Faraday's law of induction: Difference between revisions

Latest revision as of 05:39, 28 June 2025

Template:Short description Template:Use American English

In electromagnetism, Faraday's law of induction describes how a changing magnetic field can induce an electric current in a circuit. This phenomenon, known as electromagnetic induction, is the fundamental operating principle of transformers, inductors, and many types of electric motors, generators and solenoids.^[1]^[2]

"Faraday's law" is used in the literature to refer to two closely related but physically distinct statements.Template:Sfn Template:Sfn One is the Maxwell–Faraday equation, one of Maxwell's equations, which states that a time-varying magnetic field is always accompanied by a circulating electric field. This law applies to the fields themselves and does not require the presence of a physical circuit.

The other is Faraday's flux rule, or the Faraday–Lenz law, which relates the electromotive force (emf) around a closed conducting loop to the time rate of change of magnetic flux through the loop. The flux rule accounts for two mechanisms by which an emf can be generated. In transformer emf, a time-varying magnetic field induces an electric field as described by the Maxwell–Faraday equation, and the electric field drives a current around the loop. In motional emf, the circuit moves through a magnetic field, and the emf arises from the magnetic component of the Lorentz force acting on the charges in the conductor.

Historically, the differing explanations for motional and transformer emf posed a conceptual problem, since the observed current depends only on relative motion, but the physical explanations were different in the two cases. In special relativity, this distinction is understood as frame-dependent: what appears as a magnetic force in one frame may appear as an induced electric field in another. Template:Sidebar with collapsible lists

History

In 1820, Hans Christian Ørsted demonstrated that an electric current produces a magnetic field, showing that a compass needle could be deflected by a nearby current-carrying wire. This discovery prompted scientists to ask whether the reverse was also true—whether a magnetic field could generate an electric current.Template:Sfn

Initial experiments revealed that a static magnetic field had no effect on a nearby circuit: simply placing a magnet near a wire loop produced no current.Template:Sfn The breakthrough came in 1831, when Michael Faraday demonstrated that a changing magnetic field could indeed induce an electric current in a circuit. Independently, Joseph Henry made similar observations in 1832,^[3] though Faraday was the first to publish his findings.^[4]^[5]

File:Faraday emf experiment.svg

Faraday's 1831 demonstration^[6]

Faraday's notebook on August 29, 1831^[7] describes an experimental demonstration of induction.^[8] He wrapped two coils of wire around opposite sides of an iron ring, forming a primitive toroidal transformer. When he connected one coil to a battery, he observed a brief deflection in a galvanometer attached to the second coil. He concluded that a changing current in the first coil created a changing magnetic field in the ring, which in turn induced a current in the second coil. He described this as a "wave of electricity" propagated through the iron.^[9]Template:Rp

Over the following months, Faraday discovered other manifestations of electromagnetic induction.^[6] He observed transient currents when a bar magnet was rapidly moved into or out of a coil of wire. He also built a device, now known as Faraday's disk or homopolar generator, that produced a steady (DC) current by rotating a copper disk in the presence of a stationary magnet, using a sliding electrical contact.^[9]Template:Rp

File:Faraday disk generator.jpg

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

Faraday explained these phenomena using the concept of lines of force. However, his theoretical ideas were met with skepticism, as they were not formulated mathematically.^[9]Template:Rp James Clerk Maxwell later gave Faraday's insights mathematical expression, incorporating them into his broader electromagnetic theory in the early 1860s.^[9]Template:Rp^[10]^[11]

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,^[12] 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.^[13]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.^[14]^[15]

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, flux law, and FaradayTemplate:EndashLenz law,^[16] 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.^[17] 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.Template:Sfn

Mathematically, in SI units, the law is expressed 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. The magnetic flux is defined as the surface integral of the magnetic field Template:Math over a time-dependent surface Template:Math, whose boundary is the wire loop: $Φ_{B} = \iint_{Σ (t)} 𝐁 (t) \cdot d 𝐀,$ where Template:Math is an infinitesimal area vector normal to the surface. The dot product Template:Math represents the flux through the differential area element. In more visual terms, the magnetic flux is proportional to the number of magnetic field lines passing through the loop.

When the flux changes, an emf is induced around the loop. This emf corresponds to the energy per unit charge required to move it once around the loop.Template:Sfn Template:Sfn^[18] In a simple circuit with resistance $R$ , an emf $ℰ$ gives rise to a current $I$ according to the Ohm's law $ℰ = I R$ .Template:Sfn Equivalently, if the loop is broken to form an open circuit and a voltmeter is connected across the terminals, the emf is equal to the voltage measured across the open ends.^[19]

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^[20]^[21] $ℰ = - N \frac{d Φ_{B}}{d t}$ 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.^[22]

The flux can change either because the loop moves or deforms over time, or because the field itself varies in time. These two possibilities correspond to the two mechanisms described by the flux rule:Template:Sfn

Motional emf: The circuit moves through a static but non-uniform magnetic field.
Transformer emf: The circuit remains stationary while the magnetic field changes over time.

