History[edit]A diagram of Faraday's

History[edit]

A diagram of Faraday's iron ring apparatus. Change in the magnetic flux of the left coil induces a current in the right coil.[1]

Faraday's disk (see homopolar generator)
Electromagnetic induction was first discovered by Michael Faraday, who made his discovery public in 1831.[2][3] It was discovered independently by Joseph Henry in 1832.[4][5]

In Faraday's first experimental demonstration (August 29, 1831), he wrapped two wires around opposite sides of an iron ring or "torus" (an arrangement similar to a modern toroidal transformer). [6][not in citation given] Based on his assessment of recently discovered properties of electromagnets, he expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. Indeed, he saw a transient current (which he called a "wave of electricity") when he connected the wire to the battery, and another when he disconnected it.[7] This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected.[1] Within two months, Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk").[8]

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

Heinrich Lenz formulated the law named after him in 1834 to describe the "flux through the circuit". Lenz's law gives the direction of the induced EMF and current resulting from electromagnetic induction (elaborated upon in the examples below).

Following the understanding brought by these laws, many kinds of device employing magnetic induction have been invented.


Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current that flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G).[12]
Theory[edit]
Main article: Faraday's law of induction
The law of physics describing the process of electromagnetic induction is known as Faraday's law of induction and the most widespread version of this law states that the induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux enclosed by the circuit.[13][14] Or mathematically,

mathcal{E} = -{{dPhi_mathrm{B}} over dt} ,
where mathcal{E} is the electromotive force (EMF) and ΦB is the magnetic flux. The direction of the electromotive force is given by Lenz's law. This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire,[15] and is invalid in some other circumstances. A different version, the Maxwell–Faraday equation (discussed below), is valid in all circumstances.

For a tightly wound coil of wire, composed of N identical turns, each with the same magnetic flux going through them, the resulting EMF is given by[16][17]

mathcal{E} = -N {{dPhi_mathrm{B}} over dt}
Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral:

Phi_mathrm{B} = iintlimits_{Sigma(t)} mathbf{B}(mathbf{r}, t) cdot d mathbf{A} ,
where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product (the infinitesimal amount of magnetic flux). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.

When the flux changes—because 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, mathcal{E}, defined as the energy available from a unit charge that has travelled once around the wire loop.[15][18][19][20] Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.

According to the Lorentz force law (in SI units),

mathbf{F} = q left(mathbf{E} + mathbf{v} imesmathbf{B}
ight)
the EMF on a wire loop is:

mathcal{E} = frac{1}{q} oint_{mathrm{wire}}mathbf{F}cdot doldsymbol{ell} = oint_{mathrm{wire}} left(mathbf{E} + mathbf{v} imesmathbf{B}
ight)cdot doldsymbol{ell}
where E is the electric field, B is the magnetic field (aka magnetic flux density, magnetic induction), dℓ is an infinitesimal arc length along the wire, and the line integral is evaluated along the wire (along the curve coincident with the shape of the wire).

Maxwell–Faraday equation[edit]

An illustration of Kelvin-Stokes theorem with surface Σ its boundary ∂Σ and orientation n set by the right-hand rule.
The Maxwell–Faraday equation is a generalisation of Faraday's law that states that a time-varying magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice versa. The Maxwell–Faraday equation is


abla imes mathbf{E} = -frac{partial mathbf{B}} {partial t}

(in SI units) where
abla imes is the curl operator and again E(r, t) is the electric field and B(r, t) is the magnetic field. These fields can generally be functions of position r and time t.

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:[21]

oint_{partial Sigma} mathbf{E} cdot doldsymbol{ell} = - int_{Sigma} frac{partial mathbf{B}}{partial t} cdot dmathbf{A}

where, as indicated in the figure:

Σ is a surface bounded by the closed contour ∂Σ,
E is the electric field, B is the magnetic field.
dℓ is an infinitesimal vector element of the contour ∂Σ,
dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.
Both dℓ and dA 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 Σ, a positive path element dℓ of curve ∂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 n to the surface Σ.

