Electromagnetism

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Faraday's law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. The law states:^[1] This box:      Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Image File history File links Solenoid. ... Electricity (from New Latin Ä“lectricus, amberlike) is a general term for a variety of phenomena resulting from the presence and flow of electric charge. ... For other senses of this word, see magnetism (disambiguation). ... Electrostatics (also known as static electricity) is the branch of physics that deals with the phenomena arising from what seem to be stationary electric charges. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated as follows: This is analogous to Newtons third law of motion in mechanics. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... In physics, Gausss law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface. ... This box:      At a point in space, the electric potential is the potential energy per unit of charge that is associated with a static (time-invariant) electric field. ... Electrostatic induction is a method by which an electrically charged object can be used to create an electrical charge in a second object, without contact between the two objects. ... This article is about the electromagnetic phenomenon. ... Magnetostatics is the study of static magnetic fields. ... This box:      Electric current is the flow (movement) of electric charge. ... For the indie-pop band, see The Magnetic Fields. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... The Biot-Savart law is a physical law with applications in both electromagnetics and fluid dynamics. ... A bar magnet. ... Classical electrodynamics (or classical electromagnetism) is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... Lorentz force. ... Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... For magnetic induction, see Magnetic field. ... Displacement current is a quantity related to changing electric field. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ... This box:      The Liénard-Wiechert potential describes the electromagnetic effect of a moving charge. ... In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. ... As the circular plate moves down through a small region of constant magnetic field directed into the page, eddy currents are induced in the plate. ... A simple electric circuit made up of a voltage source and a resistor. ... Conduction is the movement of electrically charged particles through a transmission medium (electrical conductor). ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ... An electric current i flowing around a circuit produces a magnetic field and hence a magnetic flux Φ through the circuit. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... A resonator is a device or part that vibrates (or oscillates) with waves. ... This box:      This page is about waveguides for electromagnetic wave propagation at microwave and radio wave frequencies. ... In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... This box:      The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field of a physical system in Maxwells theory of electromagnetism. ... In physics, the electromagnetic stress-energy tensor is the portion of the stress-energy tensor due to the electromagnetic field. ... In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ... The electromagnetic four-potential is a four-vector defined in SI units (and gaussian units in parentheses) as in which φ is the electrical potential, and is the magnetic potential, a vector potential. ... André-Marie Ampère (January 20, 1775 – June 10, 1836), was a French physicist who is generally credited as one of the main discoverers of electromagnetism. ... Charles Augustin de Coulomb (born June 14, 1736, Angoulême, France - died August 23, 1806, Paris, France) was a French physicist. ... Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and... Joseph Henry Joseph Henry (December 17, 1797 – May 13, 1878) was a Scottish-American scientist who served as the first Secretary of the Smithsonian Institution. ... Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894) was the German physicist and mechanician for whom the hertz, an SI unit, is named. ... Hendrik Lorentz by Jan Veth Hendrik Antoon Lorentz (born July 18, 1853 in Arnhem, Netherlands; died February 4, 1928 in Haarlem, Netherlands) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... Wilhelm Eduard Weber (October 24, 1804 - June 23, 1891) was a noted physicist. ... For other uses, see Transformer (disambiguation). ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... This article is about machines that produce electricity. ...

The induced electromotive force or EMF in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit. Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ...

The law was discovered by Michael Faraday in 1831 and independently at the same time by Joseph Henry. Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... Joseph Henry Joseph Henry (December 17, 1797 – May 13, 1878) was a Scottish-American scientist who served as the first Secretary of the Smithsonian Institution. ...

Quantitatively, the law takes the following form:^[2]

.

where

is the electromotive force (EMF) in volts

Φ_B is the magnetic flux through the circuit (in webers).

The direction of the electromotive force (the negative sign in the above formula) is given by Lenz's law. The meaning of "flux through the circuit" is elaborated upon in the examples below. Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... Josephson junction array chip developed by NIST as a standard volt. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... This article is in need of attention. ... Lenzs law (pronounced (IPA) ) gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction. ...

Traditionally, two different ways of changing the flux through a circuit are recognized. In the case of transformer EMF the idea is to alter the field itself, for example by changing the current originating the field (as in a transformer), or by sweeping a magnet past a loop of wire. In the case of motional EMF, the idea is to move all or part of the circuit through the magnetic field, for example, as in a homopolar generator.

