1. Faraday’s Law of Induction
  • It states that a change in magnetic field induces an electromotive force (emf) in a closed loop.
  • The induced emf in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.
  • The direction of the induced emf is given by Lenz’s law, which states that the induced current or emf opposes the change that produces it.
  • The magnitude of the induced emf can be calculated using the equation: emf = -N * dΦ/dt
  • N represents the number of turns in the coil, and dΦ/dt represents the rate of change of magnetic flux.
  1. Magnetic Flux
  • Magnetic flux (Φ) is a measure of the total magnetic field passing through a given area.
  • It is directly proportional to the product of the magnetic field strength (B) and the area (A) through which the field lines pass.
  • Mathematically, magnetic flux can be calculated using the equation: Φ = B * A * Cos(θ)
  • θ represents the angle between the magnetic field and the normal to the area.
  1. Ampere’s Law
  • Ampere’s law relates the circulating magnetic field (B) around a closed loop to the current (I) passing through the loop.
  • It states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space (μ₀).
  • Mathematically, Ampere’s law can be expressed as: ∮B · dl = μ₀ * I
  • The integral is taken over the closed loop, and dl represents an infinitesimal vector element along the path.
  1. Biot-Savart Law
  • The Biot-Savart law describes the magnetic field produced by a steady current in an infinitely long straight wire.
  • It states that the magnetic field (B) at a point P at a distance r from the wire is directly proportional to the current (I), the length element (dl) of the wire, and inversely proportional to the square of the distance (r).
  • Mathematically, the Biot-Savart law can be expressed as: d𝐁 = (μ₀/4π) * (I * dl × 𝐫) / r²
  • The vector cross product (dl × 𝐫) represents the direction of the magnetic field, and μ₀ is the permeability of free space.
  1. Magnetic Force on a Current-Carrying Conductor
  • When a current-carrying conductor is placed in a magnetic field, it experiences a force.
  • The magnitude of the force can be calculated using the equation: F = |I| * |B| * sin(θ) * L
  • I represents the current in the conductor, B represents the magnetic field, θ represents the angle between the current and the magnetic field, and L represents the length of the conductor.
  • The direction of the force is given by the right-hand rule.
  1. Magnetic Field Due to a Straight Current-Carrying Conductor
  • The magnetic field produced by a straight current-carrying conductor follows a circular pattern around the wire.
  • The magnitude of the magnetic field (B) at a point P at a distance r from the wire can be calculated using the equation: B = (μ₀ * I) / (2πr)
  • μ₀ is the permeability of free space, I is the current in the wire, and π is the mathematical constant representing the ratio of a circle’s circumference to its diameter.
  1. Magnetic Field Inside a Solenoid
  • A solenoid is a tightly wound coil of wire that generates a uniform magnetic field inside when a current flows through it.
  • The magnetic field (B) inside a solenoid can be approximated as: B = μ₀ * n * I
  • μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current flowing through the solenoid.
  1. Magnetic Field due to a Circular Loop
  • The magnetic field at the center of a circular loop carrying current can be determined using Ampere’s law.
  • The magnitude of the magnetic field (B) at the center of the loop is given by the equation: B = (μ₀ * I) / (2 * R)
  • μ₀ represents the permeability of free space, I represents the current in the loop, and R is the radius of the loop.
  1. Lorentz Force
  • When a charged particle moves in a magnetic field, it experiences a force known as the Lorentz force.
  • The Lorentz force (F) acting on a particle with charge q moving with velocity v in a magnetic field B is given by the equation: F = q * v × B
  • The right-hand rule is used to determine the direction of the force.
  1. Magnetic Flux Density
  • Magnetic flux density (B) is a measure of the strength of a magnetic field at a given point.
  • B is proportional to the force experienced by a moving charge in a magnetic field.
  • The unit of magnetic flux density is the tesla (T), where 1 T = 1 N/Am.
