Faraday's law of induction: Difference between revisions

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Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current which 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).^[1]

Faraday's law of induction, or simply Faraday's law, is a law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf). This phenomenon, known as electromagnetic induction, is the fundamental operating principle of transformers, inductors, and many types of electric motors, generators and solenoids.^[2]^[3]

The Maxwell–Faraday equation (listed as one of Maxwell's equations) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that emf (electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) appears on a conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.

Once Faraday's law had been discovered, one aspect of it (transformer emf) was formulated as the Maxwell–Faraday equation. The equation of Faraday's law can be derived by the Maxwell–Faraday equation (describing transformer emf) and the Lorentz force (describing motional emf). The integral form of the Maxwell–Faraday equation describes only the transformer emf, while the equation of Faraday's law describes both the transformer emf and the motional emf.

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History

Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832.^[4] Faraday was the first to publish the results of his experiments.^[5]^[6]

File:Faraday emf experiment.svg

Faraday's 1831 demonstration^[7]

Faraday's notebook on August 29, 1831^[8] describes an experimental demonstration of electromagnetic induction^[9] that wraps two wires around opposite sides of an iron ring (like a modern toroidal transformer). His assessment of newly-discovered properties of electromagnets suggested 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. Indeed, a galvanometer's needle measured a transient current (which he called a "wave of electricity") on the right side's wire when he connected or disconnected the left side's wire to a battery.^[10]Template:Rp This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected.^[7] His notebook entry also noted that fewer wraps for the battery side resulted in a greater disturbance of the galvanometer's needle.^[8]

Within two months, Faraday had 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").^[10]Template:Rp

File:Faraday disk generator.jpg

Faraday's disk, the first electric generator, a type of homopolar generator

Michael 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.^[10]Template:Rp An exception was James Clerk Maxwell, who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory.^[10]Template:Rp^[11]^[12] In Maxwell's papers, 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 different from the original version of Faraday's law, and does not describe motional emf. Heaviside's version is the form recognized today in the group of equations known as Maxwell's equations.

Lenz's law, formulated by Emil Lenz in 1834,^[13] describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction (elaborated upon in the examples below).

The laws of induction of electric currents in mathematical form were established by Franz Ernst Neumann in 1845.^[14]Template:Non-primary source needed

According to Albert Einstein, much of the groundwork and discovery of his special relativity theory was presented by this law of induction by Faraday in 1834.^[15]^[16]

Flux rule

File:Electromagnetic induction - solenoid to loop - animation.gif

Alternating electric current flows through the solenoid on the left, producing a changing magnetic field. This field causes, by electromagnetic induction, an electric current to flow in the wire loop on the right.

Faraday's law of induction, also known as the flux rule and FaradayTemplate:EndashLenz law^[17], states that the electromotive force (emf) around a closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. This rule holds for any circuit made of thin wire and accounts for changes in flux due to variations in the magnetic field, movement of the circuit, or deformation of its shape.^[18] The direction of the induced emf is given by Lenz's law, which states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux.^[19]

Mathematically, in SI units, the law is written as $ℰ = - \frac{d Φ_{B}}{d t},$ where $ℰ$ is the electromotive force (emf) and Template:Math is the magnetic flux throught the circuit. It is defined as the surface integral of the magnetic field over a surface Template:Math, whose boundary is the wire loop: $Φ_{B} = \iint_{Σ (t)} 𝐁 (t) \cdot d 𝐀,$ where Template:Math is an element of area vector of the moving surface Template:Math, directed normal to the surface, and Template:Math is the magnetic field. The dot product Template:Math represents the element of flux through Template:Math. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic field lines that pass through the loop.

File:Surface integral illustration.svg

The definition of surface integral relies on splitting the surface Template:Math into small surface elements. Each element is associated with a vector Template:Math of magnitude equal to the area of the element and with direction normal to the element and pointing "outward" (with respect to the orientation of the surface).

When the flux changes—because Template:Math changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an emf, defined as the energy available from a unit charge that has traveled once around the wire loop.^[20]Template:Rp^[21]^[22] (Although some sources state the definition differently, this expression was chosen for compatibility with the equations of special relativity.Script error: No such module "Unsubst".) Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.Script error: No such module "Unsubst".

File:Salu's left-hand rule (magnetic induction).png

A Left Hand Rule for Faraday's Law. The sign of Template:Math, the change in flux, is found based on the relationship between the magnetic field Template:Math, the area of the loop Template:Mvar, and the normal n to that area, as represented by the fingers of the left hand. If Template:Math is positive, the direction of the emf is the same as that of the curved fingers (yellow arrowheads). If Template:Math is negative, the direction of the emf is against the arrowheads.^[23]

It is possible to find out the direction of the electromotive force (emf) directly from Faraday’s law, without invoking Lenz's law. A left hand rule helps doing that, as follows:^[23]^[24]

Align the curved fingers of the left hand with the loop (yellow line).
Stretch your thumb. The stretched thumb indicates the direction of Template:Math (brown), the normal to the area enclosed by the loop.
Find the sign of Template:Math, the change in flux. Determine the initial and final fluxes (whose difference is Template:Math) with respect to the normal Template:Math, as indicated by the stretched thumb.
If the change in flux, Template:Math, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
If Template:Math is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads).

For a tightly wound coil of wire, composed of Template:Mvar identical turns, the same magnetic field lines cross the surface Template:Mvar times. In this case, Faraday's law of induction states that^[25]^[26] $ℰ = - N \frac{d Φ_{B}}{d t}$ where Template:Mvar is the number of turns of wire and Template:Math is the magnetic flux through a single loop. The product Template:Math is known as linked flux.^[27]

Maxwell–Faraday equation

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File:Stokes' Theorem.svg

An illustration of the Kelvin–Stokes theorem with surface Template:Math, its boundary Template:Math, and orientation Template:Math set by the right-hand rule

The Maxwell–Faraday equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-conservative electric field, and vice versa. The Maxwell–Faraday equation is

Template:Equation box 1

(in SI units) where Template:Math is the curl operator and again Template:Math is the electric field and Template:Math is the magnetic field. These fields can generally be functions of position Template:Math and time Template:Mvar.^[28]

The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin–Stokes theorem:^[29]

Template:Equation box 1

where, as indicated in the figure, Template:Math is a surface bounded by the closed contour Template:Math, Template:Math is an infinitesimal vector element of the contour Template:Math, and Template:Math is an infinitesimal vector element of surface Template:Math. Its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.

Both Template:Math and Template:Math have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin–Stokes theorem. For a planar surface Template:Math, a positive path element Template:Math of curve Template:Math is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal Template:Math to the surface Template:Math.

The line integral around Template:Math is called circulation.^[20]Template:Rp A nonzero circulation of Template:Math is different from the behavior of the electric field generated by static charges. A charge-generated Template:Math-field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral.

The integral equation is true for any path Template:Math through space, and any surface Template:Math for which that path is a boundary.

If the surface Template:Math is not changing in time, the equation can be rewritten: $\oint_{\partial Σ} 𝐄 \cdot d 𝐥 = - \frac{d}{d t} \int_{Σ} 𝐁 \cdot d 𝐀 .$ The surface integral at the right-hand side is the explicit expression for the magnetic flux Template:Math through Template:Math.

The electric vector field induced by a changing magnetic flux, the solenoidal component of the overall electric field, can be approximated in the non-relativistic limit by the volume integral equation^[28]Template:Rp $𝐄_{s} (𝐫, t) \approx - \frac{1}{4 π} \underset{V}{∭} \frac{(\frac{\partial 𝐁 (𝐫^{'}, t)}{\partial t}) \times (𝐫 - 𝐫^{'})}{| 𝐫 - 𝐫^{'} |^{3}} d^{3} 𝐫^{'}$

Derivation of the flux rule from microscopic equations

The four Maxwell's equations, together with the Lorentz force law, form a complete foundation for classical electromagnetism.^[20]^[21] From these, Faraday's law can be derived directly.^[30]^[31]^[32]

The derivation begins by considering the time derivative of the magnetic flux through a surface Template:Math that may vary with time: $\frac{d Φ_{B}}{d t} = \frac{d}{d t} \int_{Σ (t)} 𝐁 (t) \cdot d 𝐀 .$ The magnetic flux can change for two reasons: the magnetic field itself may vary with time, and the surface may move or change shape, enclosing a different region of space. Both effects are captured by the three-dimensional version of the Leibniz integral rule, sometimes referred to as the "flux theorem":^[33] $\frac{d}{d t} \int_{Σ (t)} 𝐁 \cdot d 𝐀 = \int_{Σ (t)} (\frac{\partial 𝐁}{\partial t} + (\nabla \cdot 𝐁) 𝐯_{c}) \cdot d 𝐀 - \oint_{\partial Σ (t)} (𝐯_{c} \times 𝐁) \cdot d 𝐥$ Here, Template:Math is the moving boundary of the surface and $𝐯_{c}$ is the local velocity of the boundary at each point. By Gauss's law for magnetism ( $\nabla \cdot 𝐁 = 0$ ), the second term under the area integral vanishes. Applying the Maxwell–Faraday equation to the remaining term, $\int_{Σ (t)} \frac{\partial 𝐁}{\partial t} \cdot d 𝐀 = - \oint_{\partial Σ (t)} 𝐄 \cdot d 𝐥,$ and combining the two line integrals gives $\frac{d Φ_{B}}{d t} = - \oint_{\partial Σ (t)} (𝐄 + 𝐯_{c} \times 𝐁) \cdot d 𝐥 .$ This is an exact result, derived from Maxwell's equations and vector calculus.^[33]