A homopolar generator, also known as a unipolar generator, acyclic generator, or disk dynamo, is an electrical generator in which the magnetic field has the same polarity at every point, so that the armature passes through the magnetic field lines of force continually in the same direction. ...

Terminology

The phenomenon of electromagnetic induction, conneting the electromotive force with relation to the magnetic flux through the circuit, should not be confused with the electrostatic induction method for creating an electrical charge in an object with another electrically charged object. Electrostatic induction is a method by which an electrically charged object can be used to create an electrical charge in a second object, without contact between the two objects. ...

Maxwell-Faraday equation

In 1855, a curl version of "Faraday's law" was developed by James Clerk Maxwell and in 1884, Oliver Heaviside rewrote it to the following curl version: James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and...

where

E and B are the electric and magnetic fields,

∇ × denotes curl

denotes the partial time derivative holding r fixed.

This equation, called in this article the Maxwell-Faraday equation, is best known as being one of the four Maxwell's equations. In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... For the indie-pop band, see The Magnetic Fields. ... For other uses, see Curl (disambiguation). ... For thermodynamic relations, see Maxwell relations. ...

In the Maxwell-Faraday equation, Heaviside used the partial time derivative. Use of the partial time derivative, instead of the total time derivative that had been used by Maxwell, means that the Maxwell-Faraday equation does not account for motional EMF.^[3]

Nonetheless, the Maxwell-Faraday equation often simply is called "Faraday's law".^[4] In this article, however, the term "Faraday's law" refers to the flux equation and "Maxwell-Faraday equation" refers to the curl equation of Heaviside that today is one of Maxwell's equations.

Flux through a surface and EMF around a loop

Figure 1: The definition of surface integral relies on splitting the surface into small surface elements. Each element is associated with a vector dA of magnitude equal to the area of the element and with direction normal to the element and pointing outward.

Figure 2: A vector field F ( r, t ) defined throughout space, and a surface Σ bounded by curve ∂Σ moving with velocity v over which the field is integrated.

Faraday's law of induction makes use of the magnetic flux Φ_B through a surface Σ, defined by an integral over a surface: Image File history File links Size of this preview: 800 × 550 pixelsFull resolution (1164 × 800 pixel, file size: 79 KB, MIME type: image/png) % An illustration of the surface integral. ... Image File history File links Size of this preview: 800 × 550 pixelsFull resolution (1164 × 800 pixel, file size: 79 KB, MIME type: image/png) % An illustration of the surface integral. ...

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. See Figure 1. For more detail, refer to surface integral and magnetic flux. The surface is considered to have a "mouth" outlined by a closed curve denoted ∂Σ(t). See Figure 2. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ...

When the flux changes, Faraday's law of induction says that the work done (per unit charge) moving a test charge around the closed curve ∂Σ(t), called the electromotive force (EMF), is given by: Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ...

where:

is the electromotive force (emf) in volts

Φ_B is the magnetic flux in webers. The direction of the electromotive force (the negative sign in the above formula) is given by Lenz's law.

For a tightly-wound coil of wire, composed of N identical loops, each with the same Φ_B, Faraday's law of induction states that Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... Josephson junction array chip developed by NIST as a standard volt. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... This article is in need of attention. ... Lenzs law (pronounced (IPA) ) gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ...

where:

N is the number of turns of wire

Φ_B is the magnetic flux in webers through a single loop.

In choosing a path ∂Σ(t) to find EMF, the path must satisfy the basic requirements that (i) it is a closed path, and (ii) the path must capture the relative motion of the parts of the circuit (the origin of the t-dependence in ∂Σ(t) ). It is not a requirement that the path follow a line of current flow, but of course the EMF that is found using the flux law will be the EMF around the chosen path. If a current path is not followed, the EMF might not be the EMF driving the current.

Example: Spatially varying B-field

Figure 3: Closed rectangular wire loop moving along x-axis at velocity v in magnetic field B that varies with position x.

