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are a fascinating phenomenon in electromagnetism. They arise when magnetic flux changes, creating circulating electric fields that oppose the change. This process, described by , is crucial for understanding .

Calculations involving induced electric fields use Faraday's law, which relates the changing magnetic flux to the . These fields differ from electrostatic fields in their origin, behavior, and ability to maintain currents in closed loops.

Induced Electric Fields

Creation of induced electric fields

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  • Faraday's law of induction states that a changing magnetic flux induces an electric field
    • Magnetic flux is the product of the magnetic field strength and the area it passes through perpendicular to the field (measured in webers, Wb)
    • A change in magnetic flux can occur due to:
      • Change in the magnetic field strength (e.g., moving a magnet closer or farther from a coil)
      • Change in the area enclosed by the magnetic field (e.g., expanding or contracting a loop of wire in a magnetic field)
      • Change in the orientation of the area with respect to the magnetic field (e.g., rotating a coil in a magnetic field)
  • The circulates around the area where the magnetic flux is changing
    • The direction of the induced electric field is determined by
      • Lenz's law states that the induced electric field will create a current that opposes the change in magnetic flux (e.g., if the magnetic flux is increasing, the induced current will create a magnetic field that opposes the increase)
    • The magnitude of the induced electric field is proportional to the rate of change of the magnetic flux (i.e., faster changes in flux result in stronger induced electric fields)

Calculations with Faraday's law

  • Faraday's law quantifies the relationship between the changing magnetic flux and the induced electric field
    • ∮E⃗⋅dl⃗=−dΦBdt\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}
      • ∮E⃗⋅dl⃗\oint \vec{E} \cdot d\vec{l} represents the of the electric field along a closed path (measured in volts, V)
      • ΦB\Phi_B is the magnetic flux through the area enclosed by the path (measured in webers, Wb)
      • dΦBdt\frac{d\Phi_B}{dt} is the rate of change of the magnetic flux with respect to time (measured in webers per second, Wb/s)
  • To calculate the induced electric field:
    1. Determine the rate of change of the magnetic flux (e.g., by calculating the change in flux over a given time interval)
    2. Use the negative sign to indicate the direction of the induced electric field according to Lenz's law
    3. Solve for the electric field strength using the line integral (e.g., by dividing the EMF by the length of the closed path)
  • The is the voltage generated by the changing magnetic flux

Induced vs electrostatic fields

  • Induced electric fields:
    • Created by a changing magnetic flux (e.g., a moving magnet or a changing current in a nearby conductor)
    • Circulate around the area where the magnetic flux is changing (e.g., around a loop of wire in a changing magnetic field)
    • Can exist in both conducting and non-conducting materials (e.g., in a copper coil or in the space around a moving magnet)
    • Not conservative, as the work done by an induced electric field on a charge moving in a closed path is non-zero (i.e., the induced electric field can maintain a current in a closed loop)
  • Electrostatic fields:
    • Created by electric charges (e.g., the electric field around a proton or electron)
    • Originate from positive charges and terminate on negative charges (e.g., the electric field lines between a positive and negative charge)
    • Exist primarily in non-conducting materials (dielectrics) (e.g., the electric field in a capacitor filled with air or plastic)
    • Conservative, as the work done by an electrostatic field on a charge moving in a closed path is zero (i.e., the electrostatic field cannot maintain a current in a closed loop)

Time-varying fields and electromagnetic waves

  • Time-varying electric and magnetic fields are interconnected and can propagate through space as
  • occurs when a changing magnetic field in one circuit induces an EMF in a nearby circuit
  • These concepts are fundamental to understanding the behavior of electromagnetic waves in various applications

Key Terms to Review (23)