  1. Faraday’s Law Of Induction
  • Faraday’s Law of Induction states that a change in magnetic field induces an electromotive force (emf) in a closed loop.
  • The induced emf in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.
  • The direction of the induced emf is given by Lenz’s law, which states that the induced current or emf opposes the change that produces it.
  • Faraday’s law can be mathematically expressed as:
    1. emf = -N * dΦ/dt, where N represents the number of turns in the coil, and dΦ/dt represents the rate of change of magnetic flux.
  1. Induced Electric Field
  • According to Faraday’s law, a change in magnetic field induces an electric field in a closed loop.
  • The induced electric field acts in a manner to create a current that opposes the change in magnetic field.
  • Induced electric field (E) can be calculated using the equation:
    1. E = -dΦ/dt / A, where dΦ/dt represents the rate of change of magnetic flux and A is the cross-sectional area of the loop.
  • The negative sign signifies that the induced electric field acts in a direction opposite to the change in magnetic field.
  1. Example 1: Induced Emf in a Coil
  • Consider a coil with N turns and a magnetic field B passing through it. If the magnetic field changes at a constant rate, the induced emf can be calculated using:
    1. emf = -N * dΦ/dt
  • For example, if the magnetic field changes by 0.2 T/s and the number of turns is 50, the induced emf would be:
    1. emf = -(50)(0.2) = -10 V
  • The negative sign indicates that the induced emf has a polarity that opposes the change in magnetic field.
  1. Example 2: Induced Electric Field in a Loop
  • Consider a rectangular loop of dimensions 10 cm by 5 cm positioned in a magnetic field. If the magnetic flux through the loop is changing at a rate of 2 T/s, the induced electric field can be calculated using:
    1. E = -dΦ/dt / A
  • Substituting the given values, we have:
    1. E = -(2) / [(0.1)(0.05)] = -40 V/m
  • The negative sign indicates that the induced electric field opposes the change in magnetic flux.
  1. Induced Emf vs. Magnetic Flux
  • The induced emf depends on the rate of change of magnetic flux through the loop.
  • If the magnetic flux through a loop does not change, the induced emf will be zero.
  • The greater the rate of change of magnetic flux, the greater the induced emf.
  • The orientation of the loop with respect to the changing magnetic field also affects the magnitude of the induced emf.
  1. Lenz’s Law
  • Lenz’s law states that the induced current or emf in a closed loop always opposes the change that produces it.
  • This law is based on the conservation of energy principle.
  • Lenz’s law helps determine the direction of the induced current or emf.
  • It is applicable to systems involving electromagnetic induction.
  1. Example 3: Lenz’s Law
  • Consider a bar magnet being pushed towards a coil. As the magnet approaches, the magnetic flux through the coil increases.
  • According to Lenz’s law, the induced emf in the coil creates a magnetic field that opposes the approach of the magnet.
  • This means that the induced magnetic field created by the induced current acts in a direction to repel the magnet, slowing down its approach.
  • Lenz’s law ensures that energy is conserved during the process of electromagnetic induction.
  1. Self-Induction
  • Self-induction occurs when a changing current in a circuit induces an emf in the same circuit.
  • It is caused by the magnetic field produced by the changing current.
  • The self-induced emf is always in a direction to oppose the change in current.
  • Inductors are components designed to utilize self-induction and are commonly used in electronic circuits.
  1. Mutual Induction
  • Mutual induction occurs when a changing current in one circuit induces an emf in a nearby circuit.
  • It is caused by the changing magnetic field produced by the current in the first circuit.
  • Transformers are examples of devices that utilize mutual induction.
  • The primary coil in a transformer induces an emf in the secondary coil, enabling the transfer of energy between circuits.
  1. Transformer
  • A transformer is a device that transfers electrical energy between two or more circuits through the principle of mutual induction.
  • It consists of two or more coils of wire, known as the primary and secondary coils.