Consider the case in Figure 3 of a closed rectangular loop of wire in the xy-plane translated in the x-direction at velocity v. Thus, the center of the loop at x_C satisfies v = dx_C / dt. The loop has length ℓ in the y-direction and width w in the x-direction. A time-independent but spatially varying magnetic field B(x) points in the z-direction. The magnetic field on the left side is B( x_C − w / 2), and on the right side is B( x_C + w / 2). The electromotive force is to be found directly and by using Faraday's law above.

Lorentz force law method

A charge q in the wire on the left side of the loop experiences a Lorentz force q v × B k = −q v B(x_C − w / 2) j ( j, k unit vectors in the y- and z-directions; see vector cross product), leading to an EMF (work per unit charge) of v ℓ B(x_C − w / 2) along the length of the left side of the loop. On the right side of the loop the same argument shows the EMF to be v ℓ B(x_C + w / 2). The two EMF's oppose each other, both pushing positive charge toward the bottom of the loop. In the case where the B-field increases with position x, the force on the right side is largest, and the current will be clockwise: using the right-hand rule, the B-field generated by the current opposes the impressed field.^[5] The EMF driving the current must increase as we move counterclockwise (opposite to the current). Adding the EMF's in a counterclockwise tour of the loop we find In physics, the Lorentz force is the force exerted on a charged particle in an electromagnetic field. ... For the cross product in algebraic topology, see Künneth theorem. ... The left-handed orientation is shown on the left, and the right-handed on the right. ...

Faraday's law method

At any position of the loop the magnetic flux through the loop is

The sign choice is decided by whether the normal to the surface points in the same direction as B, or in the opposite direction. If we take the normal to the surface as pointing in the same direction as the B-field of the induced current, this sign is negative. The time derivative of the flux is then (using the chain rule of differentiation or the general form of Leibniz rule for differentiation of an integral): In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, Leibnizs rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form then for the derivative of this integral is thus expressible provided that and are both continuous over a region in the form // A more...

(where v = dx_C / dt is the rate of motion of the loop in the x-direction ) leading to:

as before.

The equivalence of these two approaches is general and, depending on the example, one or the other method may prove more practical.

Example: Moving loop in uniform B-field

Figure 4: Rectangular wire loop rotating at angular velocity ω in radially outward pointing magnetic field B of fixed magnitude. Current is collected by brushes attached to top and bottom discs, which have conducting rims.

Figure 4 shows a spindle formed of two discs with conducting rims and a conducting loop attached vertically between these rims. The entire assembly spins in a magnetic field that points radially outward, but is the same magnitude regardless of its direction. A radially oriented collecting return loop picks up current from the conducting rims. At the location of the collecting return loop, the B-field is in the plane of the collecting loop, so it contributes no flux to the circuit. The electromotive force is to be found directly and by using Faraday's law above.

Lorentz force law method

In this case the Lorentz force drives the current in the two vertical arms of the moving loop downward, so current flows from the top disc to the bottom disc. In the conducting rims of the discs, the Lorentz force is perpendicular to the rim, so no EMF is generated in the rims, nor in the horizontal portions of the moving loop. Current is transmitted from the bottom rim to the top rim through the external return loop, which is oriented so the B-field is in its plane. Thus, the Lorentz force in the return loop is perpendicular to the loop, and no EMF is generated in this return loop. Traversing the current path in the direction opposite to the current flow, work is done against the Lorentz force only in the vertical arms of the moving loop, where

Consequently the EMF is

where ℓ is the vertical length of the loop, and the velocity is related to the angular rate of rotation by v = r ω, with r = radius of cylinder. Notice that the same work is done on any path that rotates with the loop and connects the upper and lower rim.

Faraday's law method

See also: Faraday paradox

An intuitively appealing but mistaken approach to using the flux rule would say the flux through the circuit was just Φ_B = B w ℓ, where w = width of the moving loop. This number is time-independent, so the approach predicts incorrectly that no EMF is generated. The flaw in this argument is that it fails to consider the entire current path, which is a closed loop. In 1832 famous scientist Michael Faraday performed some very interesting experiments with magnets and conducting disks. ...

To use the flux rule, we have to look at the entire current path, which includes the path through the rims in the top and bottom discs. We can choose an arbitrary closed path through the rims and the rotating loop, and the flux law will find the EMF around the chosen path. Any path that has a segment attached to the rotating loop captures the relative motion of the parts of the circuit.