File:Lorentz force on moving conductor.svg

A conducting rod moving through a uniform magnetic field. The magnetic component of the Lorentz force pushes electrons to one end, resulting in charge separation.

Motional emf

The basic mechanism behind motional emf is illustrated by a conducting rod moving through a magnetic field that is perpendicular to both the rod and its direction of motion. Due to movement in magnetic field, the mobile electrons of the conductor experience the magnetic component (Template:Math) of the Lorentz force that drives them along the length of the rod. This leads to a separation of charge between the two ends of the rod. In the steady state, the electric field from the accumulated charge balances the magnetic force.Template:Sfn

File:Faraday's law of induction.svg

The flux rule in three cases: (a) motional emf, with moving circuit and a stationary magnetic field (b) stationary circuit, with the source of the field moving (c) time-dependent magnetic field strength

If the rod is part of a closed conducting loop moving through a nonuniform magnetic field, the same effect can drive a current around the circuit. For instance, suppose the magnetic field is confined to a limited region of space, and the loop initially lies outside this region. As it moves into the field, the area of the loop that encloses magnetic flux increases, and an emf is induced. From the Lorentz force perspective, this is because the field exerts a magnetic force on charge carriers in the parts of the loop entering the region. Once the entire loop lies in a uniform magnetic field and continues at constant speed, the total enclosed flux remains constant, and the emf vanishes. In this situation, magnetic forces on opposite sides of the loop cancel out.

Transformer emf

A complementary case is transformer emf, which occurs when the conducting loop remains stationary but the magnetic flux through it changes due to a time-varying magnetic field. This can happen in two ways: either the source of the magnetic field moves, altering the field distribution through the fixed loop, or the strength of the magnetic field changes over time at a fixed location, as in the case of a powered electromagnet.

In either situation, no magnetic force acts on the charges, and the emf is entirely due to the electric component (Template:Math) of the Lorentz force. According to the Maxwell–Faraday equation, a time-varying magnetic field produces a circulating electric field, which drives current in the loop. This phenomenon underlies the operation of electrical machines such as synchronous generators.Template:Sfn The electric field induced in this way is non-conservative, meaning its line integral around a closed loop is not zero.Template:Sfn Template:Sfn Template:Sfn

Direction of the induced current

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).

Maxwell–Faraday equation

Script error: No such module "anchor".

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 is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It states that a time-varying magnetic field always accompanies a spatially varying, non-conservative electric field. In differential form and in SI units, it reads:

Template:Equation box 1

where Template:Math is the curl operator, 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.^[25]

It can also be written in an integral form by the Kelvin–Stokes theorem:^[26]

Template:Equation box 1

where, as indicated in the figure, Template:Math is a surface bounded by the closed loop Template:Math and Template:Math is an infinitesimal vector element along that loop. The vector area element Template:Math is perpendicular to the surface and oriented according to the right-hand rule: when the thumb points in the direction of the Template:Math, the curled fingers indicate the direction of Template:Math along the boundary.

The left-hand side of the equation represents the circulation of the electric field around the loop Template:Math.^[27]Template:Rp For static electric fields, the circulation is zero, since the field can be expressed as the gradient of a scalar potential. In contrast, a time-varying magnetic field produces a non-conservative electric field with nonzero circulation. When such field acts on a conducting loop, it drives a current around the loop.

If the surface Template:Math is not changing in time, the right-hand side equation becomes the time-derivative of the magnetic flux Template:Math through the surface: $\oint_{\partial Σ} 𝐄 \cdot d 𝐥 = - \frac{d}{d t} \int_{Σ} 𝐁 \cdot d 𝐀 = - \frac{d Φ_{B}}{d t} .$ If the left-hand side is identified with the work per unit charge done by the electric field on charges in a fixed conducting loop, this equation reproduces the flux rule in the special case of a stationary circuit.

In the non-relativistic limit, the solenoidal component of the induced electric field can be approximated by the volume integral^[25]Template:Rp $𝐄_{s} (𝐫, t) \approx - \frac{1}{4 π} \underset{V}{∭} \frac{(\frac{\partial 𝐁 (𝐫^{'}, t)}{\partial t}) \times (𝐫 - 𝐫^{'})}{| 𝐫 - 𝐫^{'} |^{3}} d^{3} 𝐫^{'}$ This expression shows how changes in the magnetic field across space contribute to the induced electric field at a given point, with each contribution weighted by the inverse square of the distance.

