Electromagnetic induction

Last updated
Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current that flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G). Induction experiment.png
Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current that flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G).

Electromagnetic or magnetic induction is the production of an electromotive force (i.e., voltage) across an electrical conductor in a changing magnetic field.

Electromotive force scalar physical quantity

Electromotive force, abbreviated emf, is the electrical intensity or "pressure" developed by a source of electrical energy such as a battery or generator. A device that converts other forms of energy into electrical energy provides an emf as its output.

Electrical conductor object or material which permits the flow of electricity

In physics and electrical engineering, a conductor is an object or type of material that allows the flow of an electrical current in one or more directions. Materials made of metal are common electrical conductors. Electrical current is generated by the flow of negatively charged electrons, positively charged holes, and positive or negative ions in some cases.

Magnetic field spatial distribution of vectors allowing the calculation of the magnetic force on a test particle

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field varies with location. As such, it is an example of a vector field.

Contents

Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction. Lenz's law describes the direction of the induced field. Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism.

Michael Faraday English scientist

Michael Faraday FRS was a British scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis.

James Clerk Maxwell Scottish physicist

James Clerk Maxwell was a Scottish scientist in the field of mathematical physics. His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.

Faradays law of induction

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

Electromagnetic induction has found many applications, including electrical components such as inductors and transformers, and devices such as electric motors and generators.

Inductor passive two-terminal electrical component that stores energy in its magnetic field

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core.

Transformer electrical artefact that transfers energy through electromagnetic induction

A transformer is a static electrical device that transfers electrical energy between two or more circuits. A varying current in one coil of the transformer produces a varying magnetic flux, which, in turn, induces a varying electromotive force across a second coil wound around the same core. Electrical energy can be transferred between the two coils, without a metallic connection between the two circuits. Faraday's law of induction discovered in 1831 described the induced voltage effect in any coil due to changing magnetic flux encircled by the coil.

Electric motor electromechanical device

An electric motor is an electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and winding currents to generate force in the form of rotation. Electric motors can be powered by direct current (DC) sources, such as from batteries, motor vehicles or rectifiers, or by alternating current (AC) sources, such as a power grid, inverters or electrical generators. An electric generator is mechanically identical to an electric motor, but operates in the reverse direction, accepting mechanical energy and converting this mechanical energy into electrical energy.

History

A diagram of Faraday's iron ring apparatus. Change in the magnetic flux of the left coil induces a current in the right coil. Faraday emf experiment.svg
A diagram of Faraday's iron ring apparatus. Change in the magnetic flux of the left coil induces a current in the right coil.
Faraday's disk (see homopolar generator) Faraday disk generator.jpg
Faraday's disk (see homopolar generator)

Electromagnetic induction was discovered by Michael Faraday, published in 1831. [3] [4] It was discovered independently by Joseph Henry in 1832. [5] [6]

Joseph Henry American scientist

Joseph Henry was an American scientist who served as the first Secretary of the Smithsonian Institution. He was the secretary for the National Institute for the Promotion of Science, a precursor of the Smithsonian Institution. He was highly regarded during his lifetime. While building electromagnets, Henry discovered the electromagnetic phenomenon of self-inductance. He also discovered mutual inductance independently of Michael Faraday, though Faraday was the first to make the discovery and publish his results. Henry developed the electromagnet into a practical device. He invented a precursor to the electric doorbell and electric relay (1835). The SI unit of inductance, the henry, is named in his honor. Henry's work on the electromagnetic relay was the basis of the practical electrical telegraph, invented by Samuel F. B. Morse and Sir Charles Wheatstone, separately.

In Faraday's first experimental demonstration (August 29, 1831), he wrapped two wires around opposite sides of an iron ring or "torus" (an arrangement similar to a modern toroidal transformer).[ citation needed ] Based on his understanding of electromagnets, he expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. He saw a transient current, which he called a "wave of electricity", when he connected the wire to the battery and another when he disconnected it. [7] This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. [2] Within two months, Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk"). [8]

Torus doughnut-shaped surface of revolution

In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution.