As an example path, let's traverse the circuit in the direction of rotation in the top disc, and in the direction opposite to the direction of rotation in the bottom disc (shown by arrows in Figure 4). In this case, for the moving loop at an angle θ from the collecting loop, a portion of the cylinder of area A = r ℓ θ is part of the circuit. This area is perpendicular to the B-field, and so contributes to the flux an amount:

where the sign is negative because the right-hand rule suggests the B-field generated by the current loop is opposite in direction to the applied B field. As this is the only time-dependent portion of the flux, the flux law predicts an EMF of

in agreement with the Lorentz force law calculation.

Now let's try a different path. Follow a path traversing the rims via the opposite choice of segments. Then the coupled flux would decrease as θ increased, but the right-hand rule would suggest the current loop added to the applied B-field, so the EMF around this path is the same as for the first path. Any mixture of return paths leads to the same result for EMF, so it is actually immaterial which path is followed.

Direct evaluation of the change in flux

Figure 5: A simplified version of Figure 4. The loop slides with velocity v in a stationary, homogeneous B-field.

The use of a closed path to find EMF as done above appears to depend upon details of the path geometry. In contrast, the Lorentz-law approach is independent of such restrictions. A discussion follows intended to understand better the equivalence of paths and escape the particulars of path selection when using the flux law.

Figure 5 is an idealization of Figure 4 with the cylinder unwrapped onto a plane. The same path-related analysis works, but a simplification is suggested. The time-independent aspects of the circuit cannot affect the time-rate-of-change of flux. For example, at a constant velocity of sliding the loop, the details of current flow through the loop are not time dependent. Instead of concern over details of the closed loop selected to find the EMF, one can focus on the area of B-field swept out by the moving loop. This suggestion amounts to finding the rate at which flux is cut by the circuit.^[6] That notion provides direct evaluation of the rate of change of flux, without concern over the time-independent details of various path choices around the circuit. Just as with the Lorentz law approach, it is clear that any two paths attached to the sliding loop, but differing in how they cross the loop, produce the same rate-of-change of flux. This box:      Faradays law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. ...

In Figure 5 the area swept out in unit time is simply dA / dt = v ℓ, regardless of the details of the selected closed path, so Faraday's law of induction provides the EMF as:^[7]

This path independence of EMF shows that if the sliding loop is replaced by a solid conducting plate, or even some complex warped surface, the analysis is the same: find the flux in the area swept out by the moving portion of the circuit. In a similar way, if the sliding loop in the drum generator of Figure 4 is replaced by a 360° solid conducting cylinder, the swept area calculation is exactly the same as for the case with only a loop. That is, the EMF predicted by Faraday's law is exactly the same for the case with a cylinder with solid conducting walls or, for that matter, a cylinder with a cheese grater for walls. Notice, though, that the current that flows as a result of this EMF will not be the same because the resistance of the circuit determines the current.

The Maxwell-Faraday equation

Figure 6: An illustration of Kelvin-Stokes theorem with surface Σ its boundary ∂Σ and orientation n set by the right-hand rule.

A changing magnetic field creates an electric field; this phenomenon is described by the Maxwell-Faraday equation:^[8] The left-handed orientation is shown on the left, and the right-handed on the right. ...

where:

denotes curl

E is the electric field

B is the magnetic field

This equation appears in modern sets of Maxwell's equations and is often referred to as Faraday's law. However, because it contains only partial time derivatives, its application is restricted to situations where the test charge is stationary in a time varying magnetic field. It does not account for electromagnetic induction in situations where a charged particle is moving in a magnetic field. For other uses, see Curl (disambiguation). ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... For the indie-pop band, see The Magnetic Fields. ... For thermodynamic relations, see Maxwell relations. ...

It also can be written in an integral form by the Kelvin-Stokes theorem:^[9]

where the movement of the derivative before the integration requires a time-independent surface Σ (considered in this context to be part of the interpretation of the partial derivative), and as indicated in Figure 6:

Σ is a surface bounded by the closed contour ∂Σ; both Σ and ∂Σ are fixed, independent of time

E is the electric field,

dℓ is an infinitesimal vector element of the contour ∂Σ,

B is the magnetic field.

dA is an infinitesimal vector element of surface Σ , whose magnitude is the area of an infinitesimal patch of surface, and whose direction is orthogonal to that surface patch.