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.^[27]^[28] From these, Faraday's law can be derived directly.^[29]^[30]^[31]

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":Template:Sfn Template:Sfn $\frac{d}{d t} \int_{Σ (t)} 𝐁 \cdot d 𝐀 = \int_{Σ (t)} (\frac{\partial 𝐁}{\partial t} + (\nabla \cdot 𝐁) 𝐯_{c}) \cdot d 𝐀 - \oint_{\partial Σ (t)} (𝐯_{c} \times 𝐁) \cdot d 𝐥$ 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.Template:Sfn

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.Template:Sfn

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.Template:Sfn

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 sumTemplate:Sfn $ℰ = - \int_{Σ (t)} \frac{\partial 𝐁}{\partial t} \cdot d 𝐀 + \oint_{\partial Σ (t)} (𝐯_{c} \times 𝐁) \cdot d 𝐥 .$ 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.^[32] 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.Template:Sfn

Limitations of the flux rule

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.^[33] 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.Template:Sfn The two examples illustrated below show that one often obtains incorrect results when Faraday's law is applied too broadly.^[27]

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^[27]Template:Rp

One can analyze examples like these by taking care that the path Template:Math moves with the same velocity as the material.^[33] The electromotive force can always be correctly calculated by combining the Lorentz force law with the Maxwell–Faraday equation:^[27]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.^[34]

Flux rule and relativity

Script error: No such module "labelled list hatnote". Historically, the two distinct mechanisms encompassed by the flux rule, motional emf and transformer emf, posed a conceptual challenge.^[35] James Clerk Maxwell already recognized that electromagnetic induction could arise through different physical processes, even though the induced emf obeyed a single mathematical expression. In his 1861 paper On Physical Lines of Force,^[36] he gave separate physical explanations for each case.

In 1905, Albert Einstein highlighted this asymmetry in classical electrodynamics in his paper On the Electrodynamics of Moving Bodies.^[37] He pointed out that the physical outcome, such as the induced current, depends only on relative motion between the conductor and the magnet, yet classical theory provided different explanations depending on which object was considered to be in motion. This inconsistency suggested the absence of a preferred frame and helped motivate the development of special relativity.Template:Sfn

In modern terms, electric and magnetic fields are understood as components of a single electromagnetic field tensor. Under a change of inertial frame, the two fields transform into one another.^[38]

Notes

Template:Reflist

References

Script error: No such module "citation/CS1".
Script error: No such module "citation/CS1".
Script error: No such module "citation/CS1".
Script error: No such module "citation/CS1".
Script error: No such module "citation/CS1".
Script error: No such module "citation/CS1".

External links

Template:Michael Faraday

↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ ^a ^b Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ ^a ^b ^c ^d Script error: No such module "citation/CS1".Template:Full citation needed
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "Citation/CS1".
A partial translation of the paper is available in Script error: No such module "citation/CS1".
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ ^a ^b Script error: No such module "Citation/CS1". Video Explanation
↑ Script error: No such module "citation/CS1".
↑ ^a ^b Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ ^a ^b ^c ^d ^e Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ ^a ^b Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "citation/CS1".
↑ Script error: No such module "citation/CS1".

[Sadiku386-1] Script error: No such module "citation/CS1".

[2] Script error: No such module "citation/CS1".

[3] Script error: No such module "citation/CS1".

[4] Script error: No such module "citation/CS1".

[5] Script error: No such module "citation/CS1".

[Giancoli-6] Script error: No such module "citation/CS1".

[:1-7] Script error: No such module "citation/CS1".

[FaradayDay1999-8] Script error: No such module "citation/CS1".

[Williams-9] Script error: No such module "citation/CS1".Template:Full citation needed

[10] Script error: No such module "citation/CS1".

[IEEUK-11] Script error: No such module "citation/CS1".

[12] Script error: No such module "Citation/CS1".
A partial translation of the paper is available in Script error: No such module "citation/CS1".

[13] Script error: No such module "Citation/CS1".

[FRBS-20190301-14] Script error: No such module "citation/CS1".

[BT-20230628-15] Script error: No such module "citation/CS1".

[r127-16] Script error: No such module "citation/CS1".

[n495-17] Script error: No such module "citation/CS1".

[18] Script error: No such module "citation/CS1".

[f437-19] Script error: No such module "citation/CS1".

[20] Script error: No such module "citation/CS1".

[21] Script error: No such module "citation/CS1".

[22] Script error: No such module "citation/CS1".

[Salu2014-23] Script error: No such module "Citation/CS1". Video Explanation

[24] Script error: No such module "citation/CS1".

[Griffiths-25] Script error: No such module "citation/CS1".

[Harrington-26] Script error: No such module "citation/CS1".

[Feynman-27] Script error: No such module "citation/CS1".

[Griffiths2-28] Script error: No such module "citation/CS1".

[Davison-29] Script error: No such module "Citation/CS1".

[Krey-30] Script error: No such module "citation/CS1".

[:0-31] Script error: No such module "citation/CS1".

[32] Script error: No such module "citation/CS1".

[Stewart-33] Script error: No such module "citation/CS1".