The integral around ∂Σ is called a path integral or line integral.

Kết quả (Anh) 1:

History [edit]A diagram of Faraday's iron ring apparatus. Change in the magnetic flux of the left coil induces a current in the coil. T12Faraday's disk (see homopolar generator)Electromagnetic induction was first discovered by Michael Faraday, who made his discovery public in 1831. [2] [3] It was independently discovered by Joseph Henry in 1832. [4] [5]In Faraday's first experimental demonstration (August 29, 1831), he wrapped two wires around opposite sides of an iron ring or "torus" (an arrangement similar to a modern toroidal transformer). [6] [not in citation given] Based on his assessment of recently discovered properties of electromagnets, he expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. Indeed, he saw a transient current (which he called a "wave of electricity") when he connected the wire to the battery, and another when he disconnected it. [7] This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. [1] Within two months, Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk"). [8]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 mathematically formulated. [9] An exception was James Clerk Maxwell, who used Faraday's ideas as the basis of his quantitative electromagnetic theory. [9] [10] [11] In Maxwell's model, the time varying aspect of electromagnetic induction is expressed as a differential equation, which Oliver Heaviside referred to as Faraday's law even though it is slightly different from the original Faraday's formulation and does not describe motional EMF. Heaviside's version (see Maxwell-Faraday equation below) is the form recognized today in the group of equations known as Maxwell's equations.Heinrich Lenz formulated the law named after him in 1834 to describe the "flux through the circuit". Lenz's law gives the direction of the induced EMF and current resulting from electromagnetic induction (elaborated upon in the examples below).Following the understanding brought by these laws, many kinds of device employing magnetic induction have been invented.Faraday's experiment showing induction between coils of wire: The liquid battery (right) provides a current that flows through the small coil (A), creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil (B), the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer (G). [12]Theory[edit]Main article: Faraday's law of inductionThe law of physics describing the process of electromagnetic induction is known as Faraday's law of induction and the most widespread version of this law states that the induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux enclosed by the circuit.[13][14] Or mathematically,mathcal{E} = -{{dPhi_mathrm{B}} over dt} ,where mathcal{E} is the electromotive force (EMF) and ΦB is the magnetic flux. The direction of the electromotive force is given by Lenz's law. This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire,[15] and is invalid in some other circumstances. A different version, the Maxwell–Faraday equation (discussed below), is valid in all circumstances.For a tightly wound coil of wire, composed of N identical turns, each with the same magnetic flux going through them, the resulting EMF is given by[16][17] mathcal{E} = -N {{dPhi_mathrm{B}} over dt} Faraday's law of induction makes use of the magnetic flux ΦB through a hypothetical surface Σ whose boundary is a wire loop. Since the wire loop may be moving, we write Σ(t) for the surface. The magnetic flux is defined by a surface integral: Phi_mathrm{B} = iintlimits_{Sigma(t)} mathbf{B}(mathbf{r}, t) cdot d mathbf{A} , where dA is an element of surface area of the moving surface Σ(t), B is the magnetic field, and B·dA is a vector dot product (the infinitesimal amount of magnetic flux). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop.When the flux changes—because 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, mathcal{E}, defined as the energy available from a unit charge that has travelled once around the wire loop.[15][18][19][20] Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.According to the Lorentz force law (in SI units), mathbf{F} = q left(mathbf{E} + mathbf{v} imesmathbf{B}
ight) the EMF on a wire loop is:mathcal{E} = frac{1}{q} oint_{mathrm{wire}}mathbf{F}cdot doldsymbol{ell} = oint_{mathrm{wire}} left(mathbf{E} + mathbf{v} imesmathbf{B}
ight)cdot doldsymbol{ell}where E is the electric field, B is the magnetic field (aka magnetic flux density, magnetic induction), dℓ is an infinitesimal arc length along the wire, and the line integral is evaluated along the wire (along the curve coincident with the shape of the wire).Maxwell–Faraday equation[edit]An illustration of Kelvin-Stokes theorem with surface Σ its boundary ∂Σ and orientation n set by the right-hand rule.The Maxwell–Faraday equation is a generalisation of Faraday's law that states that a time-varying magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice versa. The Maxwell–Faraday equation is
abla imes mathbf{E} = -frac{partial mathbf{B}} {partial t}(in SI units) where
abla imes is the curl operator and again E(r, t) is the electric field and B(r, t) is the magnetic field. These fields can generally be functions of position r and time t.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:[21] oint_{partial Sigma} mathbf{E} cdot doldsymbol{ell} = - int_{Sigma} frac{partial mathbf{B}}{partial t} cdot dmathbf{A} where, as indicated in the figure:Σ is a surface bounded by the closed contour ∂Σ,E is the electric field, B is the magnetic field.dℓ is an infinitesimal vector element of the contour ∂Σ,dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.Both dℓ and dA 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 Σ, a positive path element dℓ of curve ∂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 n to the surface Σ.The integral around ∂Σ is called a path integral or line integral.

