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is all about how circuits can influence each other through magnetic fields. When current changes in one circuit, it can create a voltage in another nearby circuit. This effect is super useful in transformers, wireless chargers, and induction cooktops.

The strength of this magnetic connection is measured by the mutual inductance coefficient. It depends on things like how the circuits are arranged and what's between them. Understanding mutual inductance helps us design better power systems and wireless tech.

Mutual Inductance

Mutual inductance between circuits

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  • Mutual inductance arises when two nearby circuits are magnetically coupled
    • Each circuit generates a magnetic field when a time-varying current flows through it
    • The magnetic field generated by one circuit can induce an emf () in the other circuit
  • The in one circuit is proportional to the rate of change of current in the other circuit
    • This relationship is characterized by the mutual inductance coefficient MM
  • The mutual inductance coefficient depends on the geometry and relative position of the two circuits
    • Factors affecting MM include the number of turns in each coil, the cross-sectional area of the coils, and the distance between them
    • The of the surrounding medium also influences the mutual inductance
  • The in circuit 1 due to a changing current in circuit 2 is given by ε1=−MdI2dt\varepsilon_1 = -M \frac{dI_2}{dt}
    • Similarly, the induced emf in circuit 2 due to a changing current in circuit 1 is ε2=−MdI1dt\varepsilon_2 = -M \frac{dI_1}{dt}
    • The negative sign indicates that the induced emf opposes the change in current that produced it ()

Applications of mutual inductance

  • Transformers utilize mutual inductance to step up or step down voltages in power systems (power grids)
    • The primary and secondary coils of a are magnetically coupled, allowing efficient energy transfer between them
    • Transformers enable long-distance power transmission at high voltages and lower currents to minimize power losses
  • Wireless charging systems for electronic devices rely on mutual inductance (smartphones, smartwatches)
    • A in the charging base induces a current in a within the device, enabling wireless power transfer
    • This eliminates the need for physical connectors and allows for convenient charging of portable devices
  • Induction cooktops use mutual inductance to heat cookware directly
    • A high-frequency alternating current in a coil beneath the cooktop surface induces in the cookware, generating heat
    • This method provides efficient and precise temperature control while keeping the cooktop surface cool to the touch

Calculating induced emf

  • The induced emf in a circuit can be calculated using the equation ε=−MdIdt\varepsilon = -M \frac{dI}{dt}
    • ε\varepsilon represents the induced emf in the circuit of interest (volts)
    • MM represents the mutual inductance coefficient between the two circuits (henries)
    • dIdt\frac{dI}{dt} represents the rate of change of current in the nearby circuit (amperes per second)
  • To find the induced emf, follow these steps:
    1. Identify the mutual inductance coefficient MM between the two circuits
      • This value depends on the geometry and relative position of the circuits
    2. Determine the rate of change of current dIdt\frac{dI}{dt} in the nearby circuit
      • This can be obtained from the current waveform or given information
    3. Substitute the values of MM and dIdt\frac{dI}{dt} into the equation ε=−MdIdt\varepsilon = -M \frac{dI}{dt}
    4. Solve for the induced emf ε\varepsilon
      • The resulting value will be in volts
  • Example: If the mutual inductance between two circuits is 0.5 H and the current in the nearby circuit changes at a rate of 10 A/s, the induced emf in the circuit of interest would be ε=−0.5 H×10 A/s=−5 V\varepsilon = -0.5 \text{ H} \times 10 \text{ A/s} = -5 \text{ V}

Electromagnetic Coupling and Energy Transfer

  • occurs between when their magnetic fields interact
  • The strength of coupling depends on the and the geometry of the inductors
  • is stored in the coupled magnetic fields of the two circuits
  • The amount of mutual energy is related to the mutual inductance and the currents in both circuits

Key Terms to Review (30)

