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23.1 Induced Emf and Magnetic Flux

3 min read•june 18, 2024

measures the total passing through a surface. It's calculated using the equation Φ_B = BA cos θ, where B is the field strength, A is the surface area, and θ is the angle between them.

states that a changing induces an (emf) in a loop. The is given by = -dΦ_B/dt, where dΦ_B/dt is the rate of change of magnetic flux.

Magnetic Flux and Induced EMF

Magnetic flux calculation

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  • Magnetic flux (ΦB\Phi_B) measures the total magnetic field passing through a surface
    • Calculated using the equation: ΦB=B⃗⋅A⃗=BAcos⁡θ\Phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta
      • B⃗\vec{B}: magnetic field strength (teslas, T)
      • A⃗\vec{A}: area of the surface (square meters, m²)
      • θ\theta: angle between the magnetic field lines and the normal to the surface
  • Magnetic field perpendicular to the surface (θ=0°\theta = 0°) results in maximum flux: ΦB=BA\Phi_B = BA
    • Example: a flat coil placed perpendicular to a uniform magnetic field
  • Magnetic field parallel to the surface (θ=90°\theta = 90°) results in zero flux: ΦB=0\Phi_B = 0
    • Example: a flat coil placed parallel to a uniform magnetic field
  • : the total magnetic flux passing through all turns of a coil

Induction of electromotive force

  • Faraday's law of induction states a changing magnetic flux through a loop induces an (emf) in the loop
    • Induced emf (E\mathcal{E}) given by: E=−dΦBdt\mathcal{E} = -\frac{d\Phi_B}{dt}
      • dΦBdt\frac{d\Phi_B}{dt}: rate of change of magnetic flux
      • Negative sign indicates the induced emf opposes the change in flux ()
  • Changing magnetic flux caused by:
    • Changing the magnetic field strength (increasing or decreasing the field)
    • Changing the area of the loop (expanding or contracting the loop)
    • Changing the orientation of the loop relative to the magnetic field (rotating the loop)
  • Motion of a conductor in a magnetic field also induces an emf
    • : E=Blv\mathcal{E} = Blv
      • BB: magnetic field strength
      • ll: length of the conductor
      • vv: velocity of the conductor perpendicular to the magnetic field
    • Example: a conducting rod moving through a magnetic field

Factors affecting induced emf

  • Magnitude of induced emf depends on:
    1. Rate of change of magnetic flux
      • Faster changes in flux result in larger induced emf
      • Example: rapidly moving a magnet in and out of a coil
    2. Number of turns in a coil (for coiled conductors)
      • More turns lead to a larger induced emf
      • Example: a tightly wound coil with many turns
  • Direction of induced emf (and resulting current) determined by Lenz's law
    • Induced emf always opposes the change in magnetic flux that caused it
    • Applying the helps determine the direction of induced current
  • Practical applications:
    • Generators convert mechanical energy into electrical energy using induced emf (hydroelectric, wind turbines)
    • Transformers use induced emf to change voltage levels in electrical systems (power grids)
    • Induction cooktops use induced currents to heat cookware (efficient, safe cooking)
    • Eddy current brakes use induced currents to slow down moving objects (roller coasters, trains)

Inductance and Magnetic Properties

  • : the property of two circuits where a change in current in one induces an emf in the other
  • : the property of a circuit where a change in its own current induces an emf in itself
  • : a measure of how easily a material can be magnetized in response to an applied magnetic field
  • : a measure of the strength and orientation of a magnetic dipole, such as a current loop or a bar magnet

Key Terms to Review (32)

