Mutual Inductance

• Mutual inductance is the interaction of two coils and their magnetic fields.
• It is denoted by the symbol M and measured in Henrys (H).
• The mutual inductance between two coils depends on their geometry, distance, and the number of turns in each coil.
• The emf induced in the neighboring coil can be calculated using the equation:
  • emf = -M(dI/dt)

Self-Inductance

• Self-inductance is the effect of the magnetic field of a current-carrying coil on itself.
• It is denoted by the symbol L and measured in Henrys (H).
• The self-inductance of a coil depends on its geometry, number of turns, and the permeability of the core material.
• The emf induced in the same coil can be calculated using the equation:
  • emf = -L(dI/dt)

Example 1

Consider a coil with a self-inductance of 3 H. If the current through the coil changes at a rate of 5 A/s, calculate the induced emf. Given:

• L (self-inductance) = 3 H
• dI/dt (change in current per unit time) = 5 A/s Using the equation emf = -L(dI/dt):
• emf = -(3 H)(5 A/s) = -15 V Therefore, the induced emf in the coil is -15 V.

Example 2

Two coils are placed close to each other, and their mutual inductance is 2 H. If the current in one coil changes at a rate of 10 A/s, calculate the induced emf in the other coil. Given:

• M (mutual inductance) = 2 H
• dI/dt (change in current per unit time) = 10 A/s Using the equation emf = -M(dI/dt):
• emf = -(2 H)(10 A/s) = -20 V Therefore, the induced emf in the other coil is -20 V.

Induced Emf and Magnetic Flux

• The induced emf in a coil is directly proportional to the rate of change of magnetic flux.
• The magnetic flux (Φ) through a coil is given by the equation:
  • Φ = B × A
  • B - Magnetic field strength
  • A - Area of the coil

Faraday’s Law of Induction

• Faraday’s Law of Induction relates the induced emf to the rate of change of magnetic flux.
• It can be stated as:
  • The magnitude of the induced emf in a circuit is equal to the rate of change of magnetic flux through the circuit.
  • emf = -d(Φ)/dt

Lenz’s Law

• Lenz’s Law is a consequence of Faraday’s Law of Induction.
• It states that the direction of the induced emf and current is such as to oppose the change that caused it.
• This law ensures that energy is conserved and that the magnetic field and current oppose the change producing them.

Applications of Inductance

• Transformers: Mutual inductance is used in transformers to step up or step down voltage.
• Inductors: Self-inductance is used in components like inductors to store energy in magnetic fields.
• Generators: Faraday’s Law of Induction is the principle behind the generation of electricity in generators.
• Induction Cooktops: These cooktops utilize the principle of induction to heat the bottom of pots and pans.

Summary

• Faraday’s Law of Induction states that a change in magnetic field induces an emf in a nearby circuit.
• Mutual inductance occurs when the magnetic field of one coil induces an emf in a neighboring coil.
• Self-inductance occurs when the magnetic field of a current-carrying coil induces an emf in the same coil.
• The induced emf can be calculated using the equations: emf = -M(dI/dt) for mutual inductance and emf = -L(dI/dt) for self-inductance.
• Lenz’s Law states that the induced emf and current oppose the change that caused them.
• Inductance is used in transformers, inductors, generators, and induction cooktops.

Faraday’s Law of Induction

• Faraday’s Law of Induction states that a change in magnetic field induces an electromotive force (emf) in a nearby circuit.
• This phenomenon is the basis for the operation of many electrical devices like transformers and generators.
• There are two types of inductance: mutual inductance and self-inductance.
• Mutual inductance occurs when the magnetic field produced by one coil induces an emf in a neighboring coil.
• Self-inductance occurs when the changing magnetic field within a coil induces an emf in the same coil.