Kết quả (Anh) 2:

History [edit] A diagram of Faraday's iron ring apparatus. Change in the magnetic flux of the coil induces a current left in the right coil. [1] Faraday's disk (see homopolar generator) Electromagnetic induction Discovered by Michael Faraday was the first, who made ​​his discovery public printer 1831. [2] [3] It was Discovered independently by Joseph Henry printer 1832. [4] [5] In Faraday's first experimental demonstration (August 29, 1831), he wrapped around two opposite sides of security Wires iron ring or "torus" (an arrangement similar to a modern toroidal transformer). [6] [not print the citation given] Based on his assessment of Recently Discovered properties of electromagnets, he expected that, khi current to flow in one wire khởi, a sort of wave would travel through the ring and cause, some electrical effect on the opposite side. He plugged one wire Into a galvanometer, and watched it as he connected the wire to a battery other. Indeed, he saw a transient current (mà he gọi a "wave of Electricity") khi he connected the wire to the battery, and another khi he disconnected it. [7] This induction was Due to the magnetic flux change có lỗi khi print the battery was connected and disconnected. [1] Within two months, Faraday found Manifestations of electromagnetic induction vài other. For example, he saw transient currents khi he Quickly slid a bar magnet in and out of a coil of Wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk "). [8] Faraday's electromagnetic induction using a concept Explained he gọi lines of force. Tuy nhiên, scientists of at the time widely nhả his theoretical ideas, mainly vì They were not formulated mathematically. [9] An exception was James Clerk Maxwell, who used Faraday's ideas as the basis of his quantitative electromagnetic theory. [9] [10] [ 11] In Maxwell's model, the time varying electromagnetic induction is the aspect of Expressed as a Differential equation, Oliver Heaviside mà Faraday's law Referred to as though it is slightly thậm Faraday's original formulation the khác and does not describe motional EMF. Heaviside's version (see Maxwell-Faraday equation below) is the form recognized companies in the group today known as Maxwell's equations of equations. Lenz formulated the law named after him in 1834 to describe the print "flux through the circuit". Lenz's law Gives the direction of the induced EMF and current quả from electromagnetic induction (elaborated upon in the examples below). Following the understanding Brought by những laws, many kinds of devices employing magnetic induction được invented. Faraday's experiment showing induction giữa coils of wire: The liquid battery (right) provides a current flows through the small coil mà (A), tạo magnetic field. When the coils are stationary, no current is induced. But khi small print coil is moved out of the large or coil (B), the magnetic flux through the large coil changes, inducing a current detected by the galvanometer mà (G). [12] Theory [edit] Main article: Faraday's law of induction The law of physics describing the process of electromagnetic induction is known as Faraday's law of induction and the Most Widespread version of this law states rằng induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux enclosed by the circuit. [13] [14] Or mathematically, mathcal {E} = - {{d Phi_ mathrm {B}} over dt} , where mathcal {E} is the electromotive force (EMF ) is the magnetic flux and ΦB. The direction of the electromotive force is given by Lenz's law. This version of Faraday's law Strictly only when the closed circuit Holds is a loop of infinitely thin wire, [15] and