[HughesYoung-34] Script error: No such module "citation/CS1".

[35] Script error: No such module "citation/CS1".

[36] Script error: No such module "Citation/CS1".

[37] Script error: No such module "citation/CS1".

[Chow-38] Script error: No such module "citation/CS1".

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

Faraday's law of induction: Difference between revisions

Latest revision as of 05:39, 28 June 2025

Contents

History

Flux rule

Motional emf

Transformer emf

Direction of the induced current

Maxwell–Faraday equation

Derivation of the flux rule from microscopic equations

Limitations of the flux rule

Flux rule and relativity

See also

Notes

References

Further reading

External links

Navigation menu

@@ Line 1: / Line 1: @@
 {{Short description|Basic law of electromagnetism}}
 {{Use American English|date=March 2019}}
-[[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>
+{{multiple image|position
+| align             = right
+| direction         = horizontal
+| image1            = Elementary generator.svg
+| caption1          = Motional emf, induced by moving a conductor through a magnetic field.
+| image2            = Alternator 1.svg
+| caption2          = Transformer emf, induced by a changing magnetic field.
+| total_width       = 400
+| alt1              =
+}}
-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.
+In [[electromagnetism]], '''Faraday's law of induction''' describes how a changing [[magnetic field]] can induce an [[electric current]] in a [[electric circuit|circuit]]. 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. |url=https://books.google.com/books?id=w2ITHQAACAAJ |title=Elements of Electromagnetics |publisher=Oxford University Press |year=2007 |isbn=978-0-19-530048-2 |edition=4th |location=New York & Oxford |page=386}}</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>
-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.
+"Faraday's law" is used in the literature to refer to two closely related but physically distinct statements.{{sfn|Griffiths|2023|pp=316-317}}{{sfn|Sadiku|2018|p=422}} One is the '''Maxwell–Faraday equation''', one of [[Maxwell's equations]], which states that a time-varying magnetic field is always accompanied by a circulating [[electric field]]. This law applies to the fields themselves and does not require the presence of a physical circuit.
+The other is '''Faraday's flux rule''', or the '''Faraday–Lenz law''', which relates the [[electromotive force]] (emf) around a closed conducting loop to the time rate of change of [[magnetic flux]] through the loop. The flux rule accounts for two mechanisms by which an emf can be generated. In '''transformer emf''', a time-varying magnetic field induces an electric field as described by the Maxwell–Faraday equation, and the electric field drives a current around the loop. In '''motional emf''', the circuit moves through a magnetic field, and the emf arises from the magnetic component of the [[Lorentz force]] acting on the charges in the conductor.
+Historically, the differing explanations for motional and transformer emf posed a conceptual problem, since the observed current depends only on relative motion, but the physical explanations were different in the two cases. In [[special relativity]], this distinction is understood as frame-dependent: what appears as a magnetic force in one frame may appear as an induced electric field in another.
 {{electromagnetism|cTopic=Electrodynamics}}
 ==History==
-Electromagnetic induction was discovered independently by [[Michael Faraday]] in 1831 and [[Joseph Henry]] in 1832.<ref>{{Cite web|url=http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf|title=A Brief History of Electromagnetism}}</ref> Faraday was the first to publish the results of his experiments.<ref>{{cite book|last=Ulaby|first=Fawwaz|title=Fundamentals of applied electromagnetics|edition=5th|year=2007|url=https://www.amazon.com/exec/obidos/tg/detail/-/0132413264/ref=ord_cart_shr?%5Fencoding=UTF8&m=ATVPDKIKX0DER&v=glance|publisher=Pearson:Prentice Hall|isbn=978-0-13-241326-8|page=255}}</ref><ref>{{cite web|url=http://www.nasonline.org/member-directory/deceased-members/20001467.html |title=Joseph Henry |access-date=2016-12-30 |work=Member Directory, National Academy of Sciences}}</ref>
+In 1820, [[Hans Christian Ørsted]] demonstrated that an electric current produces a magnetic field, showing that a compass needle could be deflected by a nearby current-carrying wire. This discovery prompted scientists to ask whether the reverse was also true—whether a magnetic field could generate an electric current.{{Sfn|Darrigol|2000|pp=3-6, 32}}
+Initial experiments revealed that a static magnetic field had no effect on a nearby circuit: simply placing a magnet near a wire loop produced no current.{{Sfn|Darrigol|2000|pp=3-6, 32}} The breakthrough came in 1831, when [[Michael Faraday]] demonstrated that a changing magnetic field could indeed induce an electric current in a circuit. Independently, [[Joseph Henry]] made similar observations in 1832,<ref>{{Cite web |title=A Brief History of Electromagnetism |url=http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf}}</ref> though Faraday was the first to publish his findings.<ref>{{cite book |last=Ulaby |first=Fawwaz |url=https://www.amazon.com/exec/obidos/tg/detail/-/0132413264/ref=ord_cart_shr?%5Fencoding=UTF8&m=ATVPDKIKX0DER&v=glance |title=Fundamentals of applied electromagnetics |publisher=Pearson:Prentice Hall |year=2007 |isbn=978-0-13-241326-8 |edition=5th |page=255}}</ref><ref>{{cite web |title=Joseph Henry |url=http://www.nasonline.org/member-directory/deceased-members/20001467.html |access-date=2016-12-30 |work=Member Directory, National Academy of Sciences}}</ref>