Mutual Inductance

• Mutual inductance is the interaction of two coils and their magnetic fields.
• It is denoted by the symbol M and measured in Henrys (H).
• The mutual inductance between two coils depends on their geometry, distance, and the number of turns in each coil.
• The emf induced in the neighboring coil can be calculated using the equation:
  • emf = -M(dI/dt)
• Mutual inductance can be found using the formula:
  • M = (N₂Φ₁)/(I₁)

Example 3

Two coils have 50 turns each. The first coil is carrying a steady current of 4 A, and the second coil has a magnetic flux of 0.5 Wb. Calculate the mutual inductance between the coils. Given:

• N₁ = N₂ = 50 turns
• I₁ = 4 A
• Φ₂ = 0.5 Wb Using the formula for mutual inductance:
• M = (N₂Φ₁)/(I₁)
• M = (50 * 0.5 Wb)/(4 A)
• M = 6.25 H Therefore, the mutual inductance between the coils is 6.25 H.

Self-Inductance

• Self-inductance is the effect of the magnetic field of a current-carrying coil on itself.
• It is denoted by the symbol L and measured in Henrys (H).
• The self-inductance of a coil depends on its geometry, number of turns, and the permeability of the core material.
• The emf induced in the same coil can be calculated using the equation:
  • emf = -L(dI/dt)
• Self-inductance can be found using the formula:
  • L = (NΦ)/I

Example 4

A coil with 200 turns has a current changing at a rate of 0.02 A/s. The magnetic flux within the coil is 0.5 Wb. Calculate the self-inductance of the coil. Given:

• N = 200 turns
• dI/dt = 0.02 A/s
• Φ = 0.5 Wb Using the formula for self-inductance:
• L = (NΦ)/I
• L = (200 * 0.5 Wb)/(0.02 A/s)
• L = 5000 H Therefore, the self-inductance of the coil is 5000 H.

Faraday’s Law of Induction

• Faraday’s Law of Induction relates the induced emf to the rate of change of magnetic flux.
• It can be stated as:
  • The magnitude of the induced emf in a circuit is equal to the rate of change of magnetic flux through the circuit.
  • emf = -d(Φ)/dt
• The negative sign indicates that the induced emf and current oppose the change that caused them, as stated by Lenz’s Law.

Example 5

A circular coil of radius 0.1 m is placed in a uniform magnetic field of 0.3 T. If the magnetic field is perpendicular to the plane of the coil and reduces to zero in 0.5 s, calculate the induced emf in the coil. Given:

• r = 0.1 m
• B = 0.3 T
• d(B)/dt = -0.3 T/0.5 s (change in magnetic field per unit time) Using the equation emf = -d(Φ)/dt:
• Φ = B × A = (0.3 T)(π(0.1 m)^2) = 0.00942 Wb
• emf = -(dΦ/dt) = -(-0.3 T/0.5 s) = 0.6 V Therefore, the induced emf in the coil is 0.6 V.

Applications of Inductance

• Transformers: Mutual inductance is used in transformers to step up or step down voltage.
• Inductors: Self-inductance is used in components like inductors to store energy in magnetic fields.
• Generators: Faraday’s Law of Induction is the principle behind the generation of electricity in generators.
• Induction Cooktops: These cooktops utilize the principle of induction to heat the bottom of pots and pans.
• Solenoids: Solenoids use the magnetic field produced by a coil to control the movement of objects, such as in door locks or valves.

Applications of Inductance (contd.)

• Magnetic Resonance Imaging (MRI): MRI machines use strong magnetic fields and radio waves to create detailed images of the body’s internal structures.
• Inductive Proximity Sensors: These sensors use electromagnetic induction to detect the presence or absence of metallic objects without physical contact.
• Wireless Power Transfer: Inductive coupling is used in wireless charging systems to transfer energy between two coils placed in close proximity.
• Ignition Systems: Automobile ignition systems use an induction coil to generate a high voltage spark for igniting the fuel-air mixture in the engine.
• Electromagnetic Interference (EMI) Filters: Inductors are used in EMI filters to suppress unwanted electromagnetic interference in electronic circuits. Insert remaining slides here

Faraday’s Law of Induction

• Faraday’s Law of Induction states that the emf induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.
• Mathematically, it can be expressed as: emf = -N(dΦ/dt)
• N is the number of turns in the circuit, dΦ/dt is the change in magnetic flux per unit time.