some other Circumstances print is invalid. A Different Version, the Maxwell-Faraday equation (Discussed below), is valid print all Circumstances. For a tightly wound coil of wire, composed of N identical turns, each with magnetic flux going through the same added, the EMF is given by quả [16] [17] mathcal {E} = -N {{d Phi_ mathrm {B}} over dt} Faraday's law of induction use of the magnetic flux Makes ΦB through a hypothetical boundary surface Σ is a wire có loop. Since the wire loop lẽ moving, We write Σ (t) for the surface. The magnetic flux is defined by a surface integral: Phi_ mathrm {B} = iint limits _ { sigma (t)} mathbf {B} ( mathbf {r}, t) cdot d mathbf {A } , where dA is an element of the surface area of the moving surface Σ (t), B is the magnetic field, and B · dA is a vector dot product (the amount of magnetic flux infinitesimal). In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines có pass through the loop. When the flux changes-vì B changes, or vì wire loop is moved or deformed, or cả-Faraday's law of induction says rằng wire loop acquires an EMF, mathcal {E}, defined as the energy available from a unit charge has mà Travelled around the wire loop once. [15] [18] [19] [20] equivalently, it is the voltage would be measured by mà cutting the wire to create an open circuit, and Attaching a voltmeter to the leads. theo Lorentz force law (print SI units), mathbf {F} = q left ( mathbf {E} + mathbf {v} times mathbf {B} right) the EMF on a wire loop is: mathcal {E} = frac {1} {q} oint _ { mathrm {wire}} mathbf {F} cdot d boldsymbol { ell} = oint _ { mathrm {wire}} left ( mathbf {E} + mathbf {v} times mathbf {B} right) cdot d boldsymbol { ell} where E is the electric field, B is the magnetic field (magnetic flux density aka, magnetic induction) is an infinitesimal dℓ arc length along the wire, and the line integral along the wire is Evaluated (along the curve coincident with the shape of the wire). Maxwell-Faraday equation [edit] An illustration of Kelvin-Stokes theorem with its surface boundary Σ n ∂Σ and orientation set by the right-hand rule. The Maxwell-Faraday equation is a Faraday's law mà generalization of a time-varying states mà magnetic field is always accompanied by a spatially-varying, non-conservative electric field, and vice versa. The Maxwell-Faraday equation is Nabla times mathbf {E} = - frac { partial mathbf {B}} { partial t} (print SI units) where Nabla times is the curl operator and again E (r, t) is the electric field and B (r, t) is the magnetic field. These fields can be functions of position Generally r and time t. 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 cũng Be Written speaker in an integral form by the Kelvin-Stokes theorem: [21] oint _ { partial Sigma} mathbf {E} cdot d boldsymbol { ell} = - int _ { Sigma} frac { partial mathbf {B}} { partial t} cdot d mathbf {A} where, as indicated in the figure: Σ is a surface bounded by the closed contour ∂Σ, E is the electric field, B is the magnetic field. dℓ is an infinitesimal element of the contour ∂Σ vector, dA is an infinitesimal vector element of surface Σ. If its direction is orthogonal to mà surface patch, the magnitude is the area of an infinitesimal patch of surface. Both have a sign dA dℓ and ambiguity; to get the correct sign, the right-hand rule is used, as Explained in the article Kelvin-Stokes theorem. For a planar surface Σ, a positive path of curve ∂Σ dℓ element is defined by the right-hand rule as One That points with the fingers of the right hand thumb points khi in the direction of the normal n to the surface Σ. The integral around ∂Σ is gọi a path integral or line integral.