Curl: Curl is a vector calculus operator that describes the circulation or rotation of a vector field around a given point. It is a fundamental concept in electromagnetism and fluid dynamics, providing a measure of the local spinning motion of a vector field.
Eddy Currents: Eddy currents are circular electric currents that are induced within a conductive material when it is exposed to a changing magnetic field. These currents flow in a direction that opposes the changes in the magnetic field, creating their own opposing magnetic field.
Electric Flux: Electric flux is a measure of the total electric field passing through a given surface. It represents the number of electric field lines passing perpendicularly through a surface, and is a key concept in understanding the behavior of electric fields and charges.
Electromagnetic Induction: Electromagnetic induction is the process by which a changing magnetic field induces an electromotive force (EMF) in a conductor, causing an electric current to flow. This phenomenon is fundamental to the operation of many electrical devices and is crucial in understanding the relationship between electricity and magnetism.
Electromagnetic Waves: Electromagnetic waves are a type of energy that travels through space or a medium in the form of oscillating electric and magnetic fields. These waves are responsible for various phenomena, including heat transfer, magnetism, and the propagation of electric fields, and are fundamental to our understanding of Maxwell's equations and the electromagnetic spectrum.
Faraday's law: Faraday's law states that a change in magnetic flux through a circuit induces an electromotive force (emf) in that circuit. This principle is crucial for understanding how magnetic fields interact with electric circuits and lays the foundation for many applications in electromagnetism.
Induced electric field: An induced electric field is a non-conservative electric field generated by a changing magnetic flux. Unlike electrostatic fields, it forms closed loops and is described by Faraday's law of induction.
Induced Electric Fields: Induced electric fields refer to the electric fields that are generated by a changing magnetic field, as described by Faraday's law of electromagnetic induction. These electric fields are created without the presence of any static electric charges, and they can drive the flow of electric currents in conducting materials.
Induced emf: Induced emf is the electromotive force generated in a conductor due to a changing magnetic field. It is the principle behind electromagnetic induction and is mathematically described by Faraday's Law.
Induced EMF: Induced EMF, or induced electromotive force, is the voltage or potential difference generated in a conductor when it experiences a change in the magnetic field around it. This phenomenon is a fundamental principle of electromagnetic induction, where a changing magnetic field induces an electric field, which in turn generates a voltage or EMF in the conductor.
James Clerk Maxwell: James Clerk Maxwell was a renowned Scottish physicist who made significant contributions to the understanding of electromagnetism, the nature of light, and the foundations of modern physics. His work laid the groundwork for many of the key concepts and theories that are central to the topics of 2.4 Distribution of Molecular Speeds, 6.4 Conductors in Electrostatic Equilibrium, 11.1 Magnetism and Its Historical Discoveries, 12.3 Magnetic Force between Two Parallel Currents, 13.4 Induced Electric Fields, 13.5 Eddy Currents, and 16.1 Maxwell's Equations and Electromagnetic Waves.
Lenz's Law: Lenz's law is a fundamental principle in electromagnetic induction that describes the direction of the induced current in a conductor. It states that the direction of the induced current will be such that it opposes the change in the magnetic field that caused it, in accordance with Faraday's law of electromagnetic induction.
Line Integral: A line integral is a mathematical concept that represents the integration of a function along a specified curve or path. It is a fundamental tool in vector calculus and is used to quantify the cumulative effect of a vector field along a particular trajectory.
Maxwell's Equations: Maxwell's equations are a set of four fundamental equations in electromagnetism that describe the relationships between electric and magnetic fields and electric charges and currents. These equations form the foundation of classical electromagnetism and are essential for understanding a wide range of electromagnetic phenomena.
Michael Faraday: Michael Faraday was a pioneering scientist known for his groundbreaking work in electromagnetism and electrochemistry during the 19th century. His contributions, particularly in discovering electromagnetic induction and formulating Faraday's Law, laid the foundation for modern electrical engineering and technology.
Mutual Inductance: Mutual inductance is a measure of the magnetic coupling between two electrical circuits or coils. It describes the amount of induced voltage in one circuit due to a changing current in another circuit, and it is a key concept in understanding the principles of electromagnetic induction.
Mutual inductance (M): Mutual inductance is the phenomenon where a change in current in one coil induces an electromotive force (EMF) in a nearby coil. It is quantified by the mutual inductance coefficient, $M$, which depends on the geometry and orientation of the coils.
Solenoid: A solenoid is a coil of wire designed to create a uniform magnetic field in its interior when an electric current passes through it. It is commonly used in electromagnets, inductors, and valves.
Solenoid: A solenoid is a tightly wound coil of wire, often cylindrical in shape, that produces a magnetic field when an electric current passes through it. Solenoids are fundamental components in the study of electromagnetism and have applications in various areas of physics, including magnetic fields, magnetic force, and electromagnetic induction.
Time-Varying Fields: Time-varying fields are electromagnetic fields that change in magnitude and direction over time. These fields are fundamental in understanding the behavior of electric and magnetic phenomena, particularly in the context of electromagnetic induction and the propagation of electromagnetic waves.
Toroid: A toroid is a three-dimensional geometric shape that resembles a doughnut or an inner tube. It is characterized by a circular path or loop, with the cross-section of the loop typically circular or elliptical. Toroids are commonly encountered in various physics topics, including magnetic fields, electromagnetic induction, and self-inductance.
Weber: The weber (Wb) is the SI unit of magnetic flux, representing the quantity of magnetism. One weber is equal to one tesla meter squared ($1 \, \text{Wb} = 1 \, \text{T} \cdot m^2$).
Weber: The weber (symbol: Wb) is the unit of magnetic flux in the International System of Units (SI). It is named after the German physicist Wilhelm Eduard Weber. The weber is a fundamental unit that is closely related to the concepts of magnetic field, electromagnetic induction, and the functioning of various electrical and electronic devices.
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13.5 Eddy Currents