Faraday’s Law Of Induction- Induced emf - Example 3

(Coaxial solenoid and circular loop)

• Consider a coaxial solenoid with radius ‘a’ and length ’l’
• Place a circular conducting loop with radius ‘R’ inside the solenoid
• The magnetic field inside the solenoid is given by B = μ₀nI, where μ₀ represents permeability, n represents number of turns per unit length, and I represents current through the solenoid
• The magnetic field within the solenoid is uniform
• The magnetic field lines at every point are parallel and perpendicular to the plane of the circular loop Equation:
• The flux through the circular loop is given by Φ = B * A = B * πR²
• According to Faraday’s law, the induced emf in the loop is given by ε = -dΦ/dt Example:
• Let the magnetic field inside the solenoid be 2 T
• Consider the radius of the circular loop to be 0.5 m
• If the magnetic field changes at a rate of 0.01 T/s, find the induced emf in the loop Solution:
• Given: B = 2 T, R = 0.5 m, and dΦ/dt = 0.01 T/s
• Using the equation ε = -dΦ/dt, we can calculate the induced emf
• Substituting the values, ε = -dΦ/dt = -(2 * π * (0.5)²) * 0.01 = -0.01 π V
• Therefore, the induced emf in the circular loop is -0.01 π V Sign Convention:
• The negative sign indicates that the induced current flows in a direction to oppose the change in magnetic flux, following Lenz’s law Application:
• Transformers, generators, and induction motors are some of the devices that work based on Faraday’s law of induction and induce emf Physics Principles:
• Faraday’s law of electromagnetic induction states that a change in the magnetic field through a loop of wire induces an emf, which in turn causes a current to flow in the loop
• This law is a fundamental principle used in various fields, including electrical engineering and physics Importance:
• Faraday’s law of induction provides a basis for understanding and analyzing electromagnetic phenomena, enabling the development of numerous technologies and applications
• It forms the foundation for the study of electromagnetism and is vital in fields such as power generation, electronics, and communication

11. Flux through the circular loop: Φ = B * A = B * πR²

• B represents the magnetic field strength inside the solenoid
• A is the area of the circular loop, given by πR²

12. Induced emf in the loop: ε = -dΦ/dt

• According to Faraday’s law of induction, the induced emf is equal to the negative rate of change of magnetic flux with respect to time

13. Example: Coaxial solenoid and circular loop

• Let the magnetic field inside the solenoid be 2 T
• Consider the radius of the circular loop to be 0.5 m
• If the magnetic field changes at a rate of 0.01 T/s, find the induced emf in the loop

14. Solution:

• Given: B = 2 T, R = 0.5 m, and dΦ/dt = 0.01 T/s
• Using the equation ε = -dΦ/dt, we can calculate the induced emf
• Substituting the values, ε = -(2 * π * (0.5)²) * 0.01 = -0.01 π V
• Therefore, the induced emf in the circular loop is -0.01 π V

15. Importance of sign convention:

• The negative sign indicates that the induced current flows in a direction to oppose the change in magnetic flux, following Lenz’s law

16. Application: Transformers

• Transformers utilize Faraday’s law of induction to transfer electrical energy between separate electrical circuits
• By inducing an emf in a secondary coil, transformers can step up or step down the voltage levels in a power transmission system

17. Application: Generators

• Generators use Faraday’s law to convert mechanical energy into electrical energy
• Rotating a coil of wire in a magnetic field induces an emf, generating electricity

18. Application: Induction motors

• Induction motors use Faraday’s law to convert electrical energy into mechanical energy
• The rotating magnetic field induces a current in the rotor, creating a torque and enabling the motor to do work

19. Physics principle: Faraday’s law of electromagnetic induction

• Faraday’s law states that a change in the magnetic field passing through a loop of wire induces an emf in the loop
• This is a fundamental principle of electromagnetism that connects electricity and magnetism

20. Importance in technology:

• Faraday’s law of induction is the basis for many technologies and applications, including power generation, motors, transformers, and electric devices
• It allows for the transfer and conversion of energy, making it essential in our modern world

21. Additional Application: Magnetic Induction Cooktops

• Magnetic induction cooktops utilize Faraday’s law of induction to heat cooking vessels
• By creating an alternating magnetic field, an electric current is induced in the cooking vessel, generating heat

22. Faraday’s Law in power generation

• Large-scale power generation relies on Faraday’s law to generate electricity
• Power plants use generators that rotate conductive coils in a magnetic field to induce a current and produce electric power

23. Equation for Faraday’s law of induction

• The induced emf in a loop is given by ε = -dΦ/dt, where ε is the emf and dΦ/dt is the rate of change of magnetic flux

24. Importance of rate of change of magnetic flux

• It is the change in the magnetic field or motion of the loop that induces the emf
• A steady magnetic field or static loop does not induce an emf

25. Factors affecting induced emf

• The magnitude of the induced emf depends on the rate of change of magnetic flux
• Increased change in magnetic field or faster motion of the loop leads to a larger induced emf

26. Lenz’s law and induced current direction

• Lenz’s law states that the direction of the induced current in a loop opposes the change in magnetic flux
• The negative sign in Faraday’s law indicates this opposition

27. Mutual induction

• Mutual induction occurs between two or more coils of wire
• When the magnetic field changes in one coil, it induces an emf in the other coil

28. Self-induction

• Self-induction refers to the induction of emf in a single coil due to a changing magnetic field produced by the same coil
• It occurs when the current in a coil changes, inducing an emf that opposes the change in current

29. Faraday’s law as a basis for electromagnetic theory

• Faraday’s law is one of the fundamental principles underlying the study of electromagnetism
• It provides a foundation for understanding many other concepts, such as electromagnetic waves and Maxwell’s equations

30. Summary:

• Faraday’s law of induction states that a changing magnetic field induces an emf in a loop of wire
• The induced emf is given by ε = -dΦ/dt
• The direction of the induced current opposes the change in magnetic flux, following Lenz’s law
• Faraday’s law is essential in various applications, including power generation, transformers, and motors
• It is a fundamental principle in the study of electromagnetism