Galvanometer instrument to measure electric current

A galvanometer is an electromechanical instrument used for detecting and indicating an electric current. A galvanometer works as an actuator, by producing a rotary deflection, in response to electric current flowing through a coil in a constant magnetic field. Early galvanometers were not calibrated, but their later developments were used as measuring instruments, called ammeters, to measure the current flowing through an electric circuit.

Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically. [9] An exception was James Clerk Maxwell, who used Faraday's ideas as the basis of his quantitative electromagnetic theory. [9] [10] [11] In Maxwell's model, the time varying aspect of electromagnetic induction is expressed as a differential equation, which Oliver Heaviside referred to as Faraday's law even though it is slightly different from Faraday's original formulation and does not describe motional EMF. Heaviside's version (see Maxwell–Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations.

Oliver Heaviside electrical engineer, mathematician and physicist

Oliver Heaviside FRS was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science for years to come.

Maxwells equations set of partial differential equations that describe how electric and magnetic fields are generated and altered by each other and by charges and currents

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.

In 1834 Heinrich Lenz formulated the law named after him to describe the "flux through the circuit". Lenz's law gives the direction of the induced EMF and current resulting from electromagnetic induction.

Theory

Faraday's law of induction and Lenz's law

A solenoid Solenoid-1.png
A solenoid
The longitudinal cross section of a solenoid with a constant electrical current running through it. The magnetic field lines are indicated, with their direction shown by arrows. The magnetic flux corresponds to the 'density of field lines'. The magnetic flux is thus densest in the middle of the solenoid, and weakest outside of it. VFPt Solenoid correct2.svg
The longitudinal cross section of a solenoid with a constant electrical current running through it. The magnetic field lines are indicated, with their direction shown by arrows. The magnetic flux corresponds to the 'density of field lines'. The magnetic flux is thus densest in the middle of the solenoid, and weakest outside of it.

Faraday's law of induction makes use of the magnetic flux ΦB through a region of space enclosed by a wire loop. The magnetic flux is defined by a surface integral: [12]

where dA is an element of the surface Σ enclosed by the wire loop, B is the magnetic field. The dot product B·dA corresponds to an infinitesimal amount of magnetic flux. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.

When the flux through the surface changes, Faraday's law of induction says that the wire loop acquires an electromotive force (EMF). [note 1] The most widespread version of this law states that the induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux enclosed by the circuit: [16] [17]

,

where is the EMF and ΦB is the magnetic flux. The direction of the electromotive force is given by Lenz's law which states that an induced current will flow in the direction that will oppose the change which produced it. [18] This is due to the negative sign in the previous equation. To increase the generated EMF, a common approach is to exploit flux linkage by creating a tightly wound coil of wire, composed of N identical turns, each with the same magnetic flux going through them. The resulting EMF is then N times that of one single wire. [19] [20]

Generating an EMF through a variation of the magnetic flux through the surface of a wire loop can be achieved in several ways:

  1. the magnetic field B changes (e.g. an alternating magnetic field, or moving a wire loop towards a bar magnet where the B field is stronger),
  2. the wire loop is deformed and the surface Σ changes,
  3. the orientation of the surface dA changes (e.g. spinning a wire loop into a fixed magnetic field),
  4. any combination of the above

Maxwell–Faraday equation

In general, the relation between the EMF in a wire loop encircling a surface Σ, and the electric field E in the wire is given by

where d is an element of contour of the surface Σ, combining this with the definition of flux

we can write the integral form of the Maxwell–Faraday equation

It is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism.