Air-Core Transformer: An air-core transformer is a type of electrical transformer that uses air as the magnetic core material instead of a ferromagnetic material such as iron. This design allows for a more linear and less saturated magnetic field, making it suitable for high-frequency applications.
Coupled Circuits: Coupled circuits refer to two or more electrical circuits that are magnetically or electrically connected, allowing energy to be transferred between them. This concept is particularly important in the context of mutual inductance, where the changing current in one circuit induces a voltage in another circuit through electromagnetic induction.
Coupling Coefficient: The coupling coefficient is a dimensionless quantity that describes the degree of electromagnetic coupling between two inductively coupled circuits. It represents the fraction of the magnetic flux produced by one circuit that links with the other circuit, and it is a critical parameter in the analysis of mutual inductance.
Digital signal processing: Digital Signal Processing (DSP) involves the manipulation and analysis of signals using digital techniques, typically executed by computers or specialized processors. It is fundamental in applications that require signal filtering, data compression, and error detection.
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.
Electromagnetic Coupling: Electromagnetic coupling refers to the phenomenon where a changing magnetic field in one circuit induces an electromotive force (EMF) in a nearby circuit, allowing the transfer of energy between the two circuits without direct physical contact. This concept is fundamental to the understanding of mutual inductance, a key topic in electromagnetic theory.
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.
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 is 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 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.
Flux linkage: Flux linkage is a measure of the total magnetic flux that passes through a coil of wire, multiplied by the number of turns in the coil. It represents how much magnetic field is linked with the coil and is crucial in understanding how changes in magnetic fields can induce electromotive force (EMF) in circuits. This concept plays a central role in understanding electromagnetic induction and its applications, highlighting how electricity can be generated from magnetic fields.
Henry: The henry (H) is the unit of measurement for the physical quantity of inductance, which is a measure of the magnetic field created by an electric current passing through a coil or other inductor. It is a fundamental unit in the study of electromagnetism and is essential in understanding the behavior of circuits involving inductors.
Henry (H): The henry (H) is the SI unit of inductance. It measures how much voltage is induced in a circuit by a change in current.
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.
Inductors: Inductors are passive electronic components that store energy in the form of a magnetic field when an electric current passes through them. They are fundamental elements in electrical circuits and play a crucial role in various applications, including power supplies, filters, and timing circuits.
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.
Magnetic Field Strength: Magnetic field strength, also known as magnetic flux density or magnetic induction, is a measure of the force exerted by a magnetic field on a moving charged particle or a current-carrying conductor. It is a fundamental concept in the study of electromagnetism and is crucial in understanding the behavior of charged particles and the interactions between magnetic fields and matter.
Magnetic Permeability: Magnetic permeability is a measure of the ability of a material to support the formation of a magnetic field within itself. It describes the degree of magnetization of a material in response to an applied magnetic field, and is a fundamental property that determines the strength and behavior of magnetic fields within a material.
Mutual Energy: Mutual energy refers to the energy transfer that occurs between two inductively coupled circuits when the magnetic field created by one circuit induces a voltage in the other. This interaction is the essence of mutual inductance, where the energy exchange leads to a change in the current in each circuit, resulting in energy being stored and transferred through the magnetic field. Understanding this concept is crucial for analyzing systems where inductors are used together, particularly in transformers and coupled coils.
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.
Neumann Formula: The Neumann formula is a mathematical expression used to calculate the mutual inductance between two coaxial circular loops or coils. It provides a way to determine the magnetic flux linkage between these coils, which is a crucial concept in the study of electromagnetic induction and transformer design.
Primary Coil: The primary coil is a fundamental component in the operation of transformers, where it acts as the input winding that receives the alternating current (AC) from the power source. It is responsible for generating a varying magnetic field that induces a voltage in the secondary coil, enabling the transformation of electrical energy between the two coils.
Secondary Coil: The secondary coil is a component in a transformer that receives the induced current from the primary coil. It is responsible for transforming the voltage and current levels between the input and output of the transformer.
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.
Step-down transformer: A step-down transformer is a device that reduces the voltage from the primary winding to the secondary winding while increasing the current. It operates on the principle of electromagnetic induction and follows the transformer equation.
Transformer: A transformer is a device that uses electromagnetic induction to transfer electrical energy from one circuit to another, usually to change the voltage or current levels between the two circuits. It is a fundamental component in various applications of electricity and electronics.
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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14.2 Self-Inductance and Inductors