However, the quantity inside the integral is not the full Lorentz force per unit charge, because the velocity $𝐯_{c}$ represents the motion of loop boundary, not the actual velocity of the charge carriers. To recover the physical electromotive force, we must distinguish between these velocities. Let us choose the integration path to coincide with the physical circuit. The velocity of a charge carrier in the conductor is then given by

𝐯 (𝐫, t) = 𝐯_{c} (𝐫, t) + 𝐯_{d} (𝐫, t)

,

where $𝐯_{c}$ is the velocity of the conductor (the ions in the material), and $𝐯_{d}$ is the drift velocity of the electrons relative to the material. This decomposition assumes nonrelativistic (Galilean) addition of velocities.^[33]

The emf $ℰ$ associated with the Lorentz force is defined as $ℰ = \oint_{\partial Σ (t)} (𝐄 + 𝐯 \times 𝐁) \cdot d 𝐥 .$ Substituting the expression for the carrier velocity and the above result yields:Template:Equation box 1 Equivalently, this can be expressed as

ℰ = - \int_{\partial Σ (t)} \frac{\partial 𝐁}{\partial t} \cdot d 𝐀 + \oint_{\partial Σ (t)} (𝐯 \times 𝐁) \cdot d 𝐥,

where the first term is the "transformer emf" due to a time-varying magnetic field, and the second term is the "motional emf" due to the magnetic Lorentz force by the motion of the charges in the magnetic field.^[33]

In circuits made of thin, one-dimensional wires, the drift velocity is aligned with the wire, and hence with the integration element $d 𝐥$ . In that case, the cross product $𝐯_{d} \times 𝐁$ is perpendicular to $d 𝐥$ , and the term proportional to the drift velocity vanishes. This recovers the standard form of Faraday's law: $ℰ = - \frac{d Φ_{B}}{d t}$ In this case, the emf can also be expressed as a sum^[33] $ℰ = - \int_{Σ (t)} \frac{\partial 𝐁}{\partial t} \cdot d 𝐀 + \oint_{\partial Σ (t)} (𝐯_{c} \times 𝐁) \cdot d 𝐥 .$ In conductors that are not thin wires, the drift velocity term $\oint_{\partial Σ (t)} (𝐯_{d} \times 𝐁) \cdot d 𝐥$ may not vanish exactly. However, electrons typically drift at speeds of the order of 10^-4 m/s, and the contribution is often negligible compared to other effects.^[34] A notable exception is the Hall effect, where magnetic flux term $d Φ_{B} / d t$ vanishes, and the observed Hall voltage arises entirely from the drift velocity term.^[33]

Exceptions

Script error: No such module "Labelled list hatnote". It is tempting to generalize Faraday's law to state: If Template:Math is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through Template:Math equals the emf around Template:Math. This statement, however, is not always true. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve Template:Math matches the actual velocity of the material conducting the electricity.^[35] If the conductor is not an infinitely thin wire, one may also have take into account the velocity of charges with respect to the material.^[33] The two examples illustrated below show that one often obtains incorrect results when Faraday's law is applied too broadly.^[20]

Faraday's homopolar generator. The disc rotates with angular rate Template:Mvar, sweeping the conducting radius circularly in the static magnetic field Template:Math (which direction is along the disk surface normal). The magnetic Lorentz force Template:Math drives a current along the conducting radius to the conducting rim, and from there the circuit completes through the lower brush and the axle supporting the disc. This device generates an emf and a current, although the shape of the "circuit" is constant and thus the flux through the circuit does not change with time.

Faraday's homopolar generator. The disc rotates with angular rate Template:Mvar, sweeping the conducting radius circularly in the static magnetic field Template:Math (which direction is along the disk surface normal). The magnetic Lorentz force Template:Math drives a current along the conducting radius to the conducting rim, and from there the circuit completes through the lower brush and the axle supporting the disc. This device generates an emf and a current, although the shape of the "circuit" is constant and thus the flux through the circuit does not change with time.
A wire (solid red lines) connects to two touching metal plates (silver) to form a circuit. The whole system sits in a uniform magnetic field, normal to the page. If the abstract path ∂Σ follows the primary path of current flow (marked in red), then the magnetic flux through this path changes dramatically as the plates are rotated, yet the emf is almost zero. After Feynman Lectures on Physics: ch17

A wire (solid red lines) connects to two touching metal plates (silver) to form a circuit. The whole system sits in a uniform magnetic field, normal to the page. If the abstract path Template:Math follows the primary path of current flow (marked in red), then the magnetic flux through this path changes dramatically as the plates are rotated, yet the emf is almost zero. After Feynman Lectures on Physics^[20]Template:Rp

One can analyze examples like these by taking care that the path Template:Math moves with the same velocity as the material.^[35] The electromotive force can always be correctly calculated by combining the Lorentz force law with the Maxwell–Faraday equation:^[20]Template:Rp $ℰ = \int_{\partial Σ} (𝐄 + 𝐯 \times 𝐁) \cdot d 𝐥 = - \int_{\partial Σ (t)} \frac{\partial 𝐁}{\partial t} \cdot d 𝐀 + \oint_{\partial Σ (t)} (𝐯 \times 𝐁) \cdot d 𝐥,$ where Template:Math is the velocity of the conductor in the frame of reference in which Template:Math in described. The time derivative cannot in general be moved outside the integral since the position or shape of the loop may be a function of time.^[36]

Faraday's law and relativity

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Two phenomena

Faraday's law is a single equation describing two different phenomena: the motional emf generated by a magnetic force on a moving wire, and the transformer emf generated by an electric force due to a changing magnetic field (described by the Maxwell–Faraday equation).

James Clerk Maxwell drew attention to this fact in his 1861 paper On Physical Lines of Force.^[37] In the latter half of Part II of that paper, Maxwell gives a separate physical explanation for each of the two phenomena.

A reference to these two aspects of electromagnetic induction is made in some modern textbooks.^[38] As Richard Feynman states:

So the "flux rule" that the emf in a circuit is equal to the rate of change of the magnetic flux through the circuit applies whether the flux changes because the field changes or because the circuit moves (or both) ...
Yet in our explanation of the rule we have used two completely distinct laws for the two cases – Template:Math for "circuit moves" and Template:Math for "field changes".
We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena.
— Template:Comma separated entries

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Explanation based on four-dimensional formalism

In the general case, explanation of the motional emf appearance by action of the magnetic force on the charges in the moving wire or in the circuit changing its area is unsatisfactory. As a matter of fact, the charges in the wire or in the circuit could be completely absent, will then the electromagnetic induction effect disappear in this case? This situation is analyzed in the article, in which, when writing the integral equations of the electromagnetic field in a four-dimensional covariant form, in the Faraday’s law the total time derivative of the magnetic flux through the circuit appears instead of the partial time derivative.^[40] Thus, electromagnetic induction appears either when the magnetic field changes over time or when the area of the circuit changes. From the physical point of view, it is better to speak not about the induction emf, but about the induced electric field strength $𝐄 = - \nabla ℰ - \frac{\partial 𝐀}{\partial t}$ , that occurs in the circuit when the magnetic flux changes. In this case, the contribution to $𝐄$ from the change in the magnetic field is made through the term $- \frac{\partial 𝐀}{\partial t}$ , where $𝐀$ is the vector potential. If the circuit area is changing in case of the constant magnetic field, then some part of the circuit is inevitably moving, and the electric field $𝐄$ emerges in this part of the circuit in the comoving reference frame K’ as a result of the Lorentz transformation of the magnetic field $𝐁$ , present in the stationary reference frame K, which passes through the circuit. The presence of the field $𝐄$ in K’ is considered as a result of the induction effect in the moving circuit, regardless of whether the charges are present in the circuit or not. In the conducting circuit, the field $𝐄$ causes motion of the charges. In the reference frame K, it looks like appearance of emf of the induction $ℰ$ , the gradient of which in the form of $- \nabla ℰ$ , taken along the circuit, seems to generate the field $𝐄$ .

Einstein's view

Reflection on this apparent dichotomy was one of the principal paths that led Albert Einstein to develop special relativity:

It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor.
The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated.
But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.
Examples of this sort, together with unsuccessful attempts to discover any motion of the earth relative to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.

— Template:Comma separated entries

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References

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External links

Template:Michael Faraday

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Note that the law relating flux to emf, which this article calls "Faraday's law", is referred to in Griffiths' terminology as the "universal flux rule". Griffiths uses the term "Faraday's law" to refer to what this article calls the "Maxwell–Faraday equation". So in fact, in the textbook, Griffiths' statement is about the "universal flux rule".
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Note that the law relating flux to emf, which this article calls "Faraday's law", is referred to in Griffiths' terminology as the "universal flux rule". Griffiths uses the term "Faraday's law" to refer to what this article calls the "Maxwell–Faraday equation". So in fact, in the textbook, Griffiths' statement is about the "universal flux rule".