 [[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<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 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> He wrapped two coils of wire around opposite sides of an iron ring, forming a primitive [[toroidal transformer]]. When he connected one coil to a battery, he observed a brief deflection in a [[galvanometer]] attached to the second coil. He concluded that a changing current in the first coil created a changing magnetic field in the ring, which in turn induced a current in the second coil. He described this as a "wave of electricity" propagated through the iron.<ref name="Williams">{{cite book |last=Williams |first=L. Pearce |url=https://archive.org/details/michaelfaradaybi00will |title=Michael Faraday |publisher=New York, Basic Books |year=1965 |url-access=registration}}{{full citation needed|date=September 2018}}</ref>{{rp|182–183}}
-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}}
+Over the following months, Faraday discovered other manifestations of electromagnetic induction.<ref name="Giancoli" /> He observed transient currents when a bar magnet was rapidly moved into or out of a coil of wire. He also built a device, now known as [[Homopolar generator|Faraday's disk]] or homopolar generator, that produced a steady ([[direct current|DC]]) current by rotating a copper disk in the presence of a stationary magnet, using a sliding electrical contact.<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 is the form recognized today in the group of equations known as [[Maxwell's equations]].
+Faraday explained these phenomena using the concept of [[lines of force]]. However, his theoretical ideas were met with skepticism, as they were not formulated mathematically.<ref name=Williams/>{{rp|510}} [[James Clerk Maxwell]] later gave Faraday's insights mathematical expression, incorporating them into his broader electromagnetic theory in the early 1860s.<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).
@@ Line 31: / Line 46: @@
 ==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.]]
-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>
+Faraday's law of induction, also known as the ''flux rule'', ''flux law'', 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.{{sfn|Griffiths|2023|p=319}}
-Mathematically, in [[SI units]], the law is written as
+Mathematically, in [[SI units]], the law is expressed as
 <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 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:
+where <math>\mathcal{E}</math> is the electromotive force (emf) and {{math|Φ<sub>''B''</sub>}} is the magnetic flux throught the circuit. The magnetic flux is defined as the [[surface integral]] of the magnetic field {{math|'''B'''}} over a time-dependent [[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'')}}, 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.
+where {{math|d'''A'''}} is an infinitesimal [[area vector]] normal to the surface. The [[dot product]] {{math|'''B''' · d'''A'''}} represents the flux through the differential area element.
-[[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=]]
+In more visual terms, the magnetic flux is proportional to the number of [[field line|magnetic field lines]] passing through the loop.
-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>
+When the flux changes, an emf is induced around the loop. This emf corresponds to the energy per unit charge required to move it once around the loop.{{sfn|Feynman|Leighton|Sands|2006|loc=Ch. 17}}{{sfn|Griffiths|2023|pp=304–306}}<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> In a simple circuit with resistance <math>R</math>, an emf <math>\mathcal E</math> gives rise to a current <math>I</math> according to the [[Ohm's law]] <math>\mathcal E = IR</math>.{{sfn|Zangwill|2013|pp=462-464}} Equivalently, if the loop is broken to form an [[Electric circuit|open circuit]] and a [[voltmeter]] is connected across the terminals, the emf is equal to the voltage measured across the open ends.<ref name="f437">{{cite book | last=Paul | first=Clayton R. | last2=Scully | first2=Robert C. | last3=Steffka | first3=Mark A. | title=Introduction to Electromagnetic Compatibility | publisher=John Wiley & Sons | date=2022-11-01 | isbn=978-1-119-40434-7 | page=703}}</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 |last1=Whelan |first1=P. M. |title=Essential Principles of Physics |last2=Hodgeson |first2=M. J. |date=1978 |publisher=John Murray |isbn=0-7195-3382-1 |edition=2nd}}</ref><ref>{{cite web |last=Nave |first=Carl R. |title=Faraday's Law |url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html |access-date=2011-08-29 |work=HyperPhysics |publisher=Georgia State University}}</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. 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>
+The flux can change either because the loop moves or deforms over time, or because the field itself varies in time. These two possibilities correspond to the two mechanisms described by the flux rule:{{sfn|Griffiths|2023|pp=298-319}}
+* ''Motional emf'': The circuit moves through a static but non-uniform magnetic field.
+* ''Transformer emf'': The circuit remains stationary while the magnetic field changes over time.
+[[File:Lorentz_force_on_moving_conductor.svg|thumb|A conducting rod moving through a uniform magnetic field. The magnetic component of the Lorentz force pushes electrons to one end, resulting in charge separation.]]