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. Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... For the indie-pop band, see The Magnetic Fields. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... The left-handed orientation is shown on the left, and the right-handed on the right. ...

The integral around ∂Σ is called a path integral or line integral. The surface integral at the right-hand side of the Maxwell-Faraday equation is the explicit expression for the magnetic flux Φ_B through Σ. Notice that a nonzero path integral for E is different from the behavior of the electric field generated by charges. A charge-generated E-field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral. See gradient theorem. This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ... Rombu is the hawt. ...

The integral equation is true for any path ∂Σ through space, and any surface Σ for which that path is a boundary. Note, however, that ∂Σ and Σ are understood not to vary in time in this formula. This integral form cannot treat motional EMF because Σ is time-independent. Notice as well that this equation makes no reference to EMF , and indeed cannot do so without introduction of the Lorentz force law to enable a calculation of work.

Figure 7: Area swept out by vector element dℓ of curve ∂Σ in time dt when moving with velocity v.

Using the complete Lorentz force to calculate the EMF,

a statement of Faraday's law of induction more general than the integral form of the Maxwell-Faraday equation is (see Lorentz force): Lorentz force. ...

where ∂Σ(t) is the moving closed path bounding the moving surface Σ(t), and v is the velocity of movement. See Figure 2. Notice that the ordinary time derivative is used, not a partial time derivative, implying the time variation of Σ(t) must be included in the differentiation. In the integrand the element of the curve dℓ moves with velocity v. This box:      Faradays law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. ...

Figure 7 provides an interpretation of the magnetic force contribution to the EMF on the left side of the above equation. The area swept out by segment dℓ of curve ∂Σ in time dt when moving with velocity v is (see geometric meaning of cross-product): For the cross product in algebraic topology, see Künneth theorem. ...

so the change in magnetic flux ΔΦ_B through the this portion of the surface enclosed by ∂Σ in time dt is:

and if we add these ΔΦ_B-contributions around the loop for all segments dℓ, we obtain the magnetic force contribution to Faraday's law. That is, this term is related to motional EMF.

Example: viewpoint of a moving observer

See also: Moving magnet and conductor problem

Revisiting the example of Figure 3 in a moving frame of reference brings out the close connection between E- and B-fields, and between motional and induced EMF's.^[10] Imagine an observer of the loop moving with the loop. The observer calculates the EMF around the loop using both the Lorentz force law and Faraday's law of induction. Because this observer moves with the loop, the observer sees no movement of the loop, and zero v × B. However, because the B-field varies with position x, the moving observer sees a time-varying magnetic field, namely: Conductor moving in a magnetic field. ... This box:      Faradays law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. ...

where k is a unit vector pointing in the z-direction.^[11]

Lorentz force law version

The Maxwell-Faraday equation says the moving observer sees an electric field E_y in the y-direction given by (see curl): For other uses, see Curl (disambiguation). ...

Here the chain rule is used: In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

Solving for E_y, to within a constant that contributes nothing to an integral around the loop,

Using the Lorentz force law, which has only an electric field component, the observer finds the EMF around the loop at a time t to be:

which is exactly the same result found by the stationary observer, who sees the centroid x_C has advanced to a position x_C + v t. However, the moving observer obtained the result under the impression that the Lorentz force had only an electric component, while the stationary observer thought the force had only a magnetic component.

Faraday's law of induction

Using Faraday's law of induction, the observer moving with x_C sees a changing magnetic flux, but the loop does not appear to move: the center of the loop x_C is fixed because the moving observer is moving with the loop. The flux is then:

where the minus sign comes from the normal to the surface pointing oppositely to the applied B-field. The EMF from Faraday's law of induction is now:

the same result. The time derivative passes through the integration because the limits of integration have no time dependence. Again, the chain rule was used to convert the time derivative to an x-derivative.

The stationary observer thought the EMF was a motional EMF, while the moving observer thought it was an induced EMF.^[12]

Faraday's law as two different phenomena

Some physicists have remarked that 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. As Richard Feynman states:^[13]

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 – for "circuit moves" and 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

A similar statement is made in Griffiths.^[14]

History

Faraday's law was originally an experimental law based upon observation.^[15][16] Later it was formalized, and along with the other laws of electromagnetism a partial time derivative restricted version of it was incorporated into the modern Heaviside versions of Maxwell's equations. This box:      Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... For thermodynamic relations, see Maxwell relations. ...