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 [[Image:Induction experiment.png|thumb|upright=1.3|Faraday's experiment showing induction between coils of wire: The liquid battery ''(right)'' provides a current which 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'').<ref>{{cite book|last=Poyser|first=Arthur William|url=https://archive.org/details/bub_gb_JzBAAAAAYAAJ|title=Magnetism and Electricity: A manual for students in advanced classes|date=1892|publisher=Longmans, Green, & Co.|location=London and New York|at=Fig. 248, p. 245|access-date=2009-08-06}}</ref>]]
-'''Faraday's law of induction''' (or simply '''Faraday's law''') is a [[Scientific law|law]] of [[electromagnetism]] predicting how a [[magnetic field]] will interact with an [[electric circuit]] to produce an [[electromotive force]] (emf). This phenomenon, known as [[electromagnetic induction]], is the fundamental operating principle of [[transformer]]s, [[inductor]]s, and many types of [[electric motor]]s, [[electrical generator|generators]] and [[solenoid]]s.<ref name="Sadiku386">{{cite book|last=Sadiku|first=M. N. O.|title=Elements of Electromagnetics|year=2007|page=386|publisher=Oxford University Press|edition=4th|location=New York & Oxford|url=https://books.google.com/books?id=w2ITHQAACAAJ|isbn=978-0-19-530048-2}}</ref><ref>{{cite web|date=1999-07-22|title=Applications of electromagnetic induction|url=http://physics.bu.edu/~duffy/py106/Electricgenerators.html|publisher=[[Boston University]]}}</ref>
+'''Faraday's law of induction''', or simply '''Faraday's law''', is a [[Scientific law|law]] of [[electromagnetism]] predicting how a [[magnetic field]] will interact with an [[electric circuit]] to produce an [[electromotive force]] (emf). This phenomenon, known as [[electromagnetic induction]], is the fundamental operating principle of [[transformer]]s, [[inductor]]s, and many types of [[electric motor]]s, [[electrical generator|generators]] and [[solenoid]]s.<ref name="Sadiku386">{{cite book|last=Sadiku|first=M. N. O.|title=Elements of Electromagnetics|year=2007|page=386|publisher=Oxford University Press|edition=4th|location=New York & Oxford|url=https://books.google.com/books?id=w2ITHQAACAAJ|isbn=978-0-19-530048-2}}</ref><ref>{{cite web|date=1999-07-22|title=Applications of electromagnetic induction|url=http://physics.bu.edu/~duffy/py106/Electricgenerators.html|publisher=[[Boston University]]}}</ref>
 The '''Maxwell–Faraday equation''' (listed as one of [[Maxwell's equations]]) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that emf (electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) appears on a conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.
@@ Line 15: / Line 15: @@
 [[File:Faraday emf experiment.svg|thumb|Faraday's 1831 demonstration<ref name="Giancoli">{{cite book|last=Giancoli|first=Douglas C.|title=Physics: Principles with Applications|url=https://archive.org/details/physicsprinciple00gian|url-access=registration|year=1998|pages=[https://archive.org/details/physicsprinciple00gian/page/623 623–624]|edition=5th}}</ref>|left|227x227px]]
-Faraday's notebook on August 29, 1831<ref name=":1">{{Cite web |last=Faraday |first=Michael |date=1831-08-29 |title=Faraday's notebooks: Electromagnetic Induction |url=https://www.rigb.org/docs/faraday_notebooks__induction_0.pdf |url-status=dead |archive-url=https://web.archive.org/web/20210830003053/https://www.rigb.org/docs/faraday_notebooks__induction_0.pdf |archive-date=2021-08-30 |access-date= |website=[[The Royal Institution of Great Britain]]}}</ref> describes an experimental demonstration of electromagnetic induction (see figure)<ref name="FaradayDay1999">{{cite book|last1=Faraday|first1=Michael|last2=Day|first2=P.|title=The philosopher's tree: a selection of Michael Faraday's writings|url=https://books.google.com/books?id=ur6iKVmzYhcC&pg=PA71|access-date=28 August 2011|date=1999-02-01|publisher=CRC Press|isbn=978-0-7503-0570-9|page=71}}</ref> that wraps two wires around opposite sides of an iron [[Torus|ring]] (like a modern [[toroidal transformer]]). His assessment of newly-discovered properties of [[electromagnets]] suggested 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. Indeed, a [[galvanometer]]'s needle measured a transient current (which he called a "wave of electricity") on the right side's wire when he connected ''or'' disconnected the left side's wire to a battery.<ref name="Williams">{{cite book|title=Michael Faraday|url=https://archive.org/details/michaelfaradaybi00will|url-access=registration|first=L. Pearce|last=Williams|year=1965|publisher=New York, Basic Books}}{{full citation needed|date=September 2018}}</ref>{{rp|182–183}} This induction was due to the change in [[magnetic flux]] that occurred when the battery was connected and disconnected.<ref name="Giancoli" /> His notebook entry also noted that fewer wraps for the battery side resulted in a greater disturbance of the galvanometer's needle.<ref name=":1" />
+Faraday's notebook on August 29, 1831<ref name=":1">{{Cite web |last=Faraday |first=Michael |date=1831-08-29 |title=Faraday's notebooks: Electromagnetic Induction |url=https://www.rigb.org/docs/faraday_notebooks__induction_0.pdf |url-status=dead |archive-url=https://web.archive.org/web/20210830003053/https://www.rigb.org/docs/faraday_notebooks__induction_0.pdf |archive-date=2021-08-30 |access-date= |website=[[The Royal Institution of Great Britain]]}}</ref> describes an experimental demonstration of electromagnetic induction<ref name="FaradayDay1999">{{cite book|last1=Faraday|first1=Michael|last2=Day|first2=P.|title=The philosopher's tree: a selection of Michael Faraday's writings|url=https://books.google.com/books?id=ur6iKVmzYhcC&pg=PA71|access-date=28 August 2011|date=1999-02-01|publisher=CRC Press|isbn=978-0-7503-0570-9|page=71}}</ref> that wraps two wires around opposite sides of an iron [[Torus|ring]] (like a modern [[toroidal transformer]]). His assessment of newly-discovered properties of [[electromagnets]] suggested 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. Indeed, a [[galvanometer]]'s needle measured a transient current (which he called a "wave of electricity") on the right side's wire when he connected ''or'' disconnected the left side's wire to a battery.<ref name="Williams">{{cite book|title=Michael Faraday|url=https://archive.org/details/michaelfaradaybi00will|url-access=registration|first=L. Pearce|last=Williams|year=1965|publisher=New York, Basic Books}}{{full citation needed|date=September 2018}}</ref>{{rp|182–183}} This induction was due to the change in [[magnetic flux]] that occurred when the battery was connected and disconnected.<ref name="Giancoli" /> His notebook entry also noted that fewer wraps for the battery side resulted in a greater disturbance of the galvanometer's needle.<ref name=":1" />
 Within two months, Faraday had 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 ([[direct current|DC]]) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("[[Homopolar generator|Faraday's disk]]").<ref name="Williams" />{{rp|191–195}}
 [[File:Faraday disk generator.jpg|thumb|Faraday's disk, the first [[electric generator]], a type of [[homopolar generator]]|left]]
-[[Michael 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.<ref name=Williams/>{{rp|510}} An exception was [[James Clerk Maxwell]], who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory.<ref name=Williams/>{{rp|510}}<ref>{{cite book|last=Clerk Maxwell |first=James |date=1904 |title=A Treatise on Electricity and Magnetism |volume=2 |edition=3rd |publisher=Oxford University Press |pages=178–179, 189}}</ref><ref name="IEEUK">{{cite web|url=http://www.theiet.org/resources/library/archives/biographies/faraday.cfm |title=Archives Biographies: Michael Faraday |publisher=The Institution of Engineering and Technology}}</ref> In Maxwell's papers, 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 different from the original version of Faraday's law, and does not describe [[#Two phenomena|motional emf]]. Heaviside's version (see [[#Maxwell–Faraday equation|Maxwell–Faraday equation below]]) is the form recognized today in the group of equations known as [[Maxwell's equations]].
+[[Michael 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.<ref name=Williams/>{{rp|510}} An exception was [[James Clerk Maxwell]], who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory.<ref name=Williams/>{{rp|510}}<ref>{{cite book|last=Clerk Maxwell |first=James |date=1904 |title=A Treatise on Electricity and Magnetism |volume=2 |edition=3rd |publisher=Oxford University Press |pages=178–179, 189}}</ref><ref name="IEEUK">{{cite web|url=http://www.theiet.org/resources/library/archives/biographies/faraday.cfm |title=Archives Biographies: Michael Faraday |publisher=The Institution of Engineering and Technology}}</ref> In Maxwell's papers, 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 different from the original version of Faraday's law, and does not describe [[#Two phenomena|motional emf]]. Heaviside's version is the form recognized today in the group of equations known as [[Maxwell's equations]].
-[[Lenz's law]], formulated by [[Emil Lenz]] in 1834,<ref>{{cite journal|last=Lenz |first=Emil |date=1834 |url=http://gallica.bnf.fr/ark:/12148/bpt6k151161/f499.image.r=lenz.langEN |title=Ueber<!--[sic]--> die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme |journal=Annalen der Physik und Chemie |volume=107 |issue=31 |pages=483–494|bibcode=1834AnP...107..483L |doi=10.1002/andp.18341073103 }}<br>A partial translation of the paper is available in {{cite book|last=Magie |first=W. M. |date=1963 |title=A Source Book in Physics |publisher=Harvard Press |location=Cambridge, MA |pages=511–513}}</ref> describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction (elaborated upon in the examples below).
