Electromagnetic Induction 2

Electromagnetic Induction

Electromagnetic Induction is a fundamental principle in electromagnetism that describes how a magnetic field interacts with an electric circuit to produce an electromotive force (EMF), and consequently, a current. This phenomenon was first discovered by Michael Faraday in the 1830s.

Discovery and Basic Principle

• When a DC current passes through a long straight conductor, a magnetizing force and a static magnetic field are developed around it.
• Faraday observed that moving a permanent magnet in and out of a coil or a single loop of wire induced an electromotive force (voltage) and thus a current. This is called electromagnetic induction.
• Similarly, moving a coil back and forth within a stationary magnetic field also induces a voltage and current.
• This principle forms the basis of operation for transformers, motors, and generators.

Faraday's Laws of Electromagnetic Induction

Faraday's Laws consist of two main laws:

1. Faraday's First Law

• Statement: Any change in the magnetic field (magnetic flux) of a coil of wire will cause an electromotive force (EMF) to be induced in the coil. If the conductor circuit is closed, an induced current will circulate through it.
• Methods to change the magnetic field:
  1. Moving a magnet towards or away from the coil.
  2. Moving the coil into or out of the magnetic field.
  3. Changing the area of a coil placed in the magnetic field.
  4. Rotating the coil relative to the magnet.
• Observation: The needle of a galvanometer deflects when there is relative motion between the magnet and the coil. When the magnet or coil is stationary, the needle returns to zero. The direction of deflection changes with the direction of relative motion, indicating a change in polarity.

2. Faraday's Second Law

• Statement: The magnitude of the EMF induced in the coil is equal to the rate of change of flux linkage. The flux linkage of the coil is the product of the number of turns in the coil and the magnetic flux associated with the coil.
• Mathematical Formula:

  E = N (dΦ / dt)

  Where:

  • E is the induced electromotive force (EMF).
  • N is the number of turns in the coil.
  • dΦ / dt is the rate of change of magnetic flux (Φ) with respect to time (t).
• Derivation (Simplified):

  If a magnet approaches a coil, flux linkage at time T₁ is NΦ₁ and at T₂ is NΦ₂. The change in flux linkage is N(Φ₂ - Φ₁) = NΔΦ. If this change occurs over time Δt, the rate of change of flux linkage is NΔΦ/Δt. Taking the derivative, E = N dΦ/dt.

How to Increase Induced EMF in a Coil

• Increasing the number of turns (N) in the coil: A higher number of turns leads to greater flux linkage and thus increased induced EMF.
• Increasing magnetic field strength (B): A stronger magnetic field produces more magnetic flux (Φ = BA), leading to a greater rate of change of flux and increased induced EMF.
• Increasing the speed of relative motion: A faster relative movement between the coil and the magnet causes the coil to cut magnetic lines of flux at a quicker rate, resulting in a larger induced EMF.

Applications of Faraday's Law

• Transformers: Electrical equipment that uses mutual induction to change AC voltage levels.
• Induction Cookers: Operate on the principle of mutual induction to generate heat in cookware.
• Electromagnetic Flowmeters: Induce an EMF in a flowing fluid to measure its velocity.
• Electric Musical Instruments: Electric guitars and violins use electromagnetic induction to convert string vibrations into electrical signals.
• Maxwell's Equations: The converse of Faraday's law, stating that a changing magnetic field brings a change in the electric field, is integral to Maxwell's equations.

Fleming's Left and Right-Hand Rules

These rules are used to determine the direction of force, current, or magnetic field when two of the three are known. They do not determine magnitude.

Fleming's Left-Hand Rule (for Motors)

• Purpose: Determines the direction of the force acting on a current-carrying conductor placed in a magnetic field.
• Principle: When a current-carrying conductor is placed in a magnetic field, a force acts on the conductor, perpendicular to both the direction of the current and the magnetic field.
• Hand Orientation:
  • Thumb: Direction of Force (Motion).
  • Forefinger: Direction of Magnetic Field.
  • Middle finger: Direction of Current.
• Formula for Force Magnitude: F = BiL (for a conductor perpendicular to the field)
  • F is the force.
  • B is the magnetic field strength (flux density).
  • i is the current.
  • L is the length of the conductor.
• Interaction Explanation: The magnetic field produced by the current in the conductor interacts with the external magnetic field. Where the fields are in the same direction, they reinforce; where they are opposite, they weaken. This creates a region of higher magnetic field concentration on one side, resulting in a force that pushes the conductor from the more concentrated field to the less concentrated field.

