Faraday’s Law of Induction- Mutual and Self-Inductance - Mutual inductance

  • Faraday’s law of induction states that a changing magnetic field induces an electromotive force (EMF) in a closed circuit.
  • Mutual inductance is the ability of two circuits to induce an electromotive force in each other.
  • Self-inductance is the ability of a circuit to induce an electromotive force in itself.
  • Mutual inductance is denoted by M and is measured in henries (H).
  • Self-inductance is denoted by L and is also measured in henries (H).

Mutual Inductance

  • Mutual inductance occurs when two circuits, A and B, are in close proximity to each other.
  • The change in current in circuit A induces a magnetic field, which in turn induces an EMF in circuit B.
  • The induced EMF in circuit B is given by the formula:
    • EMF_B = -M * (dI_A / dt)

Example of Mutual Inductance

  • Consider two coils, coil A and coil B, placed close to each other.
  • When the current in coil A changes, it induces an EMF in coil B.
  • Suppose the current in coil A is increasing at a rate of 2 A/s.
  • If the mutual inductance between the two coils is 5 H, the induced EMF in coil B is given by:
    • EMF_B = -5 * (2 / 1) = -10 V

Self-Inductance

  • Self-inductance occurs when a changing current in a circuit induces an EMF in the same circuit.
  • The induced EMF is proportional to the rate of change of current through the circuit.
  • The induced EMF is given by the formula:
    • EMF = -L * (dI / dt)

Example of Self-Inductance

  • Consider a coil with self-inductance L.
  • If the current through the coil changes at a rate of 3 A/s, the induced EMF in the coil is given by:
    • EMF = -L * (3 / 1)

Induced EMF and Faraday’s Law

  • Faraday’s law of induction relates the induced EMF to the change in magnetic flux.
  • The induced EMF is given by the formula:
    • EMF = -N * (dΦ / dt), where N is the number of turns in the coil.

Example of Faraday’s Law

  • Suppose the magnetic flux through a coil changes at a rate of 4 Wb/s.
  • If the coil has 100 turns, the induced EMF in the coil is given by:
    • EMF = -100 * (4 / 1) = -400 V

Lenz’s Law

  • Lenz’s law states that the induced current in a circuit will always oppose the change that caused it.
  • This law is based on the principle of conservation of energy.
  • The negative sign in the formulas for induced EMF indicates the opposition to the change.

Applications of Inductance

  • Mutual inductance is used in transformers to transfer electrical energy from one circuit to another.
  • Self-inductance is used in inductors to store and release energy in a circuit.
  • Inductors are commonly used in power supplies, electric motors, and electronic filters.

Summary

  • Faraday’s law of induction describes how a changing magnetic field induces an EMF in a closed circuit.
  • Mutual inductance occurs when two circuits induce EMF in each other.
  • Self-inductance occurs when a circuit induces EMF in itself.
  • The induced EMF is determined by the rate of change of current or magnetic flux.
  1. Mutual Inductance (Continued)
  • The magnitude of mutual inductance, M, depends on the physical arrangement of the two circuits.
  • It is given by the equation:
    • M = (μ₀ * N₁ * N₂ * A) / l where μ₀ is the permeability of free space, N₁ and N₂ are the number of turns in the two coils, A is the area of overlap between the two coils, and l is the distance between the two coils.
  1. Example of Mutual Inductance (Continued)
  • Suppose two coils, A and B, have 200 turns each.
  • The area of overlap between them is 0.04 m², and the distance between them is 0.1 m.
  • If the permeability of free space, μ₀, is 4π × 10⁻⁷ T m/A, the mutual inductance, M, is given by:
    • M = (4π × 10⁻⁷ * 200 * 200 * 0.04) / 0.1
    • M = 5.04 × 10⁻⁶ H
  1. Self-Inductance (Continued)
  • Self-inductance is determined by the geometry of the circuit and the material it is made of.
  • The self-inductance of a solenoid is given by the equation:
    • L = (μ₀ * N² * A) / l where N is the number of turns in the solenoid, A is the cross-sectional area of the solenoid, and l is the length of the solenoid.
  1. Example of Self-Inductance (Continued)
  • Consider a solenoid with 500 turns, a cross-sectional area of 0.001 m², and a length of 0.2 m.
  • If the permeability of free space, μ₀, is 4π × 10⁻⁷ T m/A, the self-inductance, L, is given by:
    • L = (4π × 10⁻⁷ * 500² * 0.001) / 0.2
    • L = 0.05 H
  1. Inductance in Series Circuits
  • When inductors are connected in series, the total inductance is given by the sum of the individual inductances.
  • L_total = L₁ + L₂ + L₃ + …
  1. Example of Inductance in Series Circuits
  • Suppose we have three inductors with inductances of 2 H, 4 H, and 6 H connected in series.
  • The total inductance of the circuit is given by:
    • L_total = 2 + 4 + 6
    • L_total = 12 H
  1. Inductance in Parallel Circuits
  • When inductors are connected in parallel, the reciprocal of the total inductance is given by the sum of the reciprocals of the individual inductances.
  • 1/L_total = 1/L₁ + 1/L₂ + 1/L₃ + …
  1. Example of Inductance in Parallel Circuits
  • Suppose we have three inductors with inductances of 2 H, 4 H, and 6 H connected in parallel.
  • The reciprocal of the total inductance of the circuit is given by:
    • 1/L_total = 1/2 + 1/4 + 1/6
    • 1/L_total ≈ 0.694 H⁻¹
    • L_total ≈ 1.442 H
  1. Energy Stored in an Inductor
  • Inductors store energy in their magnetic fields.
  • The energy stored in an inductor is given by the equation:
    • U = (1/2) * L * I² where U is the energy stored, L is the inductance, and I is the current through the inductor.
  1. Example of Energy Stored in an Inductor
  • Consider an inductor with an inductance of 5 H and a current of 3 A.
  • The energy stored in the inductor is given by:
    • U = (1/2) * 5 * 3²
    • U = 22.5 J