+[[Lenz's law]], formulated by [[Emil Lenz]] in 1834,<ref>{{cite journal|last=Lenz |first=Emil |date=1834 |url=http://gallica.bnf.fr/ark:/12148/bpt6k151161/f499.image.r=lenz.langEN |title=Ueber<!--[sic]--> die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme |journal=Annalen der Physik und Chemie |volume=107 |issue=31 |pages=483–494|bibcode=1834AnP...107..483L |doi=10.1002/andp.18341073103 }}<br/>A partial translation of the paper is available in {{cite book|last=Magie |first=W. M. |date=1963 |title=A Source Book in Physics |publisher=Harvard Press |location=Cambridge, MA |pages=511–513}}</ref> describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction (elaborated upon in the examples below).
+The laws of induction of electric currents in mathematical form were established by [[Franz Ernst Neumann]] in 1845.<ref>{{cite journal |last=Neumann |first=Franz Ernst |year=1846 |title=Allgemeine Gesetze der inducirten elektrischen Ströme |url=https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf |journal=Annalen der Physik |volume=143 |pages=31–44 |bibcode=1846AnP...143...31N |doi=10.1002/andp.18461430103 |archive-url=https://web.archive.org/web/20200312012028/https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf |archive-date=12 March 2020 |number=1}}</ref>{{Non-primary source needed|date=June 2025}}
 According to [[Albert Einstein]], much of the groundwork and discovery of his [[special relativity]] theory was presented by this law of induction by Faraday in 1834.<ref name="FRBS-20190301">{{cite news |last=Siegel |first=Ethan |authorlink=Ethan Siegel |title=Relativity Wasn't Einstein's Miracle; It Was Waiting In Plain Sight For 71 Years |url=https://www.forbes.com/sites/startswithabang/2019/03/01/relativity-wasnt-einsteins-miracle-it-was-waiting-in-plain-sight-for-71-years/ |date=1 March 2019 |work=[[Forbes]] |url-status=live |archiveurl=https://archive.today/20230703120353/https://www.forbes.com/sites/startswithabang/2019/03/01/relativity-wasnt-einsteins-miracle-it-was-waiting-in-plain-sight-for-71-years/?sh=5d837e5b644c  |archivedate=3 July 2023  |accessdate=3 July 2023 }}</ref><ref name="BT-20230628">{{cite news |last=Siegel |first=Ethan |authorlink=Ethan Siegel |title=71 years earlier, this scientist beat Einstein to relativity - Michael Faraday's 1834 law of induction was the key experiment behind the eventual discovery of relativity. Einstein admitted it himself. |url=https://bigthink.com/starts-with-a-bang/scientist-beat-einstein-relativity/ |date=28 June 2023 |work=[[Big Think]] |url-status=live |archiveurl=https://archive.today/20230628183501/https://bigthink.com/starts-with-a-bang/scientist-beat-einstein-relativity/ |archivedate=28 June 2023 |accessdate=3 July 2023 }}</ref>
 {{clear}}
-==Faraday's law==
+==Flux rule==
 [[File:Electromagnetic_induction_-_solenoid_to_loop_-_animation.gif|thumb|Alternating electric current flows through the solenoid on the left, producing a changing magnetic field. This field causes, by electromagnetic induction, an electric current to flow in the wire loop on the right.]]
-The most widespread version of Faraday's law states:
+Faraday's law of induction, also known as the '''flux rule''' and '''Faraday{{endash}}Lenz law'''<ref name="r127">{{cite book | last=Fujimoto | first=Minoru | title=Physics of Classical Electromagnetism | publisher=Springer Science & Business Media | publication-place=New York | date=2007-09-06 | isbn=978-0-387-68018-7 | page=105}}</ref>, states that the [[electromotive force]] (emf) around a closed circuit is equal to the negative [[Time derivative|rate of change]] of the [[magnetic flux]] through the circuit. This rule holds for any circuit made of thin wire and accounts for changes in flux due to variations in the [[magnetic field]], movement of the circuit, or deformation of its shape.<ref name="n495">{{cite book | last=Landau| first=Lev Davidovich | last2=Lifshitz | first2=Evgeniĭ Mikhaĭlovich | last3=Pitaevskiĭ | first3=Lev Petrovich | title=Electrodynamics of Continuous Media | publisher=Pergamon press | publication-place=Oxford | date=1984 | isbn=0-08-030276-9 | page=219}}</ref> The direction of the induced emf is given by [[Lenz's law]], which states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux.<ref>{{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics | publisher=Cambridge University Press | date=2023 | isbn=978-1-009-39773-5 | doi=10.1017/9781009397735 | url=https://www.cambridge.org/highereducation/product/9781009397735/book|page=319}}</ref>
-{{Blockquote|The electromotive force around a closed path is equal to the negative of the time rate of change of the [[magnetic flux]] enclosed by the path.<ref name="Jordan & Balmain (1968)">{{cite book| last1=Jordan|first1= Edward |last2=Balmain|first2=Keith G.| title = Electromagnetic Waves and Radiating Systems | edition = 2nd| page = 100| publisher = Prentice-Hall| date = 1968|quote=Faraday's Law, which states that the electromotive force around a closed path is equal to the negative of the time rate of change of magnetic flux enclosed by the path.}}</ref><ref name="Hayt (1989)">{{cite book| last = Hayt| first = William| title = Engineering Electromagnetics| edition = 5th| page = [https://archive.org/details/engineeringelect5thhayt/page/312 312]| isbn = 0-07-027406-1| publisher = McGraw-Hill| date = 1989| quote = The magnetic flux is that flux which passes through any and every surface whose perimeter is the closed path.| url = https://archive.org/details/engineeringelect5thhayt/page/312}}</ref>}}
-===Mathematical statement===
+Mathematically, in [[SI units]], the law is written as
-[[Image:Surface integral illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface {{math|Σ}} into small surface elements. Each element is associated with a vector {{math|d'''A'''}} of magnitude equal to the area of the element and with direction normal to the element and pointing "outward" (with respect to the orientation of the surface).]]
+<math display="block">\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}, </math>
-For a loop of wire in a [[magnetic field]], the [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} is defined for any [[Surface (mathematics)|surface]] {{math|Σ}} whose [[Boundary (topology)|boundary]] is the given loop. Since the wire loop may be moving, we write {{math|Σ(''t'')}} for the surface. The magnetic flux is the [[surface integral]]:
+where <math>\mathcal{E}</math> is the electromotive force (emf) and {{math|Φ<sub>''B''</sub>}} is the magnetic flux throught the circuit. It is defined as the [[surface integral]] of the magnetic field over a [[Surface (mathematics)|surface]] {{math|Σ(''t'')}}, whose [[Boundary (topology)|boundary]] is the wire loop:
 <math display="block"> \Phi_B = \iint_{\Sigma(t)} \mathbf{B}(t) \cdot \mathrm{d} \mathbf{A}\, , </math>
-where {{math|d'''A'''}} is an element of [[area vector]] of the moving surface {{math|Σ(''t'')}}, {{math|'''B'''}} is the magnetic field, and {{math|'''B''' · d'''A'''}} is a [[dot product|vector dot product]] representing the element of flux through {{math|d'''A'''}}. In more visual terms, the magnetic flux through the wire loop is proportional to the number of [[field line|magnetic field lines]] that pass through the loop.
+where {{math|d'''A'''}} is an element of [[area vector]] of the moving surface {{math|Σ(''t'')}}, directed normal to the surface, and {{math|'''B'''}} is the magnetic field. The [[dot product]] {{math|'''B''' · d'''A'''}} represents the element of flux through {{math|d'''A'''}}. In more visual terms, the magnetic flux through the wire loop is proportional to the number of [[field line|magnetic field lines]] that pass through the loop.
+[[Image:Surface integral illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface {{math|Σ}} into small surface elements. Each element is associated with a vector {{math|d'''A'''}} of magnitude equal to the area of the element and with direction normal to the element and pointing "outward" (with respect to the orientation of the surface).]]When the flux changes—because {{math|'''B'''}} changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an emf, defined as the energy available from a unit charge that has traveled once around the wire loop.<ref name="Feynman">{{Cite web| last=Feynman|first=Richard P. |title=The Feynman Lectures on Physics Vol. II |url=https://feynmanlectures.caltech.edu/II_toc.html|access-date=2020-11-07 |website=feynmanlectures.caltech.edu}}</ref>{{Rp|ch17}}<ref name="Griffiths2">{{cite book|last=Griffiths|first=David J. | title=Introduction to Electrodynamics | url=https://archive.org/details/introductiontoel00grif_0/page/301 | edition=3rd |pages=[https://archive.org/details/introductiontoel00grif_0/page/301 301–303] | publisher=Prentice Hall| year=1999 | location=Upper Saddle River, NJ | isbn=0-13-805326-X}}</ref><ref>{{cite book |last1=Tipler|last2=Mosca |title=Physics for Scientists and Engineers |year=2004|page=795|publisher=Macmillan |isbn=9780716708100 |url=https://books.google.com/books?id=R2Nuh3Ux1AwC&pg=PA795}}</ref> (Although some sources state the definition differently, this expression was chosen for compatibility with the equations of [[special relativity]].{{Cn|date=June 2025}}) Equivalently, it is the voltage that would be measured by cutting the wire to create an [[Electric circuit|open circuit]], and attaching a [[voltmeter]] to the leads.{{Cn|date=June 2025|reason=The voltmeter closes the circuit. Does the voltage depend on the shape of the circuit which open circuit and the voltmeter form?}}[[File:Salu's left-hand rule (magnetic induction).png|thumb|A Left Hand Rule for Faraday's Law. The sign of {{math|ΔΦ<sub>''B''</sub>}}, the change in flux, is found based on the relationship between the magnetic field {{math|'''B'''}}, the area of the loop {{mvar|A}}, and the normal n to that area, as represented by the fingers of the left hand. If {{math|ΔΦ<sub>''B''</sub>}} is positive, the direction of the emf is the same as that of the curved fingers (yellow arrowheads). If {{math|ΔΦ<sub>''B''</sub>}} is negative, the direction of the emf is against the arrowheads.<ref name=Salu2014/>|alt=]]