Faraday's law and relativity

Faraday's law describes two different phenomena: the motional EMF generated by a magnetic force on a moving wire (see Lorentz force), and the transformer EMF this is generated by an electric force due to a changing magnetic field (due to the differential form of the Maxwell–Faraday equation). James Clerk Maxwell drew attention to the separate physical phenomena in 1861. [21] [22] This is believed to be a unique example in physics of where such a fundamental law is invoked to explain two such different phenomena. [23]

Einstein noticed that the two situations both corresponded to a relative movement between a conductor and a magnet, and the outcome was unaffected by which one was moving. This was one of the principal paths that led him to develop special relativity. [24]

Applications

The principles of electromagnetic induction are applied in many devices and systems, including:


Electrical generator

Rectangular wire loop rotating at angular velocity ω in radially outward pointing magnetic field B of fixed magnitude. The circuit is completed by brushes making sliding contact with top and bottom discs, which have conducting rims. This is a simplified version of the drum generator. Spindle.PNG
Rectangular wire loop rotating at angular velocity ω in radially outward pointing magnetic field B of fixed magnitude. The circuit is completed by brushes making sliding contact with top and bottom discs, which have conducting rims. This is a simplified version of the drum generator.

The EMF generated by Faraday's law of induction due to relative movement of a circuit and a magnetic field is the phenomenon underlying electrical generators. When a permanent magnet is moved relative to a conductor, or vice versa, an electromotive force is created. If the wire is connected through an electrical load, current will flow, and thus electrical energy is generated, converting the mechanical energy of motion to electrical energy. For example, the drum generator is based upon the figure to the bottom-right. A different implementation of this idea is the Faraday's disc, shown in simplified form on the right.

In the Faraday's disc example, the disc is rotated in a uniform magnetic field perpendicular to the disc, causing a current to flow in the radial arm due to the Lorentz force. Mechanical work is necessary to drive this current. When the generated current flows through the conducting rim, a magnetic field is generated by this current through Ampère's circuital law (labelled "induced B" in the figure). The rim thus becomes an electromagnet that resists rotation of the disc (an example of Lenz's law). On the far side of the figure, the return current flows from the rotating arm through the far side of the rim to the bottom brush. The B-field induced by this return current opposes the applied B-field, tending to decrease the flux through that side of the circuit, opposing the increase in flux due to rotation. On the near side of the figure, the return current flows from the rotating arm through the near side of the rim to the bottom brush. The induced B-field increases the flux on this side of the circuit, opposing the decrease in flux due to rotation. Thus, both sides of the circuit generate an EMF opposing the rotation. The energy required to keep the disc moving, despite this reactive force, is exactly equal to the electrical energy generated (plus energy wasted due to friction, Joule heating, and other inefficiencies). This behavior is common to all generators converting mechanical energy to electrical energy.

Electrical transformer

When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, d ΦB / d t. Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF. If the two ends of this loop are connected through an electrical load, current will flow.

Current clamp

A current clamp Current Clamp.jpg
A current clamp

A current clamp is a type of transformer with a split core which can be spread apart and clipped onto a wire or coil to either measure the current in it or, in reverse, to induce a voltage. Unlike conventional instruments the clamp does not make electrical contact with the conductor or require it to be disconnected during attachment of the clamp.

Magnetic flow meter

Faraday's law is used for measuring the flow of electrically conductive liquids and slurries. Such instruments are called magnetic flow meters. The induced voltage ℇ generated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by:

where ℓ is the distance between electrodes in the magnetic flow meter.

Eddy currents

Conductors (of finite dimensions) moving through a uniform magnetic field, or stationary within a changing magnetic field, will have currents induced within them. These induced eddy currents can be undesirable, since they dissipate energy in the resistance of the conductor. There are a number of methods employed to control these undesirable inductive effects.

Electromagnet laminations

Hawkins Electrical Guide - Figure 292 - Eddy currents in a solid armature.jpg

Eddy currents occur when a solid metallic mass is rotated in a magnetic field, because the outer portion of the metal cuts more lines of force than the inner portion, hence the induced electromotive force not being uniform, tends to set up currents between the points of greatest and least potential. Eddy currents consume a considerable amount of energy and often cause a harmful rise in temperature. [25]

Hawkins Electrical Guide - Figure 293 - Armature core with a few laminations showing effect on eddy currents.jpg

Only five laminations or plates are shown in this example, so as to show the subdivision of the eddy currents. In practical use, the number of laminations or punchings ranges from 40 to 66 per inch, and brings the eddy current loss down to about one percent. While the plates can be separated by insulation, the voltage is so low that the natural rust/oxide coating of the plates is enough to prevent current flow across the laminations. [25]

Small DC Motor pole laminations and overview.jpg

This is a rotor approximately 20mm in diameter from a DC motor used in a CD player. Note the laminations of the electromagnet pole pieces, used to limit parasitic inductive losses.