+=== Motional emf ===
+The basic mechanism behind motional emf is illustrated by a conducting rod moving through a magnetic field that is perpendicular to both the rod and its direction of motion. Due to movement in magnetic field, the mobile electrons of the conductor experience the magnetic component ({{math|''q'''''v''' × '''B'''}}) of the Lorentz force that drives them along the length of the rod. This leads to a separation of charge between the two ends of the rod. In the steady state, the electric field from the accumulated charge balances the magnetic force.{{sfn|Griffiths|2023|p=307}}
+[[File:Faraday's_law_of_induction.svg|thumb|The flux rule in three cases: (a) motional emf, with moving circuit and a stationary magnetic field (b) stationary circuit, with the source of the field moving (c) time-dependent magnetic field strength]]
+If the rod is part of a closed conducting loop moving through a nonuniform magnetic field, the same effect can drive a current around the circuit. For instance, suppose the magnetic field is confined to a limited region of space, and the loop initially lies outside this region. As it moves into the field, the area of the loop that encloses magnetic flux increases, and an emf is induced. From the Lorentz force perspective, this is because the field exerts a magnetic force on charge carriers in the parts of the loop entering the region. Once the entire loop lies in a uniform magnetic field and continues at constant speed, the total enclosed flux remains constant, and the emf vanishes. In this situation, magnetic forces on opposite sides of the loop cancel out.
+=== Transformer emf ===
+A complementary case is transformer emf, which occurs when the conducting loop remains stationary but the magnetic flux through it changes due to a time-varying magnetic field. This can happen in two ways: either the source of the magnetic field moves, altering the field distribution through the fixed loop, or the strength of the magnetic field changes over time at a fixed location, as in the case of a powered electromagnet.
+In either situation, no magnetic force acts on the charges, and the emf is entirely due to the  electric component ({{math|''q'''''E'''}}) of the Lorentz force. According to the Maxwell–Faraday equation, a time-varying magnetic field produces a circulating electric field, which drives current in the loop. This phenomenon underlies the operation of  [[Electrical machine|electrical machines]] such as [[Synchronous generator|synchronous generators]].{{sfn|Sadiku|2018|pp=424-427}} The electric field induced in this way is [[Conservative force#Mathematical description|non-conservative]], meaning its line integral around a closed loop is not zero.{{sfn|Purcell|Morin|2013|p=259}}{{sfn|Sadiku|2018|pp=424-425}}{{sfn|Griffiths|2023|p=317}}
+=== Direction of the induced current ===
+[[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">{{cite journal |author=Yehuda Salu |year=2014 |title=A Left Hand Rule for Faraday's Law |url=https://www.researchgate.net/publication/262986189 |journal=[[The Physics Teacher]] |volume=52 |issue=1 |pages=48 |bibcode=2014PhTea..52...48S |doi=10.1119/1.4849156}} [https://www.youtube.com/watch?v=ipUD9VcAd9o Video Explanation]</ref>|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" /><ref>{{cite web |last1=Salu |first1=Yehuda |date=17 January 2017 |title=Bypassing Lenz's Rule - A Left Hand Rule for Faraday's Law |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 |access-date=30 July 2017 |website=www.PhysicsForArchitects.com}}</ref>
 * Align the curved fingers of the left hand with the loop (yellow line).
 * Stretch your thumb. The stretched thumb indicates the direction of {{math|'''n'''}} (brown), the normal to the area enclosed by the loop.
@@ Line 45: / Line 86: @@
 * If the change in flux, {{math|ΔΦ<sub>''B''</sub>}}, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
 * 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, 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. 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 ==
@@ Line 54: / Line 91: @@
 [[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]]]]
-The Maxwell–Faraday equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-[[Conservative vector field|conservative]] electric field, and vice versa. The Maxwell–Faraday equation is
+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 states that a time-varying magnetic field always accompanies a spatially varying, non-[[Conservative vector field|conservative]] electric field. In differential form and in [[SI units]], it reads:
 {{Equation box 1
@@ Line 64: / Line 101: @@
 |background colour = #ECFCF4}}
-(in [[SI units]]) where {{math|∇ ×}} is the [[Curl (mathematics)|curl]] [[linear operator|operator]] and again {{math|'''E'''('''r''', ''t'')}} is the [[electric field]] and {{math|'''B'''('''r''', ''t'')}} is the [[magnetic field]]. These fields can generally be functions of position {{math|'''r'''}} and time {{mvar|t}}.<ref name="Griffiths">{{cite book
+where {{math|∇ ×}} is the [[Curl (mathematics)|curl]] [[linear operator|operator]], {{math|'''E'''('''r''', ''t'')}} is the [[electric field]] and {{math|'''B'''('''r''', ''t'')}} is the [[magnetic field]]. These fields can generally be functions of position {{math|'''r'''}} and time {{mvar|t}}.<ref name="Griffiths">{{cite book
   |last = Griffiths
   |first = David J.
@@ Line 77: / Line 114: @@
 }}</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>
+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 87: / Line 124: @@