Sure! Here are slides 21 to 30 on the topic of “Faraday’s Law of Induction- Mutual and Self-Inductance - Mutual inductance”:

  1. Applications of Mutual Inductance
  • Transformers: Mutual inductance is used in transformers to transfer electrical energy from one circuit to another with different voltages.
  • Inductive Coupling: Mutual inductance is employed in wireless power transfer systems and communication devices.
  • Magnetic Resonance Imaging (MRI): Mutual inductance is utilized in MRI machines to generate strong and uniform magnetic fields.
  1. Coefficient of Coupling
  • The coefficient of coupling, denoted by k, is a measure of the magnetic coupling between two circuits.
  • It varies between 0 and 1, where 0 indicates no coupling and 1 indicates maximum coupling.
  • The coefficient of coupling can be determined using the formula:
    • k = M / √(L₁ * L₂)
  1. Leakage Inductance
  • Leakage inductance refers to the portion of mutual inductance that does not transfer energy between the two circuits.
  • It occurs due to the imperfect magnetic coupling between the coils.
  • Leakage inductance can cause voltage spikes and affect the overall performance of a transformer or inductor.
  1. Factors Affecting Mutual Inductance
  • The number of turns: Increasing the number of turns increases mutual inductance.
  • The cross-sectional area: Larger area leads to greater mutual inductance.
  • The distance between coils: Decreasing the distance between coils increases mutual inductance.
  1. Mutual Inductance and Magnetic Fields
  • The magnetic field produced by one coil induces a magnetic field in the other coil, resulting in mutual inductance.
  • This magnetic field aids or opposes the change in current depending on the direction of the induced field.
  • The direction of the induced field and the resulting mutual inductance depend on the relative orientation of the coils.
  1. Maxwell’s Equations and Inductance
  • Faraday’s law of induction, an important equation in electromagnetism, is one of Maxwell’s equations.
  • Maxwell’s equations describe the fundamental relationship between electric and magnetic fields.
  • The concept of inductance is closely related to these equations and plays a crucial role in the analysis of electrical circuits.
  1. Calculation of Mutual Inductance
  • Mutual inductance can be experimentally determined by measuring the induced EMF in one circuit when the current in the other circuit changes.
  • Various methods and setups, such as the transformer ratio method and the time-varying magnetic field method, can be used.
  • Calculations can be performed using equations involving the number of turns, the magnetic field, and the physical dimensions of the coils.
  1. Back EMF in Inductive Circuits
  • In inductive circuits, when the current changes, a back EMF is induced that opposes the applied voltage.
  • This back EMF is a manifestation of Lenz’s law, which states that the induced current will always work against the change causing it.
  • Back EMF is responsible for the transient behavior observed in inductive circuits.
  1. Energy Transfer in Mutual Inductance
  • In a transformer, mutual inductance transfers energy from the primary coil to the secondary coil.
  • The power transferred is given by the equation:
    • P = V₂ * I₂ = V₁ * I₁ * (N₂ / N₁)² where V₁ and I₁ are the primary voltage and current, and V₂ and I₂ are the secondary voltage and current.
  1. Summary and Key Points
  • Faraday’s law of induction describes the relationship between a changing magnetic field and induced EMF.
  • Mutual inductance occurs when two circuits induce EMF in each other due to their proximity.
  • Mutual inductance can be determined experimentally and calculated using equations.
  • The coefficient of coupling, leakage inductance, and back EMF are important aspects of mutual inductance.
  • Mutual inductance is utilized in transformers, inductive coupling, and various applications in electrical engineering.