Example 1

A circuit with 100 turns is located in a magnetic field. The magnetic field changes from 0.5 T to 0.2 T in 2 seconds. Calculate the induced emf in the circuit. Given:

• N = 100 turns
• dΦ/dt = (final flux - initial flux) / time = (0.2 T - 0.5 T) / 2 s = -0.15 T/s Using the equation emf = -N(dΦ/dt):
• emf = -(100 turns)(-0.15 T/s) = 15 V/s Therefore, the induced emf in the circuit is 15 V/s.

Faraday’s Law of Induction (contd.)

• The negative sign in Faraday’s Law of Induction indicates that the direction of the induced current is such that it opposes the change in magnetic flux.
• This is known as Lenz’s Law.
• Lenz’s Law ensures that energy is conserved and the magnetic field and current oppose the change producing them.

Examples of Faraday’s Law in Everyday Life

• Electric generators: Faraday’s Law is used to generate electricity in power plants and turbines through the rotation of magnetic fields.
• Bicycle dynamos: Moving a magnet near a coil induces an emf, which is used to power lights on bicycles.
• Microphones: Electromagnetic induction is used to convert sound waves into electrical signals in microphones.

Self-Inductance

• Self-inductance is caused by the magnetic field generated by a current-carrying coil interacting with itself.
• It is denoted by the symbol L and is measured in Henrys (H).
• The emf induced in the same coil is given by the equation: emf = -L(dI/dt).
• Self-inductance depends on the size, shape, and number of turns in the coil.

Example 2

A coil with an inductance of 0.1 H has a current changing at a rate of 5 A/s. Calculate the induced emf in the coil. Given:

• L = 0.1 H
• dI/dt = 5 A/s Using the equation emf = -L(dI/dt):
• emf = -(0.1 H)(5 A/s) = -0.5 V Therefore, the induced emf in the coil is -0.5 V.

Mutual Inductance

• Mutual inductance occurs when the magnetic field produced by one coil induces an emf in a neighboring coil.
• Mutual inductance is denoted by the symbol M and is measured in Henrys (H).
• It depends on the geometry, distance, and number of turns in each coil.
• The induced emf in the neighboring coil is given by the equation: emf = -M(dI/dt).

Example 3

Two coils have mutual inductance of 0.05 H. If the current in one coil changes at a rate of 10 A/s, calculate the induced emf in the other coil. Given:

• M = 0.05 H
• dI/dt = 10 A/s Using the equation emf = -M(dI/dt):
• emf = -(0.05 H)(10 A/s) = -0.5 V Therefore, the induced emf in the other coil is -0.5 V.

Applications of Inductance

• Transformers: Mutual inductance is used in transformers to step up or step down voltage.
• Inductors: Self-inductance is utilized in components like inductors to store energy in magnetic fields.
• Generators: Faraday’s Law of Induction is the principle behind the generation of electricity in generators.
• Induction Cooktops: These cooktops utilize the principle of induction to heat the bottom of pots and pans.
• Wireless Charging: Inductive coupling is used in wireless charging systems for mobile devices.

Summary

• Faraday’s Law of Induction states that the induced emf in a circuit is proportional to the rate of change of magnetic flux through the circuit.
• Self-inductance is caused by the magnetic field of a current-carrying coil interacting with itself.
• Mutual inductance occurs when the magnetic field of one coil induces an emf in a neighboring coil.
• Lenz’s Law ensures that the induced emf and current oppose the change in magnetic flux.
• Inductance is used in many applications such as transformers, inductors, generators, and induction cooktops.