraday's law starting with Φ_B = Blx

✓ Solve numerical problems: finding emf, current, flux change with different loop geometries

✓ Analyze rotating coils: flux changes as coil rotates through magnetic field

✓ Explain energy conservation in context of Lenz's law (work done vs heat dissipated)

✓ Distinguish between motional emf and transformer emf (both follow Faraday's law)

✓ Multiple choice: recognizing when flux changes vs when it doesn't