Parasitic induction within conductors

Hawkins Electrical Guide - Figure 291 - Formation of eddy currents in a solid bar inductor.jpg

In this illustration, a solid copper bar conductor on a rotating armature is just passing under the tip of the pole piece N of the field magnet. Note the uneven distribution of the lines of force across the copper bar. The magnetic field is more concentrated and thus stronger on the left edge of the copper bar (a,b) while the field is weaker on the right edge (c,d). Since the two edges of the bar move with the same velocity, this difference in field strength across the bar creates whorls or current eddies within the copper bar. [25]

High current power-frequency devices, such as electric motors, generators and transformers, use multiple small conductors in parallel to break up the eddy flows that can form within large solid conductors. The same principle is applied to transformers used at higher than power frequency, for example, those used in switch-mode power supplies and the intermediate frequency coupling transformers of radio receivers.

See also

Related Research Articles

Electromagnetic field physical field produced by electrically charged objects

An electromagnetic field is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature.

Lorentz force mutual force exerted by two punctual charges in relative motion

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

Inductive coupling

In electrical engineering, two conductors are said to be inductively coupled or magnetically coupled when they are configured such that a change in current through one wire induces a voltage across the ends of the other wire through electromagnetic induction. A changing current through the first wire creates a changing magnetic field around it by Ampere's circuital law. The changing magnetic field induces an electromotive force in the second wire by Faraday's law of induction. The amount of inductive coupling between two conductors is measured by their mutual inductance.

Flux measure of the flow of something through a surface, in some cases per surface area

Flux describes any effect that appears to pass or travel through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying mathematics when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.

Magnetic flux physical quantity

In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B passing through that surface. The SI unit of magnetic flux is the weber (Wb), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage in the measuring coils to calculate the magnetic flux.

Electromagnet type of magnet in which the magnetic field is produced by the flow of electric current

An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. The magnetic field disappears when the current is turned off. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the hole in the center of the coil. The wire turns are often wound around a magnetic core made from a ferromagnetic or ferrimagnetic material such as iron; the magnetic core concentrates the magnetic flux and makes a more powerful magnet.

Inductance electrical property

In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in electric current through it induces an electromotive force (voltage) in the conductor. It is more accurately called self-inductance. The same property causes a current in one conductor to induce an electromotive force in nearby conductors; this is called mutual inductance.

Lenzs law

Lenz's law, named after the physicist Emil Lenz who formulated it in 1834, states that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field. Or as informally, yet concisely summarised by D.J. Griffiths:

Nature abhors a change in flux.

Eddy currents are loops of electrical current induced within conductors by a changing magnetic field in the conductor according to Faraday's law of induction. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. They can be induced within nearby stationary conductors by a time-varying magnetic field created by an AC electromagnet or transformer, for example, or by relative motion between a magnet and a nearby conductor. The magnitude of the current in a given loop is proportional to the strength of the magnetic field, the area of the loop, and the rate of change of flux, and inversely proportional to the resistivity of the material.

Electrodynamic suspension

Electrodynamic suspension (EDS) is a form of magnetic levitation in which there are conductors which are exposed to time-varying magnetic fields. This induces eddy currents in the conductors that creates a repulsive magnetic field which holds the two objects apart.

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

Counter-electromotive force, also known as back electromotive force, is the electromotive force or "voltage" that opposes the change in current which induced it. CEMF is the EMF caused by magnetic induction.

Faraday paradox

The Faraday paradox or Faraday's paradox is any experiment in which Michael Faraday's law of electromagnetic induction appears to predict an incorrect result. The paradoxes fall into two classes:

Toroidal inductors and transformers

Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal shape. They are passive electronic components, consisting of a circular ring or donut shaped magnetic core of ferromagnetic material such as laminated iron, iron powder, or ferrite, around which wire is wound.