  • The primary coil is connected to an alternating current (AC) source, creating a changing magnetic field.
  • The changing magnetic field induces an emf in the secondary coil, which can be used to power other devices.
  • Transformers are essential in the transmission and distribution of electrical energy.
  1. Faraday’s Law Of Induction - Induced emf
  • To calculate the induced emf in a coil, use the equation:
    • emf = -N * dΦ/dt
      • N: Number of turns in the coil
      • dΦ/dt: Rate of change of magnetic flux
  • Example: If a coil with 200 turns has a magnetic flux changing at a rate of 0.02 T/s, the induced emf can be calculated as:
    • emf = -(200)(0.02) = -4 V
  1. Faraday’s Law Of Induction - Induced electric field
  • The induced electric field (E) can be calculated using the equation:
    • E = -dΦ/dt / A
      • dΦ/dt: Rate of change of magnetic flux
      • A: Cross-sectional area of the loop
  • Example: If the magnetic flux changes at a rate of 5 T/s and the loop has an area of 0.02 m², the induced electric field can be calculated as:
    • E = -(5) / (0.02) = -250 V/m
  1. Lenz’s Law and Conservation of Energy
  • Lenz’s law is based on the conservation of energy principle.
  • It states that the induced current or emf always opposes the change that produces it.
  • Lenz’s law ensures that energy is conserved during electromagnetic induction.
  • This principle is used to determine the direction of induced current and emf.
  1. Lenz’s Law - Example: Bar Magnet Approach
  • When a bar magnet approaches a coil, the changing magnetic field induces an emf in the coil.
  • According to Lenz’s law, the induced emf creates a magnetic field that opposes the approach of the magnet.
  • The induced magnetic field repels the magnet, slowing down its approach.
  • Lenz’s law ensures that energy is conserved during this process.
  1. Self-Induction and Self-Induced emf
  • Self-induction occurs when a changing current in a circuit induces an emf in the same circuit.
  • This is caused by the magnetic field produced by the changing current.
  • The self-induced emf is always in a direction to oppose the change in current.
  • Inductors are components designed to utilize self-induction.
  1. Mutual Induction and Mutual Induced emf
  • Mutual induction occurs when a changing current in one circuit induces an emf in a nearby circuit.
  • This is caused by the changing magnetic field produced by the current in the first circuit.
  • Transformers are examples of devices that utilize mutual induction.
  • The primary coil in a transformer induces an emf in the secondary coil, enabling the transfer of energy between circuits.
  1. Transformers and Energy Transfer
  • Transformers are devices used to transfer electrical energy between two or more circuits.
  • They operate on the principle of mutual induction.
  • Transformers consist of primary and secondary coils, connected by a shared magnetic field.
  • The changing magnetic field induces an emf in the secondary coil, allowing for energy transfer between circuits.
  1. Transformers - Step-up and Step-down
  • Transformers can step up or step down the input voltage, depending on the turns ratio of the primary and secondary coils.
  • A step-up transformer has more turns in the secondary coil than the primary coil, resulting in an increased output voltage.
  • A step-down transformer has fewer turns in the secondary coil, resulting in a decreased output voltage.
  1. Transformers - Turns Ratio and Voltage Ratio
  • The turns ratio of a transformer is the ratio of the number of turns in the secondary coil to the number of turns in the primary coil.
  • The voltage ratio of a transformer is equal to the turns ratio.
  • For example, a transformer with a turns ratio of 2:1 will have a voltage ratio of 2:1, meaning the output voltage is twice the input voltage.
  1. Transformers - Power and Efficiency
  • The power in a transformer is given by the equation:
    • P = VI
      • P: Power
      • V: Voltage
      • I: Current
  • Transformers are designed to be highly efficient, with minimal energy losses.
  • Efficiency is calculated as the ratio of output power to input power, expressed as a percentage.
  • Efficient transformers minimize energy wastage and are crucial in power transmission and distribution systems.