 |background colour = #ECFCF4}}
-where, as indicated in the figure, {{math|'''Σ'''}} is a surface bounded by the closed contour {{math|∂'''Σ'''}}, {{math|d'''l'''}} is an [[infinitesimal]] vector element of the contour {{math|'''∂Σ'''}}, and {{math|d'''A'''}} is an infinitesimal vector element of surface {{math|'''Σ'''}}. Its direction is [[orthogonal]] to that surface patch, the magnitude is the area of an infinitesimal patch of surface.
+where, as indicated in the figure, {{math|'''Σ'''}} is a surface bounded by the closed loop {{math|∂'''Σ'''}} and {{math|d'''l'''}} is an [[infinitesimal]] vector element along that loop. The vector area element {{math|d'''A'''}} is perpendicular to the surface and oriented according to the [[right-hand rule]]: when the thumb points in the direction of the {{math|d'''A'''}}, the curled fingers indicate the direction of {{math|d'''l'''}} along the boundary.
-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 left-hand side of the equation represents the [[Circulation (physics)|circulation]] of the electric field around the loop {{math|∂'''Σ'''}}.<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|ch3}} For static electric fields, the circulation is zero, since the field can be expressed as the gradient of a scalar potential. In contrast, a time-varying magnetic field produces a non-conservative electric field with nonzero circulation. When such field acts on a conducting loop, it drives a current around the loop.
-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.
+If the surface {{math|'''Σ'''}} is not changing in time, the right-hand side equation becomes the time-derivative of the [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} through the surface:
+<math display="block"> \oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{A} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}. </math>If the left-hand side is identified with the work per unit charge done by the electric field on charges in a fixed conducting loop, this equation reproduces the flux rule in the special case of a stationary circuit.
-The integral equation is true for ''any'' path {{math|∂'''Σ'''}} through space, and any surface {{math|'''Σ'''}} for which that path is a boundary.
+In the non-relativistic limit, the [[solenoidal vector field|solenoidal component]] of the induced electric field can be approximated by the [[volume integral]]<ref name="Griffiths" />{{Rp|321}}
+<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>This expression shows how changes in the magnetic field across space contribute to the induced electric field at a given point, with each contribution weighted by the inverse square of the distance.
-If the surface {{math|'''Σ'''}} is not changing in time, the equation can be rewritten:
-<math display="block"> \oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{A}. </math>
-The [[surface integral]] at the right-hand side is the explicit expression for the [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} through {{math|'''Σ'''}}.
-The electric vector field induced by a changing magnetic flux, the [[solenoidal vector field|solenoidal component]] of the overall electric field, can be approximated in the non-relativistic limit by the [[volume integral]] equation<ref name="Griffiths"/>{{Rp|321}}
-<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>
 ==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]].<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 four [[Maxwell's equations]], together with the Lorentz force law, form a complete foundation for [[classical electromagnetism]].<ref name=Feynman/><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> 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 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"> \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 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>
+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":{{sfn|Zangwill|2013|p=10}}{{sfn|Zangwill|2013|pp=462–464}}
 <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>
 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,
@@ Line 113: / Line 145: @@
 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" />
+This is an exact result, derived from Maxwell's equations and vector calculus.{{sfn|Zangwill|2013|pp=462–464}}
 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>\mathbf v(\mathbf r, t) = \mathbf v_c(\mathbf r, t) + \mathbf v_d(\mathbf r, t)</math>,
-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/>
+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.{{sfn|Zangwill|2013|pp=462–464}}
 The emf <math>\mathcal{E}</math> associated with the Lorentz force is defined as
@@ Line 124: / Line 156: @@
 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" />
+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.{{sfn|Zangwill|2013|pp=462–464}}
 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">\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}</math>
-In this case, the emf can also be expressed as a sum<ref name="zangvill" />
+In this case, the emf can also be expressed as a sum{{sfn|Zangwill|2013|pp=462–464}}
-<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" />
+<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.{{sfn|Zangwill|2013|pp=462–464}}
-==Exceptions==
+==Limitations of the flux rule<span class="anchor" id="Exceptions"></span>==
 {{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. 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/>
+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.{{sfn|Zangwill|2013|pp=462–464}} 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 144: / Line 176: @@