-When the flux changes—because {{math|'''B'''}} changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an [[electromotive force|emf]], defined as the energy available from a unit charge that has traveled once around the wire loop.<ref name="Feynman">{{Cite web| last=Feynman|first=Richard P. |title=The Feynman Lectures on Physics Vol. II |url=https://feynmanlectures.caltech.edu/II_toc.html|access-date=2020-11-07 |website=feynmanlectures.caltech.edu}}</ref>{{Rp|ch17}}<ref name=Griffiths2>{{cite book|last=Griffiths|first=David J. | title=Introduction to Electrodynamics | url=https://archive.org/details/introductiontoel00grif_0/page/301 | edition=3rd |pages=[https://archive.org/details/introductiontoel00grif_0/page/301 301–303] | publisher=Prentice Hall| year=1999 | location=Upper Saddle River, NJ | isbn=0-13-805326-X}}</ref><ref>{{cite book |last1=Tipler|last2=Mosca |title=Physics for Scientists and Engineers |year=2004|page=795|publisher=Macmillan |isbn=9780716708100 |url=https://books.google.com/books?id=R2Nuh3Ux1AwC&pg=PA795}}</ref> (Although some sources state the definition differently, this expression was chosen for compatibility with the equations of [[special relativity]].) Equivalently, it is the voltage that would be measured by cutting the wire to create an [[Electric circuit|open circuit]], and attaching a [[voltmeter]] to the leads.
-Faraday's law states that the emf is also given by the [[time derivative|rate of change]] of the magnetic flux:
-<math display="block">\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}, </math>
-where <math>\mathcal{E}</math> is the [[electromotive force]] (emf) and {{math|Φ<sub>''B''</sub>}} is the [[magnetic flux]].
-The direction of the electromotive force is given by [[Lenz's law]].
-The laws of induction of electric currents in mathematical form were established by [[Franz Ernst Neumann]] in 1845.<ref>{{cite journal|first=Franz Ernst|last=Neumann |title=Allgemeine Gesetze der inducirten elektrischen Ströme|journal=Annalen der Physik |volume=143|number=1|pages=31–44 |year=1846|doi=10.1002/andp.18461430103 |url=https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf |archive-url=https://web.archive.org/web/20200312012028/https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf |archive-date=12 March 2020 |bibcode=1846AnP...143...31N}}</ref>
-Faraday's law contains the information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula.
-[[File:Salu's left-hand rule (magnetic induction).png|thumb|A Left Hand Rule for Faraday's Law. The sign of {{math|ΔΦ<sub>''B''</sub>}}, the change in flux, is found based on the relationship between the magnetic field {{math|'''B'''}}, the area of the loop {{mvar|A}}, and the normal n to that area, as represented by the fingers of the left hand. If {{math|ΔΦ<sub>''B''</sub>}} is positive, the direction of the emf is the same as that of the curved fingers (yellow arrowheads). If {{math|ΔΦ<sub>''B''</sub>}} is negative, the direction of the emf is against the arrowheads.<ref name=Salu2014/>|alt=]]
 It is possible to find out the direction of the electromotive force (emf) directly from Faraday’s law, without invoking Lenz's law. A left hand rule helps doing that, as follows:<ref name="Salu2014">{{cite journal|year=2014 |url=https://www.researchgate.net/publication/262986189 |title=A Left Hand Rule for Faraday's Law | journal=[[The Physics Teacher]] | volume=52|pages=48 |doi=10.1119/1.4849156 |author=Yehuda Salu| issue=1 |bibcode=2014PhTea..52...48S}} [https://www.youtube.com/watch?v=ipUD9VcAd9o Video Explanation]</ref><ref>{{cite web |url=http://Physicsforarchitects.com/bypassing-lenzs-rule |archive-url=https://web.archive.org/web/20200507170609/http://physicsforarchitects.com/bypassing-lenzs-rule |archive-date=7 May 2020 |title=Bypassing Lenz's Rule - A Left Hand Rule for Faraday's Law |website=www.PhysicsForArchitects.com |last1=Salu|first1=Yehuda |date=17 January 2017 |access-date=30 July 2017}}</ref>
 * Align the curved fingers of the left hand with the loop (yellow line).
@@ Line 57: / Line 46: @@
 * If {{math|ΔΦ<sub>''B''</sub>}} is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads).
-For a tightly wound [[inductor|coil of wire]], composed of {{mvar|N}} identical turns, each with the same {{math|Φ<sub>''B''</sub>}}, Faraday's law of induction states that<ref>{{cite book| title=Essential Principles of Physics| first1=P. M.|last1=Whelan|first2=M. J.|last2=Hodgeson|edition=2nd|date=1978|publisher=John Murray|isbn=0-7195-3382-1}}</ref><ref>{{cite web|last=Nave|first=Carl R. | title=Faraday's Law | url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html | work=HyperPhysics |publisher=Georgia State University |access-date=2011-08-29}}</ref>
+For a tightly wound [[inductor|coil of wire]], composed of {{mvar|N}} identical turns, the same magnetic field lines cross the surface {{mvar|N}} times. In this case, Faraday's law of induction states that<ref>{{cite book| title=Essential Principles of Physics| first1=P. M.|last1=Whelan|first2=M. J.|last2=Hodgeson|edition=2nd|date=1978|publisher=John Murray|isbn=0-7195-3382-1}}</ref><ref>{{cite web|last=Nave|first=Carl R. | title=Faraday's Law | url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html | work=HyperPhysics |publisher=Georgia State University |access-date=2011-08-29}}</ref>
 <math display="block"> \mathcal{E} = -N \frac{\mathrm{d}\Phi_B}{\mathrm{d}t} </math>
-where {{mvar|N}} is the number of turns of wire and {{math|Φ<sub>''B''</sub>}} is the magnetic flux through a single loop.
+where {{mvar|N}} is the number of turns of wire and {{math|Φ<sub>''B''</sub>}} is the magnetic flux through a single loop. The product {{math|''N''Φ<sub>''B''</sub>}} is known as [[Flux linkage|linked flux]].<ref>{{Cite web |title=121-11-77: "linked flux" |url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-77 |access-date=2025-06-20 |website=IEC 60050 - International Electrotechnical Vocabulary}}</ref>
-===Maxwell–Faraday equation===
+== Maxwell–Faraday equation ==
 {{anchor|Faraday equation}}
 [[Image:Stokes' Theorem.svg|thumb|right|An illustration of the Kelvin–Stokes theorem with surface {{math|'''Σ'''}}, its boundary {{math|∂'''Σ'''}}, and orientation {{math|'''n'''}} set by the [[right-hand rule]]]]
@@ Line 88: / Line 77: @@
 }}</ref>
-The Maxwell–Faraday equation is one of the four [[Maxwell's equations]], and therefore plays a fundamental role in the theory of [[classical electromagnetism]]. It can also be written in an '''integral form''' by the [[Kelvin–Stokes theorem]],<ref name=Harrington>{{cite book|first=Roger F.| last=Harrington|title=Introduction to electromagnetic engineering |year=2003 |page=56 |publisher=Dover Publications |location=Mineola, NY |isbn=0-486-43241-6 |url=https://books.google.com/books?id=ZlC2EV8zvX8C&q=%22faraday%27s+law+of+induction%22&pg=PA57}}</ref> thereby reproducing Faraday's law:
+The Maxwell–Faraday equation is one of the four [[Maxwell's equations]], and therefore plays a fundamental role in the theory of [[classical electromagnetism]]. It can also be written in an integral form by the [[Kelvin–Stokes theorem]]:<ref name=Harrington>{{cite book|first=Roger F.| last=Harrington|title=Introduction to electromagnetic engineering |year=2003 |page=56 |publisher=Dover Publications |location=Mineola, NY |isbn=0-486-43241-6 |url=https://books.google.com/books?id=ZlC2EV8zvX8C&q=%22faraday%27s+law+of+induction%22&pg=PA57}}</ref>
 {{Equation box 1
@@ Line 102: / Line 91: @@
 Both {{math|d'''l'''}} and {{math|d'''A'''}} have a sign ambiguity; to get the correct sign, the [[right-hand rule]] is used, as explained in the article [[Kelvin–Stokes theorem]]. For a planar surface {{math|'''Σ'''}}, a positive path element {{math|d'''l'''}} of curve {{math|∂'''Σ'''}} is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal {{math|'''n'''}} to the surface {{math|'''Σ'''}}.
-The [[line integral]] around {{math|∂'''Σ'''}} is called [[Circulation (physics)|circulation]].<ref name=Feynman />{{Rp|ch3}} A nonzero circulation of {{math|'''E'''}} is different from the behavior of the electric field generated by static charges. A charge-generated {{math|'''E'''}}-field can be expressed as the gradient of a [[scalar field]] that is a solution to [[Poisson's equation]], and has a zero path integral. See [[gradient theorem]].
+The [[line integral]] around {{math|∂'''Σ'''}} is called [[Circulation (physics)|circulation]].<ref name=Feynman />{{Rp|ch3}} A nonzero circulation of {{math|'''E'''}} is different from the behavior of the electric field generated by static charges. A charge-generated {{math|'''E'''}}-field can be expressed as the gradient of a [[scalar field]] that is a solution to [[Poisson's equation]], and has a zero path integral.