Fleming's Right-Hand Rule (for Generators)

• Purpose: Determines the direction of the induced current in a conductor moving inside a magnetic field.
• Principle: As per Faraday's law, when a conductor moves inside a magnetic field, an induced current is generated. This rule gives the direction of that current.
• Hand Orientation:
  • Thumb: Direction of Motion (Applied Force).
  • Forefinger: Direction of Magnetic Field.
  • Middle finger: Direction of Induced Current.

Lenz's Law of Electromagnetic Induction

• Statement: "The direction of an induced EMF is such that it will always oppose the change that is causing it." In other words, an induced current will always oppose the motion or change that started the induction.
• Relation to Faraday's Law: Lenz's Law provides the negative sign in Faraday's formula, indicating the opposing nature of the induced EMF:

  E = -N (dΦ / dt)

• Conservation of Energy: Lenz's Law is a consequence of the law of conservation of energy. If the induced current aided the change, it would create a perpetual motion machine, violating energy conservation.
• Magnetic Flux Change: If magnetic flux increases, the induced EMF generates a magnetic flux that opposes this increase. If magnetic flux decreases, the induced EMF generates a magnetic flux that adds to the original flux, opposing the decrease.

Eddy Currents

• Definition: Eddy currents are circulating currents induced in a bulk conductor when it is subjected to a changing magnetic field. They flow in closed loops in a plane perpendicular to the magnetic field. By Lenz's law, these currents create a magnetic field that opposes the change causing them.
• Energy Loss: Eddy currents cause energy loss by transforming useful forms of energy (like kinetic energy) into heat, which is generally undesirable.
• In Transformers:
  • The changing magnetic flux in a transformer's iron core induces EMF not only in the windings but also in the core itself.
  • Since iron is a good conductor, these induced currents (eddy currents) in a solid iron core can be large.
  • By Lenz's law, these eddy currents flow in a direction that opposes the flux created by the primary coil, effectively weakening it.
  • This requires the primary coil to draw more current to produce the desired magnetic field, leading to increased power loss and "fatter" hysteresis curves.
  • To reduce eddy currents, transformer cores are typically laminated (made of thin insulated sheets) rather than solid.

Induction - Self Induction and Mutual Induction

What is Induction?

Induction is the process by which:

• An electrical conductor becomes electrified when near a charged body.
• A magnetizable body becomes magnetized when in a magnetic field.
• An electromotive force is produced in a circuit by varying the magnetic field linked with the circuit.

Induction, in the context of coils, is also known as Inductance (symbol L, SI unit Henry).

• 1 Henry: Defined as the amount of inductance required to produce an EMF of 1 volt when the current changes at the rate of 1 Ampere per second.
• Factors Affecting Inductance:
  1. The number of turns of the wire in the inductor.
  2. The material used in the core (e.g., its permeability).
  3. The shape and geometry of the core.
• Faraday's Law for Inductance:

  EMF = -L (ΔI / Δt)

  This defines inductance as the EMF generated to oppose the change in current.

Types of Inductance:

1. Self Induction
2. Mutual Induction

Self Induction

• Definition: When there is a change in the current or magnetic flux of a coil, an opposed induced electromotive force is produced in the same coil. The magnetic flux (Φ) becomes directly proportional to the current (I) passing through the circuit.
• Relation: Φ = LI
• Where L is the self-inductance of the coil (coefficient of self-inductance).
• Factors affecting Self-Inductance: Cross-sectional area, permeability of the material, and number of turns in the coil.
• Rate of change of magnetic flux (induced EMF):

  e = - (dΦ / dt) = - (d(LI) / dt)

  e = -L (dI / dt)

• Self Inductance Formula:

  L = N (Φ / I)

  Where:

  • L is self-inductance in Henry.
  • N is the number of turns.
  • Φ is the magnetic flux.
  • I is the current in amperes.
• Derivation of Inductance (for a solenoid):
  • From NΦ = LI and Φ = BA, we get N(BA) = LI.
  • Also, for a solenoid, magnetizing force Hl = Ni, so H = Ni/l.
  • And B = μH = μ(Ni/l), where μ = μ₀μᵣ (permeability).
  • Substituting B into LI = N(BA):

    LI = N (μ(Ni/l)) A

    L = (μN²A) / l

    If A = πr², then L = (μN²πr²) / l

Mutual Induction

• Definition: When a change in current flowing through one coil (primary coil) produces an EMF in a neighboring coil (secondary coil). This induced EMF is called mutually induced EMF.
• Relation: The magnetic flux (Φ) linked with the secondary coil due to the current in the primary coil is proportional to that current.

  Φ = MI

• Where M is the mutual inductance of the two coils.
• Rate of change of magnetic flux (induced EMF):

  e = - (dΦ / dt) = - (d(MI) / dt)

  e = -M (dI / dt)

• Mutual Inductance Formula (for two coils):

  M = (μ₀μᵣN₁N₂A) / l

  Where:

  • μ₀ is the permeability of free space.
  • μᵣ is the relative permeability of the soft iron core.
  • N₁, N₂ are the number of turns in coil 1 and coil 2 respectively.
  • A is the cross-sectional area.
  • l is the length of the coil.

Difference between Self and Mutual Inductance

Examples

Example 1 (Faraday's Law)

Problem: A circular coil of 4 turns with radius 3×10⁻² mm is subjected to a varying magnetic field that changes uniformly from 0.4 T to 3.4 T in an interval of 27 s. The axis of the solenoid makes an angle of 35° to the magnetic field. Find the induced EMF.

Solution:

• N = 4 turns
• r = 3 × 10⁻² mm = 3 × 10⁻⁵ m
• Bᵢ = 0.4 T
• Bբ = 3.4 T
• Δt = 27 s
• θ = 35°
• Area A = πr² = π(3 × 10⁻⁵ m)²
• Induced EMF E = N (ΔΦ / Δt) = N (Δ(BA cosθ) / Δt)
• E = N * A * cosθ * (ΔB / Δt)
• E = N * A * cosθ * ((Bբ - Bᵢ) / Δt)
• Plugging in values: E = 4 * π(3×10⁻⁵)² * cos(35°) * ((3.4 - 0.4) / 27)
• E ≈ 1.03 × 10⁻¹⁰ V (Note: The provided document's example had 3x10⁻² mm, but the solution implies 3x10⁻² m for a reasonable EMF value, or the original problem statement might have intended mm to be m. Assuming 3x10⁻² m results in a larger EMF, matching the provided order of magnitude in the solution for a similar setup: 1.03x10⁻³ V if radius was 3x10⁻²m, not mm)

Example 2 (Self-Inductance)

Problem: A solenoid with 500 turns is wound on an iron core whose relative permeability is 800. The length of the solenoid is 40 cm, and its radius is 3 cm. The current changes from 0 to 3 A in 0.4 seconds. Calculate the average EMF induced.

Solution:

• Given:
  • N = 500 turns
  • μᵣ = 800
  • l = 40 cm = 0.4 m
  • r = 3 cm = 0.03 m
  • Δi = 3 A - 0 A = 3 A
  • Δt = 0.4 s
  • μ₀ = 4π × 10⁻⁷ T m A⁻¹
  • Permeability μ = μ₀μᵣ
  • Area A = πr² = π(0.03 m)²
• Self-inductance L = (μN²A) / l = (μ₀μᵣN²πr²) / l
• Substitute values:

  L = (4π × 10⁻⁷ * 800 * 500² * π * (0.03)²) / 0.4

  L ≈ 1.77 H

• Magnitude of induced EMF ε = L (Δi / Δt)
• ε = 1.77 H * (3 A / 0.4 s)
• ε ≈ 13.275 V