14.2 Self-Inductance and Inductors
3 min read•june 24, 2024
is a crucial concept in electromagnetic circuits. When current changes in a coil, it creates a that opposes the change, inducing a voltage. This is key to understanding how inductors work.
Inductors, devices that exploit self-inductance, come in various shapes like solenoids and toroids. Their properties, such as number of turns and core material, affect their inductance and capacity. Understanding these factors is essential for circuit design and analysis.
Self-Inductance and Inductors
Self-induced emf and current change
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- of induction states that a changing magnetic flux through a loop induces an electromotive force (emf) in the loop
- Induced emf opposes the change in magnetic flux, according to (back emf)
- In a circuit with an , a changing current generates a changing magnetic flux
- Changing magnetic flux induces an emf in the , called the self-induced emf (voltage drop across inductor)
- Self-induced emf is proportional to the rate of change of current in the circuit
- Constant of proportionality is the self-inductance, denoted by (, H)
- Self-induced emf is given by:
- is the self-induced emf (volts, V)
- is the self-inductance (henries, H)
- is the rate of change of current (amperes per second, A/s)
- Examples of devices with self-inductance include , , and solenoids
Self-inductance of cylindrical solenoids
- Self-inductance of a depends on its physical properties
- Number of turns,
- Length of the , (meters, m)
- Cross-sectional area, (square meters, m²)
- Permeability of the core material, (henries per meter, H/m)
- Self-inductance of a cylindrical is given by:
- is the self-inductance (henries, H)
- is the permeability of the core material (henries per meter, H/m)
- For an air-core solenoid,
- is the number of turns
- is the cross-sectional area (square meters, m²)
- is the length of the solenoid (meters, m)
- Increasing the number of turns or cross-sectional area increases the self-inductance
- Increasing the length of the solenoid decreases the self-inductance
- Examples of cylindrical solenoids include and inductors in electronic circuits
- The magnetic field inside the solenoid is directly related to its self-inductance
Self-inductance in rectangular toroids
- is a doughnut-shaped inductor with a rectangular or circular cross-section
- Self-inductance of a depends on its geometry and material characteristics
- Number of turns,
- Mean radius of the toroid, (meters, m)
- Height of the rectangular cross-section, (meters, m)
- Width of the rectangular cross-section, (meters, m)
- Permeability of the core material, (henries per meter, H/m)
- Self-inductance of a rectangular toroid is given by:
- is the self-inductance (henries, H)
- is the permeability of the core material (henries per meter, H/m)
- is the number of turns
- is the height of the rectangular cross-section (meters, m)
- is the width of the rectangular cross-section (meters, m)
- is the mean radius of the toroid (meters, m)
- Increasing the number of turns, height, or width of the cross-section increases the self-inductance
- Increasing the mean radius of the toroid decreases the self-inductance
- Examples of rectangular toroids include transformers and inductors in power electronics
Energy and Time Characteristics of Inductors
- Inductors store energy in their magnetic field
- The energy stored in an inductor is given by:
- is the energy stored (joules, J)
- is the inductance (henries, H)
- is the current (amperes, A)
- The of a material affects its energy storage capacity
- The of an inductor in an RL circuit is given by:
- is the time constant (seconds, s)
- is the inductance (henries, H)
- is the resistance (ohms, Ω)
- Inductors with a have higher inductance and energy storage capacity compared to air-core inductors
Key Terms to Review (32)
Camera flashes: Camera flashes are devices that produce a brief burst of light to illuminate a scene. They often use capacitors and inductors to rapidly discharge stored energy, creating a bright flash.
Cylindrical Solenoid: A cylindrical solenoid is a type of inductor, which is a device that stores energy in the form of a magnetic field when an electric current flows through it. The cylindrical solenoid is a coil of wire wound around a cylindrical core, typically made of a ferromagnetic material, that produces a uniform magnetic field inside the coil when an electric current is applied.
DI/dt: The term dI/dt represents the rate of change of current over time in an electrical circuit. It is a crucial concept that describes how quickly the electric current is increasing or decreasing, which directly relates to the behavior of inductors and self-inductance. Understanding dI/dt helps explain how inductors resist changes in current and how RL circuits respond when connected to a voltage source or switched on and off.
Electric Motors: An electric motor is a device that converts electrical energy into mechanical energy, allowing for the rotation of a shaft or rotor. Electric motors are fundamental components in a wide range of applications, from household appliances to industrial machinery, playing a crucial role in the conversion and transmission of power.
Electromagnets: Electromagnets are devices that generate a magnetic field when an electric current flows through a coil of wire. They are widely used in various applications, including electric motors, generators, and electronic devices, due to their ability to create a controlled and adjustable magnetic field.
Energy Storage: Energy storage refers to the ability of a system or device to store energy in a form that can be readily accessed and utilized when needed. It is a crucial concept in the context of self-inductance and inductors, where energy is stored in the magnetic field created by the flow of electric current.