 The integral equation is true for ''any'' path {{math|∂'''Σ'''}} through space, and any surface {{math|'''Σ'''}} for which that path is a boundary.
@@ Line 113: / Line 102: @@
 <math display="block"> \mathbf E_s (\mathbf r,t) \approx -\frac{1}{4\pi}\iiint_V \ \frac{\left(\frac{\partial \mathbf{B}(\mathbf{r}',t)}{\partial t} \right) \times \left(\mathbf{r}-\mathbf{r}' \right) }{|\mathbf {r} - \mathbf{r}'|^3} d^3\mathbf{r'}</math>
-==Proof==
+==Derivation of the flux rule from microscopic equations==
-The four [[Maxwell's equations]] (including the Maxwell–Faraday equation), along with Lorentz force law, are a sufficient foundation to derive ''everything'' in [[classical electromagnetism]].<ref name=Feynman/><ref name=Griffiths2/> Therefore, it is possible to "prove" Faraday's law starting with these equations.<ref name=Davison>{{Cite journal | last1 = Davison | first1 = M. E. | title = A Simple Proof that the Lorentz Force, Law Implied Faraday's Law of Induction, when '''B''' is Time Independent | doi = 10.1119/1.1987339 | journal = American Journal of Physics | volume = 41 | issue = 5 | page = 713| year = 1973 |bibcode = 1973AmJPh..41..713D }}</ref><ref name=Krey>{{cite book|title=Basic Theoretical Physics: A Concise Overview |last1=Krey |last2=Owen |date=14 August 2007 |page=155 |publisher=Springer |isbn=9783540368052 | url=https://books.google.com/books?id=xZ_QelBmkxYC&pg=PA155}}</ref>
+The four [[Maxwell's equations]], together with the Lorentz force law, form a complete foundation for [[classical electromagnetism]].<ref name=Feynman/><ref name=Griffiths2/> From these, Faraday's law can be derived directly.<ref name="Davison">{{Cite journal |last1=Davison |first1=M. E. |year=1973 |title=A Simple Proof that the Lorentz Force, Law Implied Faraday's Law of Induction, when '''B''' is Time Independent |journal=American Journal of Physics |volume=41 |issue=5 |page=713 |bibcode=1973AmJPh..41..713D |doi=10.1119/1.1987339}}</ref><ref name="Krey">{{cite book |last1=Krey |url=https://books.google.com/books?id=xZ_QelBmkxYC&pg=PA155 |title=Basic Theoretical Physics: A Concise Overview |last2=Owen |date=14 August 2007 |publisher=Springer |isbn=9783540368052 |page=155}}</ref><ref name=":0">{{cite book|title=Theoretische Elektrotechnik|last=Simonyi|first=K.|date=1973|publisher=VEB Deutscher Verlag der Wissenschaften|edition=5th|location=Berlin|at=eq.&nbsp;20, p.&nbsp;47}}</ref>
-The starting point is the time-derivative of flux through an arbitrary surface {{math|Σ}} (that can be moved or deformed) in space:
-<math display="block">\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma(t)} \mathbf{B}(t) \cdot \mathrm{d}\mathbf{A}</math>
-(by definition). This total time derivative can be evaluated and simplified with the help of the Maxwell–Faraday equation and some vector identities; the details are in the box below:
-{| class="wikitable"
-|Consider the time-derivative of magnetic flux through a closed boundary (loop) that can move or be deformed. The area bounded by the loop is denoted as {{math|Σ(''t'')}}), then the time-derivative can be expressed as
-<math display="block">\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma(t)} \mathbf{B}(t) \cdot \mathrm{d}\mathbf{A}</math>
-The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore:
-<!-- not clear at all how to derive such formula from the previous one! --><math display="block">\left. \frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right|_{t=t_0} = \left( \int_{\Sigma(t_0)} \left. \frac{\partial\mathbf{B}}{\partial t}\right|_{t=t_0} \cdot \mathrm{d}\mathbf{A}\right) + \left( \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t_0) \cdot \mathrm{d}\mathbf{A} \right)</math>
-where {{math|''t''<sub>0</sub>}} is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer emf, the second to motional emf (from the magnetic Lorentz force on charge carriers due to the motion or deformation of the conducting loop in the magnetic field). The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation:
-<math display="block"> \int_{\Sigma(t_0)} \left. \frac{\partial \mathbf{B}}{\partial t}\right|_{t=t_0} \cdot \mathrm{d}\mathbf{A} = - \oint_{\partial \Sigma(t_0)} \mathbf{E}(t_0) \cdot \mathrm{d}\mathbf{l} </math>
-Next, we analyze the second term on the right-hand side:
-<math display="block">\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t_0) \cdot \mathrm{d}\mathbf{A}</math>
-[[File:Derivation of Faraday Equation Wikipedia 20181127 - 4.png|alt=|thumb|upright=1.2|The area swept out by a vector element {{math|d'''l'''}} of a loop {{math|∂'''Σ'''}} in time {{math|d''t''}} when it has moved with velocity {{math|'''v'''<sub>'''l'''</sub>}} .]]The proof of this is a little more difficult than the first term; more details and alternate approaches for the proof can be found in the references.<ref name="Davison" /><ref name="Krey" /><ref name=":0">{{cite book|title=Theoretische Elektrotechnik|last=Simonyi|first=K.|date=1973|publisher=VEB Deutscher Verlag der Wissenschaften|edition=5th|location=Berlin|at=eq.&nbsp;20, p.&nbsp;47}}</ref> As the loop moves and/or deforms, it sweeps out a surface (see the right figure). As a small part of the loop {{math|d'''l'''}} moves with velocity {{math|'''v'''<sub>'''l'''</sub>}} over a short time {{math|d''t''}}, it sweeps out an area whose vector is {{math|1=d'''A'''<sub>sweep</sub> = '''v'''<sub>'''l'''</sub> d''t'' × d'''l'''}} (note that this vector is toward out from the display in the right figure). Therefore, the change of the magnetic flux through the loop due to the deformation or movement of the loop over the time {{math|d''t''}} is <math display="block">\mathrm{d}\Phi_B = \int \mathbf{B} \cdot \mathrm{d}\mathbf{A}_\text{sweep} = \int \mathbf{B} \cdot (\mathbf{v}_{\mathbf{l}} \mathrm{d}t \times \mathrm{d}\mathbf{l}) = -\int \mathrm{d}t \, \mathrm{d}\mathbf{l} \cdot (\mathbf{v}_{\mathbf{l}}\times\mathbf{B})</math>
-Here, [[Triple product|identities of triple scalar products]] are used. Therefore,
-<math display="block">\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t_0) \cdot \mathrm{d}\mathbf{A} = -\oint_{\partial \Sigma(t_0)} (\mathbf{v}_{\mathbf{l}}(t_0)\times \mathbf{B}(t_0))\cdot \mathrm{d}\mathbf{l}</math>
-where {{math|'''v'''<sub>'''l'''</sub>}} is the velocity of a part of the loop {{math|∂'''Σ'''}}.
-Putting these together results in,
+The derivation begins by considering the time derivative of the magnetic flux through a surface {{math|Σ(t)}} that may vary with time:
-<math display="block">\begin{align}
+<math display="block"> \frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t) \cdot \mathrm{d}\mathbf{A}. </math>
-\left. \frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right|_{t=t_0} &= \left(- \oint_{\partial \Sigma(t_0)} \mathbf{E}(t_0) \cdot \mathrm{d}\mathbf{l}\right) + \left(- \oint_{\partial \Sigma(t_0)} \left[\mathbf{v}_{\mathbf{l}}(t_0)\times \mathbf{B}(t_0)\right] \cdot \mathrm{d}\mathbf{l} \right) \\[1ex]
+The magnetic flux can change for two reasons: the magnetic field itself may vary with time, and the surface may move or change shape, enclosing a different region of space. Both effects are captured by the three-dimensional version of the [[Leibniz integral rule#Three-dimensional, time-dependent case|Leibniz integral rule]], sometimes referred to as the "flux theorem":<ref name="zangvill">{{cite book |last=Zangwill |first=Andrew |title=Modern Electrodynamics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-89697-9 |publication-place=Cambridge |pages=10, 462–464}} Proof of the theorem is found on page 10.</ref>
-&= - \oint_{\partial \Sigma(t_0)} \left[ \mathbf{E}(t_0) + \mathbf{v}_{\mathbf{l}}(t_0)\times \mathbf{B}(t_0) \right] \cdot \mathrm{d}\mathbf{l}.
+<math display="block"> \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B} \cdot \mathrm{d}\mathbf{A} = \int_{\Sigma(t)} \left(\frac{\partial \mathbf{B}}{\partial t} + (\nabla\cdot\mathbf B)\mathbf{v}_c \right)\cdot \mathrm{d}\mathbf{A} - \oint_{\partial \Sigma(t)} (\mathbf{v}_c \times \mathbf{B}) \cdot \mathrm{d}\mathbf{l} </math>
-\end{align}</math>
+Here, {{math|∂Σ(t)}} is the moving boundary of the surface and <math>\mathbf{v}_c</math> is the local velocity of the boundary at each point. By [[Gauss's law for magnetism]] (<math>\nabla\cdot\mathbf B = 0</math>), the second term under the area integral vanishes. Applying the [[Maxwell–Faraday equation]] to the remaining term,
-|}
+<math display="block"> \int_{\Sigma(t)} \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{A} = - \oint_{\partial \Sigma(t)} \mathbf{E} \cdot \mathrm{d}\mathbf{l},</math>
+and combining the two line integrals gives
+<math display="block"> \frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = - \oint_{\partial \Sigma(t)} \left( \mathbf{E} + \mathbf{v}_c \times \mathbf{B} \right) \cdot \mathrm{d}\mathbf{l}.</math>
+This is an exact result, derived from Maxwell's equations and vector calculus.<ref name="zangvill" />
-The result is:
+However, the quantity inside the integral is not the full Lorentz force per unit charge, because the velocity <math>\mathbf{v}_c</math> represents the motion of loop boundary, not the actual velocity of the charge carriers. To recover the physical electromotive force, we must distinguish between these velocities. Let us choose the integration path to coincide with the physical circuit. The velocity of a charge carrier in the conductor is then given by
-<math display="block">\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = - \oint_{\partial \Sigma} \left( \mathbf{E} + \mathbf{v}_{\mathbf{l}} \times \mathbf{B} \right) \cdot \mathrm{d}\mathbf{l}.</math>