Eddy Currents: Eddy currents are electric currents that are induced within a conductive material when it is exposed to a changing magnetic field. These circulating currents create their own magnetic fields that oppose the original changing magnetic field, leading to various effects that are important in various applications.
Electric and magnetic fields: Electric and magnetic fields are two interdependent fields that propagate as waves through space. They form the basis of electromagnetic waves, where oscillations in one field induce oscillations in the other.
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 the fundamental principle behind the operation of many electrical devices and systems, including transformers, generators, and motors.
Electromotive force: Electromotive force (emf) is the energy provided by a source per unit charge to move electrons through a circuit. It is measured in volts (V) and drives the current around the circuit.
Electromotive Force: Electromotive force (EMF) is the voltage or potential difference generated in an electrical circuit, typically by a source of electrical energy such as a battery or generator. It represents the driving force that causes electric charge to flow through a circuit, enabling the conversion of other forms of energy into electrical energy.
Faraday's Law: Faraday's law describes the relationship between a changing magnetic field and the electric field it induces. It states that the magnitude of the induced electromotive force (emf) in a circuit is proportional to the rate of change of the magnetic flux through the circuit.
Flux linkage: Flux linkage is a measure of the total magnetic flux that passes through a coil of wire, taking into account the number of turns in the coil. It is expressed as the product of the magnetic flux and the number of turns, indicating how effectively a coil can link with magnetic fields. This concept is crucial in understanding how induced electromotive force (emf) is generated when there is a change in magnetic flux through the coil, which can also lead to phenomena like eddy currents.
Heinrich Lenz: Heinrich Lenz was a Russian physicist who discovered the fundamental principle that the direction of the induced current in a conductor is always such that it opposes the change in the magnetic field that caused it. This principle, known as Lenz's law, is a crucial concept in understanding the behavior of electromagnetic induction and its applications.
Induced EMF: Induced EMF, or electromotive force, refers to the voltage that is generated when a conductor, such as a wire, experiences a change in the magnetic flux passing through it. This phenomenon is a fundamental principle in the field of electromagnetism and is central to understanding the operation of various electrical devices and systems.
Lenz's Law: Lenz's law is a fundamental principle in electromagnetism that describes the direction of the induced current or electromotive force (emf) generated by electromagnetic induction. It states that the direction of the induced current is always such that it opposes the change in the magnetic field that caused it, in accordance with Faraday's law of induction.
Magnetic Dipole Moment: The magnetic dipole moment is a vector quantity that describes the strength and orientation of a magnetic dipole, which is a pair of equal and opposite magnetic poles separated by a small distance. It is a fundamental property of certain particles and systems that exhibit a magnetic field.
Magnetic Field: A magnetic field is a region in space where magnetic forces can be detected. It is a vector field that describes the magnetic influence of electric currents and magnetized materials on the space around them. The magnetic field is a fundamental concept in electromagnetism and is essential for understanding various phenomena in physics, including the behavior of ferromagnets, the motion of charged particles, and the production of electromagnetic waves.
Magnetic field strength inside a solenoid: Magnetic field strength inside a solenoid is the intensity of the magnetic field created within a coil of wire when an electric current passes through it. It is uniform and parallel to the axis of the solenoid.
Magnetic flux: Magnetic flux is the measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. It is calculated as the product of the magnetic field and the area through which it passes, perpendicular to the field.
Magnetic Flux: Magnetic flux is a measure of the total amount of magnetic field passing through a given surface or area. It represents the strength and distribution of a magnetic field and is a fundamental concept in the study of electromagnetism and its applications.
Magnetic Permeability: Magnetic permeability is a measure of the ability of a material to support the formation of a magnetic field within itself. It is a fundamental property that describes the degree of magnetization of a material in response to an applied magnetic field.
Michael Faraday: Michael Faraday was a renowned British scientist who made significant contributions to the fields of electricity and electromagnetism. His groundbreaking discoveries and inventions laid the foundation for many modern electrical and electromagnetic technologies.
Motional EMF: Motional EMF, also known as the electromotive force (EMF) induced by motion, is the voltage generated when a conductor moves through a magnetic field. This phenomenon is a fundamental principle in the field of electromagnetic induction, which describes the generation of electric currents and voltages due to changing magnetic fields.
Mutual inductance: Mutual inductance is the phenomenon where a change in the current in one coil induces a voltage in another coil that is magnetically coupled to it. It is quantified by the mutual inductance coefficient, which depends on factors like the number of turns in each coil and their relative positioning.
Mutual Inductance: Mutual inductance is a measure of the magnetic coupling between two electrical circuits or coils. It describes the amount of induced electromotive force (emf) in one circuit due to a changing current in a neighboring circuit, and is a fundamental concept in the study of electromagnetic induction.
Right-hand rule: The right-hand rule is a mnemonic used to determine the direction of angular momentum vectors. It states that if you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the angular momentum vector.
Right-Hand Rule: The right-hand rule is a mnemonic device used to determine the direction of various vector quantities in physics, such as magnetic fields, angular momentum, and the force on a moving charge in a magnetic field. It is a simple and intuitive way to visualize the relationship between these vectors and their associated directions.
Self-inductance: Self-inductance is the property of a coil (or circuit) that allows it to oppose the change in current flowing through it by generating an electromotive force (EMF). This EMF is proportional to the rate of change of current.
Self-Inductance: Self-inductance is a property of an electrical circuit or component that describes the ability of the circuit or component to generate an opposing electromotive force (emf) within itself when the current flowing through it changes. It is a measure of how much the magnetic field created by a changing current in a circuit opposes changes in that current.
Solenoid: A solenoid is a type of electromagnet consisting of a coil of wire wound into a tight spiral. When an electric current flows through the coil, it creates a magnetic field inside the solenoid, which can be used to produce a strong and uniform magnetic field in a specific region of space.
Step-down transformer: A step-down transformer is a device that decreases the voltage from the primary coil to the secondary coil while increasing the current. It is commonly used to convert high voltage electricity from power lines to a lower, safer voltage for use in homes and businesses.
Tesla: The tesla (T) is the SI unit of magnetic field strength or magnetic flux density. It measures how much force a magnetic field exerts on moving charges or current-carrying wires.
Tesla: The tesla (T) is the unit of magnetic flux density or magnetic induction in the International System of Units (SI). It is named after the Serbian-American inventor and electrical engineer Nikola Tesla, who made significant contributions to the design of the modern alternating-current (AC) electrical supply system.
Transformer: A transformer is a device that transfers electrical energy from one alternating current (AC) circuit to another through the process of electromagnetic induction, without the need for direct electrical connection. It is a crucial component in the transmission and distribution of electrical power, as well as in various electronic devices.
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 used to quantify the amount of magnetic flux present in a magnetic field, and it plays a crucial role in understanding various electromagnetic phenomena.
ε: The term 'ε' is a Greek letter that represents the concept of electromotive force (emf) in the context of electrical circuits and electromagnetic induction. It is a fundamental quantity that describes the voltage or potential difference generated within a system, such as a battery or a conductor experiencing a changing magnetic field.
Φ (Phi): Phi, also known as the magnetic flux, is a fundamental concept in electromagnetism that describes the amount of magnetic field passing through a given surface or area. It is a scalar quantity that represents the total magnetic field lines that intersect a particular surface, and it is a crucial factor in understanding the phenomenon of electromagnetic induction.
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23.2 Faraday’s Law of Induction: Lenz’s Law