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.
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.
Inductance Density: Inductance density is a measure of the magnetic field strength and energy storage capability within an inductor. It quantifies the amount of inductance per unit volume or per unit length of the inductor, providing a way to characterize and compare the magnetic properties of different inductor designs.
Inductive time constant: The inductive time constant, denoted as $\tau_L$, is the time required for the current in an RL circuit to change significantly (about 63.2%) towards its final value after a change in voltage. It is calculated as $\tau_L = \frac{L}{R}$, where $L$ is the inductance and $R$ is the resistance.
Inductor: An inductor is a passive electrical component that stores energy in its magnetic field when electric current flows through it. It typically consists of a coil of wire and exhibits property known as inductance.
Inductor: An inductor is a passive electronic component that is used to store energy in the form of a magnetic field. It is a fundamental element in various electrical circuits and plays a crucial role in the behavior and functioning of these circuits.
Joseph Henry: Joseph Henry was a pioneering American scientist known for his foundational work in electromagnetism and self-induction in the 19th century. His discoveries laid the groundwork for understanding how inductors function and the principle of self-inductance, demonstrating how a changing current in a coil induces an electromotive force (EMF) in itself. Henry's contributions are vital for comprehending the behavior of inductors in electrical circuits, as he was one of the first to explore the relationship between electric currents and magnetic fields.
L: In the context of electrical circuits, 'L' represents inductance, which is a property of an electrical conductor that quantifies its ability to store energy in a magnetic field when an electric current flows through it. Inductance is a critical aspect of inductors, which are components designed to exploit this property for various applications such as filtering, energy storage, and in tuning circuits. The unit of inductance is the henry (H), named after Joseph Henry, who contributed to the understanding of electromagnetic induction.
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 core: A magnetic core is a solid or hollow cylinder made from ferromagnetic material that is used to enhance the magnetic field produced by electrical components such as coils and inductors. It serves to increase the inductance by concentrating the magnetic field lines, which results in more efficient energy storage and transfer. Magnetic cores are essential in devices like transformers, solenoids, and inductors to improve performance by reducing losses associated with magnetic fields.
Magnetic Field: A magnetic field is a region of space where magnetic forces can be detected. It is a fundamental concept in electromagnetism, describing the invisible lines of force that surround and permeate magnetic materials, electric currents, and changing electric fields. The magnetic field plays a crucial role in various topics within the study of college physics.
Metal detectors: Metal detectors are devices that use electromagnetic fields to detect the presence of metallic objects. They operate based on the principles of electromagnetic induction and eddy currents.
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.
Rectangular toroid: A rectangular toroid is a three-dimensional, doughnut-shaped object with a rectangular cross-section. It serves as a specific type of inductor where the magnetic field is confined within its core, providing efficient inductance properties for electrical circuits. The design of a rectangular toroid allows for enhanced magnetic flux and reduced leakage of magnetic fields, making it ideal for various applications in inductors and transformers.
Self-Induced EMF: Self-induced EMF (electromotive force) is the voltage generated within an inductor due to the changing magnetic field created by the current flowing through the inductor itself. This self-induced voltage opposes any change in the current, as described by Lenz's law, and is a fundamental property of inductors.
Self-inductance: Self-inductance is the property of a conductor by which a change in current induces an electromotive force (EMF) in the same conductor. It is quantified by the inductance \$L\$ and measured in Henrys (H).
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 Constant: The time constant is a fundamental concept that describes the rate of change in various electrical and physical systems. It represents the time required for a system to reach approximately 63% of its final value when undergoing a step change in input.
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.
Traffic signals: Traffic signals are electronic devices that manage and control the flow of vehicles and pedestrians at intersections. They use a combination of lights (typically red, yellow, and green) to indicate when it is safe to proceed, stop, or prepare to stop.
Transformers: A transformer is a device that transfers electrical energy from one electrical circuit to another through the process of electromagnetic induction. It is a crucial component in the transmission and distribution of electrical power, allowing for the efficient transfer of energy over long distances.
ε = -L(dI/dt): The equation ε = -L(dI/dt) describes the relationship between the induced electromotive force (ε) and the rate of change of current (dI/dt) in a circuit with an inductor. This equation is a fundamental principle in the study of electromagnetic induction and self-inductance.
μ: The Greek letter μ is a commonly used symbol in physics that typically represents the coefficient of friction, a dimensionless quantity that describes the ratio of the friction force between two surfaces to the normal force pressing them together. This term is particularly relevant in the context of self-inductance and inductors, where it plays a crucial role in understanding the behavior of these electrical components.
μ₀: μ₀, also known as the permeability of free space or the vacuum permeability, is a fundamental physical constant that represents the magnetic permeability of free space or a vacuum. It is a crucial parameter in the study of electromagnetism and is used in various electromagnetic laws and equations.