+:<math>\mathbf v(\mathbf r, t) = \mathbf v_c(\mathbf r, t) + \mathbf v_d(\mathbf r, t)</math>,
-where {{math|∂Σ}} is the boundary (loop) of the surface {{math|Σ}}, and {{math|'''v'''<sub>'''l'''</sub>}} is the velocity of a part of the boundary.
+where <math>\mathbf v_c</math> is the velocity of the conductor (the ions in the material), and <math>\mathbf v_d</math> is the drift velocity of the electrons relative to the material. This decomposition assumes nonrelativistic (Galilean) addition of velocities.<ref name=zangvill/>
-In the case of a conductive loop, emf (Electromotive Force) is the electromagnetic work done on a unit charge when it has traveled around the loop once, and this work is done by the [[Lorentz force law|Lorentz force]].  Therefore, emf is expressed as
+The emf <math>\mathcal{E}</math> associated with the Lorentz force is defined as
-<math display="block">\mathcal{E} = \oint \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math>
+<math display="block">\mathcal{E} = \oint_{\partial \Sigma(t)} \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}.</math>
-where <math>\mathcal{E}</math> is emf and {{math|'''v'''}} is the unit charge velocity.
+Substituting the expression for the carrier velocity and the above result yields:{{Equation box 1|cellpadding|border|indent=:|equation=<math>\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} + \oint_{\partial \Sigma(t)} \left( \mathbf{v}_d\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}.</math>|border colour=#50C878|background colour=#ECFCF4}}
+Equivalently, this can be expressed as
+:<math>\mathcal{E} = -\int_{\partial \Sigma(t)} \frac{\partial \mathbf B}{\partial t}\cdot{\rm d}\mathbf A + \oint_{\partial \Sigma(t)} \left( \mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l},</math>
+where the first term is the "transformer emf" due to a time-varying magnetic field, and the second term is the "motional emf" due to the magnetic Lorentz force by the motion of the charges in the magnetic field.<ref name="zangvill" />
-In a macroscopic view, for charges on a segment of the loop, {{math|'''v'''}} consists of two components in average; one is the velocity of the charge along the segment {{math|'''v'''<sub>'''t'''</sub>}}, and the other is the velocity of the segment {{math|'''v'''<sub>'''l'''</sub>}} (the loop is deformed or moved). {{math|'''v'''<sub>'''t'''</sub>}} does not contribute to the work done on the charge since the direction of {{math|'''v'''<sub>'''t'''</sub>}} is same to the direction of <math>\mathrm{d}\mathbf{l}</math>. Mathematically,
+In circuits made of thin, one-dimensional wires, the drift velocity is aligned with the wire, and hence with the integration element <math display="inline">{\rm d}\mathbf l</math>. In that case, the cross product <math display="inline">\mathbf v_d\times\mathbf B</math> is perpendicular to <math display="inline">{\rm d}\mathbf l</math>, and the term proportional to the drift velocity vanishes. This recovers the standard form of Faraday's law:
-<math display="block">(\mathbf{v}\times \mathbf{B})\cdot \mathrm{d}\mathbf{l} = ((\mathbf{v}_t + \mathbf{v}_l) \times \mathbf{B}) \cdot \mathrm{d}\mathbf{l}=(\mathbf{v}_t\times \mathbf{B}+\mathbf{v}_l\times \mathbf{B})\cdot \mathrm{d}\mathbf{l} = (\mathbf{v}_l\times \mathbf{B})\cdot \mathrm{d}\mathbf{l}</math>
+<math display="block">\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}</math>
-since <math>(\mathbf{v}_t\times \mathbf{B})</math> is perpendicular to <math>\mathrm{d}\mathbf{l}</math> as <math>\mathbf{v}_t</math> and <math>\mathrm{d}\mathbf{l}</math> are along the same direction. Now we can see that, for the conductive loop, emf is same to the time-derivative of the magnetic flux through the loop except for the sign on it. Therefore, we now reach the equation of Faraday's law (for the conductive loop) as
+In this case, the emf can also be expressed as a sum<ref name="zangvill" />
-<math display="block">\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = -\mathcal{E}</math>
+<math display="block">\mathcal{E} = -\int_{\Sigma(t)} \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{A} + \oint_{\partial \Sigma(t)} \left(\mathbf{v}_c\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}.</math>In conductors that are not thin wires, the drift velocity term <math display="inline">\oint_{\partial \Sigma(t)} \left( \mathbf{v}_d\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math> may not vanish exactly. However, electrons typically drift at speeds of the order of 10<sup>-4</sup> m/s, and the contribution is often negligible compared to other effects.<ref>{{Cite web |last=Ling |first=Samuel J. |last2=Moebs |first2=William |last3=Sanny |first3=Jeff |date=2016-10-06 |title=9.2 Model of Conduction in Metals - University Physics Volume 2 {{!}} OpenStax |url=https://openstax.org/books/university-physics-volume-2/pages/9-2-model-of-conduction-in-metals |access-date=2025-06-19 |website=openstax.org |language=English}}</ref> A notable exception is the [[Hall effect]], where magnetic flux term <math display="inline">\mathrm{d}\Phi_B/\mathrm{d}t</math> vanishes, and the observed Hall voltage arises entirely from the drift velocity term.<ref name="zangvill" />
-where <math display="inline">\mathcal{E} = \oint \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math>. With breaking this integral, <math display="inline">\oint\mathbf{E}\cdot\mathrm{d}\mathbf{l}</math> is for the transformer emf (due to a time-varying magnetic field) and <math display="inline">\oint \left(\mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l} = \oint \left(\mathbf{v}_l\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math> is for the motional emf (due to the magnetic Lorentz force on charges by the motion or deformation of the loop in the magnetic field).
 ==Exceptions==
 {{See also|Faraday paradox|Hering's Paradox}}
-It is tempting to generalize Faraday's law to state: ''If ''{{math|∂Σ}}'' is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through ''{{math|Σ}}'' equals the emf around ''{{math|∂Σ}}''.'' This statement, however, is not always true and the reason is not just from the obvious reason that emf is undefined in empty space when no conductor is present. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve {{math|∂Σ}} matches the actual velocity of the material conducting the electricity.<ref name=Stewart>{{cite book |title=Intermediate Electromagnetic Theory |first1=Joseph V. |last1=Stewart |page=396 |quote=This example of Faraday's Law [the homopolar generator] makes it very clear that in the case of extended bodies care must be taken that the boundary used to determine the flux must not be stationary but must be moving with respect to the body.}}</ref> The two examples illustrated below show that one often obtains incorrect results when the motion of {{math|∂Σ}} is divorced from the motion of the material.<ref name=Feynman/>
+It is tempting to generalize Faraday's law to state: ''If ''{{math|∂Σ}}'' is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through ''{{math|Σ}}'' equals the emf around ''{{math|∂Σ}}''.'' This statement, however, is not always true. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve {{math|∂Σ}} matches the actual velocity of the material conducting the electricity.<ref name=Stewart>{{cite book |title=Intermediate Electromagnetic Theory |first1=Joseph V. |last1=Stewart |page=396 |quote=This example of Faraday's Law [the homopolar generator] makes it very clear that in the case of extended bodies care must be taken that the boundary used to determine the flux must not be stationary but must be moving with respect to the body.}}</ref> If the conductor is not an infinitely thin wire, one may also have take into account the velocity of charges with respect to the material.<ref name="zangvill" /> The two examples illustrated below show that one often obtains incorrect results when Faraday's law is applied too broadly.<ref name=Feynman/>
 <gallery widths="300">
@@ Line 169: / Line 140: @@
 </gallery>
-One can analyze examples like these by taking care that the path {{math|∂Σ}} moves with the same velocity as the material.<ref name=Stewart/> Alternatively, one can always correctly calculate the emf by combining Lorentz force law with the Maxwell–Faraday equation:<ref name=Feynman/>{{Rp|ch17}}<ref name=HughesYoung>{{cite book|title=The Electromagnetodynamics of Fluid|first1=W. F.|last1=Hughes|first2=F. J.|last2=Young|publisher=John Wiley|date=1965|at=Eq. (2.6–13) p. 53}}</ref>
+One can analyze examples like these by taking care that the path {{math|∂Σ}} moves with the same velocity as the material.<ref name=Stewart/> The electromotive force can always be correctly calculated by combining the Lorentz force law with the Maxwell–Faraday equation:<ref name=Feynman/>{{Rp|ch17}}
-<math display="block">\mathcal{E} = \int_{\partial \Sigma} (\mathbf{E} + \mathbf{v}_m \times \mathbf{B}) \cdot \mathrm{d}\mathbf{l} = -\int_\Sigma \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\Sigma + \oint_{\partial \Sigma} (\mathbf{v}_m\times\mathbf{B}) \cdot \mathrm{d}\mathbf{l}</math>
+<math display="block">\mathcal{E} = \int_{\partial \Sigma} (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \mathrm{d}\mathbf{l} = -\int_{\partial \Sigma(t)} \frac{\partial \mathbf B}{\partial t}\cdot{\rm d}\mathbf A + \oint_{\partial \Sigma(t)} \left( \mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l},</math>