 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==
+==Flux rule and relativity==
 {{Further|Moving magnet and conductor problem}}
-{{rewrite section|date=June 2025}}
+Historically, the two distinct mechanisms encompassed by the flux rule, motional emf and transformer emf, posed a conceptual challenge.<ref>{{Cite web |title=The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction |url=https://www.feynmanlectures.caltech.edu/II_17.html |website=www.feynmanlectures.caltech.edu}}</ref> [[James Clerk Maxwell]] already recognized that electromagnetic induction could arise through different physical processes, even though the induced emf obeyed a single mathematical expression. In his 1861 paper ''[[On Physical Lines of Force]]'',<ref>{{cite journal |last=Clerk Maxwell |first=James |author-link=James Clerk Maxwell |year=1861 |title=On physical lines of force |journal=[[Philosophical Magazine]] |publisher=[[Taylor & Francis]] |volume=90 |pages=11–23 |doi=10.1080/14786431003659180 |s2cid=135524562}}</ref> he gave separate physical explanations for each case.
-===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|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.
-A reference to these two aspects of electromagnetic induction is made in some modern textbooks.<ref name=Griffiths1>{{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/301 301–3]|publisher=Prentice Hall|year=1999|location=Upper Saddle River, NJ|isbn=0-13-805326-X}}<br>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".</ref> As Richard Feynman states:
-{{Blockquote|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 – {{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>{{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 | 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===
-Reflection on this apparent dichotomy was one of the principal paths that led [[Albert Einstein]] to develop [[special relativity]]:
-{{Blockquote|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.
+In 1905, [[Albert Einstein]] highlighted this asymmetry in classical electrodynamics in his paper ''On the Electrodynamics of Moving Bodies''.<ref>{{cite web |last=Einstein |first=Albert |author-link=Albert Einstein |title=On the Electrodynamics of Moving Bodies |url=http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf}}</ref> He pointed out that the physical outcome, such as the induced current, depends only on relative motion between the conductor and the magnet, yet classical theory provided different explanations depending on which object was considered to be in motion. This inconsistency suggested the absence of a preferred frame and helped motivate the development of [[special relativity]].{{sfn|Griffiths|2023|pp=316-318}}
-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.
+In modern terms, electric and magnetic fields are understood as components of a single [[electromagnetic field tensor]]. Under a change of [[Inertial frame of reference|inertial frame]], the two fields transform into one another.<ref name="Chow">{{cite book |author=Tai L. Chow |url=https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153 |title=Electromagnetic theory |publisher=Jones and Bartlett |year=2006 |isbn=0-7637-3827-1 |location=Sudbury, Massachusetts |page=395}}</ref>
-| [[Albert Einstein]], ''[[On the Electrodynamics of Moving Bodies]]''<ref>{{cite web|first=Albert|last=Einstein|author-link=Albert Einstein|url=http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf|title=On the Electrodynamics of Moving Bodies}}</ref>}}
 ==See also==
@@ Line 178: / Line 188: @@
 * [[Eddy current]]
 * [[Inductance]]
-* [[Maxwell's equations]]
 * [[Crosstalk]]
-* [[Faraday paradox]]
 * [[Felici's law]]
 }}{{Clear}}
-==References==
+==Notes==
 {{Reflist|30em}}
+== References ==
+* {{cite book | last=Darrigol | first=Olivier | title=Electrodynamics from Ampère to Einstein | publisher=Clarendon Press | publication-place=Oxford ; New York | date=2000 | isbn=0-19-850594-9}}
+* {{cite book |first1 = Richard Phillips |last1 = Feynman |author-link = Richard Feynman |first2 = Robert B. |last2 = Leighton | first3 = Matthew L. |last3 = Sands |title = The Feynman lectures on physics |publisher = Pearson / Addison-Wesley | year = 2006 |isbn = 0-8053-9047-2  |volume=2|url=https://www.feynmanlectures.caltech.edu/II_toc.html}}
+* {{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}}
+* {{cite book | last=Purcell | first=Edward M. | last2=Morin | first2=David J. | title=Electricity and Magnetism: | publisher=Cambridge University Press | date=2013 | isbn=978-1-139-01297-3 | doi=10.1017/cbo9781139012973 | url=https://www.cambridge.org/core/product/identifier/9781139012973/type/book}}
+* {{cite book | first=Matthew N. O. |last=Sadiku | title=Elements of electromagnetics | year= 2018 | edition=7th | publisher=Oxford University Press | location=New York/Oxford | isbn = 978-0-19-069861-4 | url=https://lccn.loc.gov/2017046497}}
+* {{cite book | last=Zangwill | first=Andrew | title=Modern Electrodynamics | publisher=Cambridge University Press | date=2013 | isbn=978-0-521-89697-9 | doi=10.1017/CBO9781139034777}}
 ==Further reading==

Faraday's law of induction: Difference between revisions

Latest revision as of 05:39, 28 June 2025

History

Flux rule

Motional emf

Transformer emf

Direction of the induced current

Maxwell–Faraday equation

Derivation of the flux rule from microscopic equations

Limitations of the flux rule

Flux rule and relativity

See also

Notes

References

Further reading

External links

Navigation menu

Search