References

Notes

  1. The EMF is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads. Mathematically, is defined as the energy available from a unit charge that has traveled once around the wire loop. [13] [14] [15]

References

  1. Poyser, A. W. (1892). Magnetism and Electricity: A Manual for Students in Advanced Classes. London and New York: Longmans, Green, & Co. p. 285.
  2. 1 2 Giancoli, Douglas C. (1998). Physics: Principles with Applications (Fifth ed.). pp. 623–624.
  3. Ulaby, Fawwaz (2007). Fundamentals of applied electromagnetics (5th ed.). Pearson:Prentice Hall. p. 255. ISBN   0-13-241326-4.
  4. "Joseph Henry". Distinguished Members Gallery, National Academy of Sciences. Archived from the original on 2013-12-13. Retrieved 2006-11-30.
  5. "A Brief History of Electromagnetism" (PDF).
  6. "Electromagnetism". Smithsonian Institution Archives.
  7. Michael Faraday, by L. Pearce Williams, p. 182-3
  8. Michael Faraday, by L. Pearce Williams, p. 191–5
  9. 1 2 Michael Faraday, by L. Pearce Williams, p. 510
  10. Maxwell, James Clerk (1904), A Treatise on Electricity and Magnetism, Vol. II, Third Edition. Oxford University Press, pp. 178–9 and 189.
  11. "Archives Biographies: Michael Faraday", The Institution of Engineering and Technology.
  12. Good, R. H. (1999). Classical Electromagnetism. Saunders College Publishing. p. 107. ISBN   0-03-022353-9.
  13. Feynman, R. P.; Leighton, R. B.; Sands, M. L. (2006). The Feynman Lectures on Physics, Volume 2. Pearson/Addison-Wesley. pp. 17–2. ISBN   0-8053-9049-9.
  14. Griffiths, D. J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 301–303. ISBN   0-13-805326-X.
  15. Tipler, P. A.; Mosca, G. (2003). Physics for Scientists and Engineers (5th ed.). W.H. Freeman. p. 795. ISBN   978-0716708100.
  16. Jordan, E.; Balmain, K. G. (1968). Electromagnetic Waves and Radiating Systems (2nd ed.). Prentice-Hall. p. 100.
  17. Hayt, W. (1989). Engineering Electromagnetics (5th ed.). McGraw-Hill. p. 312. ISBN   0-07-027406-1.
  18. Schmitt, R. (2002). Electromagnetics Explained. p. 75.
  19. Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN   0-7195-3382-1.
  20. Nave, C. R. "Faraday's Law". HyperPhysics . Georgia State University . Retrieved 2011-08-29.
  21. Maxwell, J. C. (1861). "On physical lines of force". Philosophical Magazine . 90: 11–23. doi:10.1080/1478643100365918.
  22. Griffiths, D. J. (1999). Introduction to Electrodynamics (3rd ed.). Prentice Hall. pp. 301–303. ISBN   0-13-805326-X. Note that the law relating flux to EMF, which this article calls "Faraday's law", is referred to by Griffiths as the "universal flux rule". He uses the term "Faraday's law" to refer to what this article calls the "Maxwell–Faraday equation".
  23. "The flux rule" is the terminology that Feynman uses to refer to the law relating magnetic flux to EMF. Feynman, R. P.; Leighton, R. B.; Sands, M. L. (2006). The Feynman Lectures on Physics, Volume II. Pearson/Addison-Wesley. p. 17-2. ISBN   0-8053-9049-9.
  24. Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik . 17 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.
    Translated in Einstein, A. (1923). "On the Electrodynamics of Moving Bodies" (PDF). The Principle of Relativity. Jeffery, G.B.; Perret, W. (transl.). London: Methuen and Company.
  25. 1 2 3 Images and reference text are from the public domain book: Hawkins Electrical Guide, Volume 1, Chapter 19: Theory of the Armature, pp. 270–273, Copyright 1917 by Theo. Audel & Co., Printed in the United States

Further reading