-where "it is very important to notice that (1) {{math|['''v'''<sub>''m''</sub>]}} is the velocity of the conductor ... not the velocity of the path element {{math|d'''l'''}} and (2) in general, the partial derivative with respect to time cannot be moved outside the integral since the area is a function of time."<ref name=HughesYoung/>
+where {{math|'''v'''}} is the velocity of the conductor in the frame of reference in which {{math|'''B'''}} in described. The time derivative cannot in general be moved outside the integral since the position or shape of the loop may be a function of time.<ref name="HughesYoung">{{cite book |last1=Hughes |first1=W. F. |title=The Electromagnetodynamics of Fluid |last2=Young |first2=F. J. |date=1965 |publisher=John Wiley |at=Eq. (2.6–13) p. 53}}</ref>
 ==Faraday's law and relativity==
 {{Further|Moving magnet and conductor problem}}
+{{rewrite section|date=June 2025}}
 ===Two phenomena===
-Faraday's law is a single equation describing two different phenomena: the ''motional emf'' generated by a magnetic force on a moving wire (see the [[Lorentz force#Force on a current-carrying wire|Lorentz force]]), and the ''transformer emf'' generated by an electric force due to a changing magnetic field (described by the [[#Maxwell–Faraday equation|Maxwell–Faraday equation]]).
+Faraday's law is a single equation describing two different phenomena: the ''motional emf'' generated by a magnetic force on a moving wire, and the ''transformer emf'' generated by an electric force due to a changing magnetic field (described by the [[#Maxwell–Faraday equation|Maxwell–Faraday equation]]).
 [[James Clerk Maxwell]] drew attention to this fact in his 1861 paper ''[[On Physical Lines of Force]]''.<ref>{{cite journal|author-link = James Clerk Maxwell|last=Clerk Maxwell|first= James|journal = [[Philosophical Magazine]]|doi = 10.1080/14786431003659180 |pages = 11–23|publisher = [[Taylor & Francis]]|title = On physical lines of force|volume = 90|year = 1861|s2cid=135524562}}</ref> In the latter half of Part II of that paper, Maxwell gives a separate physical explanation for each of the two phenomena.
@@ Line 187: / Line 158: @@
 Yet in our explanation of the rule we have used two completely distinct laws for the two cases – {{math|'''v''' × '''B'''}} for "circuit moves" and {{math|∇ × '''E''' {{=}} −∂<sub>''t''</sub>'''B'''}} for "field changes".
-We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of ''two different phenomena''.|Richard P. Feynman, ''[[The Feynman Lectures on Physics]]''<ref>[https://feynmanlectures.caltech.edu/II_17.html#Ch17-S1-p10 The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction]</ref>}}{{Dubious|reason=Feynman Lectures sometimes uses intentionally misleading statements to set up later revelations.  This may be one of those times.  Also see talk section "Two phenomena"|date=June 2023}}
+We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of ''two different phenomena''.|Richard P. Feynman, ''[[The Feynman Lectures on Physics]]''<ref>{{Cite web|url=https://www.feynmanlectures.caltech.edu/II_17.html|title=The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction|website=www.feynmanlectures.caltech.edu}}</ref>}}{{Dubious|reason=Feynman Lectures sometimes uses intentionally misleading statements to set up later revelations. This may be one of those times. Also see talk section "Two phenomena"|date=June 2023}}
 === Explanation based on four-dimensional formalism ===
-In the general case, explanation of the ''motional emf'' appearance by action of the magnetic force on the charges in the moving wire or in the circuit changing its area is unsatisfactory. As a matter of fact, the charges in the wire or in the circuit could be completely absent, will then the electromagnetic induction effect disappear in this case? This situation is analyzed in the article, in which, when writing the integral equations of the electromagnetic field in a four-dimensional covariant form, in the Faraday’s law the total time derivative of the magnetic flux through the circuit appears instead of the partial time derivative.<ref>{{cite journal | last1 = Fedosin | first1 = Sergey G. | title = On the Covariant Representation of Integral Equations of the Electromagnetic Field | journal = Progress in Electromagnetics Research C | volume = 96 | pages = 109–122| year = 2019 | url = https://rdcu.be/ccV9o| doi = 10.2528/PIERC19062902|arxiv=1911.11138| bibcode=2019arXiv191111138F| s2cid = 208095922 }}</ref> Thus, electromagnetic induction appears either when the magnetic field changes over time or when the area of the circuit changes. From the physical point of view, it is better to speak not about the induction emf, but about the induced electric field strength <math display="inline"> \mathbf E = - \nabla \mathcal{E}  - \frac{ \partial \mathbf A}{ \partial t}</math>, that occurs in the circuit when the magnetic flux changes. In this case, the contribution to <math> \mathbf E</math> from the change in the magnetic field is made through the term <math display="inline"> - \frac{ \partial \mathbf A}{ \partial t}</math> , where <math> \mathbf A</math> is the vector potential. If the circuit area is changing in case of the constant magnetic field, then some part of the circuit is inevitably moving, and the electric field <math> \mathbf E</math> emerges in this part of the circuit in the comoving reference frame K’ as a result of the Lorentz transformation of the magnetic field <math> \mathbf B</math>, present in the stationary reference frame K, which passes through the circuit. The presence of the field <math> \mathbf E</math> in K’ is considered as a result of the induction effect in the moving circuit, regardless of whether the charges are present in the circuit or not. In the conducting circuit, the field <math> \mathbf E</math> causes motion of the charges. In the reference frame K, it looks like appearance of emf of the induction <math> \mathcal{E} </math>, the gradient of which in the form of <math> - \nabla \mathcal{E}  </math>, taken along the circuit, seems to generate the field <math> \mathbf E</math>.
+In the general case, explanation of the ''motional emf'' appearance by action of the magnetic force on the charges in the moving wire or in the circuit changing its area is unsatisfactory. As a matter of fact, the charges in the wire or in the circuit could be completely absent, will then the electromagnetic induction effect disappear in this case? This situation is analyzed in the article, in which, when writing the integral equations of the electromagnetic field in a four-dimensional covariant form, in the Faraday’s law the total time derivative of the magnetic flux through the circuit appears instead of the partial time derivative.<ref>{{cite journal | last1 = Fedosin | first1 = Sergey G. | title = On the Covariant Representation of Integral Equations of the Electromagnetic Field | journal = Progress in Electromagnetics Research C | volume = 96 | pages = 109–122| year = 2019 | url = https://rdcu.be/ccV9o| doi = 10.2528/PIERC19062902|arxiv=1911.11138 | s2cid = 208095922 }}</ref> Thus, electromagnetic induction appears either when the magnetic field changes over time or when the area of the circuit changes. From the physical point of view, it is better to speak not about the induction emf, but about the induced electric field strength <math display="inline"> \mathbf E = - \nabla \mathcal{E}  - \frac{ \partial \mathbf A}{ \partial t}</math>, that occurs in the circuit when the magnetic flux changes. In this case, the contribution to <math> \mathbf E</math> from the change in the magnetic field is made through the term <math display="inline"> - \frac{ \partial \mathbf A}{ \partial t}</math> , where <math> \mathbf A</math> is the vector potential. If the circuit area is changing in case of the constant magnetic field, then some part of the circuit is inevitably moving, and the electric field <math> \mathbf E</math> emerges in this part of the circuit in the comoving reference frame K’ as a result of the Lorentz transformation of the magnetic field <math> \mathbf B</math>, present in the stationary reference frame K, which passes through the circuit. The presence of the field <math> \mathbf E</math> in K’ is considered as a result of the induction effect in the moving circuit, regardless of whether the charges are present in the circuit or not. In the conducting circuit, the field <math> \mathbf E</math> causes motion of the charges. In the reference frame K, it looks like appearance of emf of the induction <math> \mathcal{E} </math>, the gradient of which in the form of <math> - \nabla \mathcal{E}  </math>, taken along the circuit, seems to generate the field <math> \mathbf E</math>.
 ===Einstein's view===

Faraday's law of induction: Difference between revisions

Revision as of 05:24, 20 June 2025

Contents

History

Flux rule

Maxwell–Faraday equation

Derivation of the flux rule from microscopic equations

Exceptions

Faraday's law and relativity

Two phenomena

Explanation based on four-dimensional formalism

Einstein's view

See also

References

Further reading

External links

Navigation menu

Faraday's law of induction: Difference between revisions

Revision as of 05:24, 20 June 2025

History

Flux rule

Maxwell–Faraday equation

Derivation of the flux rule from microscopic equations

Exceptions

Faraday's law and relativity

Two phenomena

Explanation based on four-dimensional formalism

Einstein's view

See also

References

Further reading

External links

Navigation menu

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