Faraday's law of induction is based on Michael Faraday's experiments in 1831. The effect was also discovered by Joseph Henry at about the same time, but Faraday published first.^[17][18] Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... Leopold I 1831 (MDCCCXXXI) was a common year starting on Saturday (see link for calendar). ... Joseph Henry Joseph Henry (December 17, 1797 – May 13, 1878) was a Scottish-American scientist who served as the first Secretary of the Smithsonian Institution. ...

See Maxwell's original discussion of induced electromotive force.^[19]

Lenz's law, formulated by Estonian physicist Heinrich Lenz in 1834, gives the direction of the induced electromotive force and current resulting from electromagnetic induction. Lenzs law (pronounced (IPA) ) gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction. ... Heinrich Friedrich Emil Lenz (February 12, 1804 - February 10, 1865) was a Baltic German physicist most famous for formulating Lenzs law in 1833. ...

Electrical generator

Figure 8: Faraday's disc electric generator. The disc rotates with angular rate ω, sweeping the conducting radius circularly in the static magnetic field B. The magnetic Lorentz force v × B drives the 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. Thus, current is generated from mechanical motion.

Main article: electrical generator

The EMF generated by Faraday's law of induction due to relative movement of a circuit and a magnetic field is the phenomenon underlying electrical generators. When a permanent magnet is moved relative to a conductor, or vice versa, an electromotive force is created. If the wire is connected through an electrical load, current will flow, and thus electrical energy is generated, converting the mechanical energy of motion to electrical energy. For example, the drum generator is based upon Figure 4. A different implementation of this idea is the Faraday's disc, shown in simplified form in Figure 8. Note that either the analysis of Figure 5, or direct application of the Lorentz force law, shows that a solid conducting disc works the same way. This article is about machines that produce electricity. ... This article is about machines that produce electricity. ... For other uses, see Magnet (disambiguation). ... If an electric circuit has a well-defined output terminal, the circuit connected to this terminal (or its input impedance) is the load. ... Electrical energy can refer to several closely related things. ... This box:      Faradays law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. ... A homopolar generator, also known as a unipolar generator, acyclic generator, or disk dynamo, is an electrical generator in which the magnetic field has the same polarity at every point, so that the armature passes through the magnetic field lines of force continually in the same direction. ... This box:      Faradays law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. ...

In the Faraday's disc example, when the generated current flows through the wire loop, a magnetic field is generated through Ampere's circuital law. The electromagnet thus created resists rotation of the disc (an example of Le Chatelier's principle). The energy required to keep the disc moving, despite this reactive force, is exactly equal to the electrical energy generated (plus energy wasted due to friction, Joule heating, and other inefficiencies). This behavior is common to all generators converting mechanical energy to electrical energy. An electromagnet is a type of magnet in which the magnetic field is produced by the flow of an electric current. ... In chemistry, Le Chateliers principle, also called the Le Chatelier-Braun principle, can be used to predict the effect of a change in conditions on a chemical equilibrium. ... For other uses, see Friction (disambiguation). ... In physics, mechanical energy describes the potential energy and kinetic energy present in the components of a mechanical system. ...

Although Faraday's law always describes the working of electrical generators, the detailed mechanism can differ in different cases. When the magnet is rotated around a stationary conductor, the changing magnetic field creates an electric field, as described by the the Maxwell-Faraday equation, and that electric field pushes the charges through the wire. This case is called an induced EMF. On the other hand, when the magnet is stationary and the conductor is rotated, the moving charges experience a magnetic force (as described by the Lorentz force law), and this magnetic force pushes the charges through the wire. This case is called motional EMF. (For more information on motional EMF, induced EMF, Faraday's law, and the Lorentz force, see above example, and see Griffiths^[20].) This box:      Faradays law of induction describes an important basic law of electromagnetism, which is involved in the working of transformers, inductors, and many forms of electrical generators. ...

Electrical motor

Main article: electrical motor

An electrical generator can be run "backwards" to become a motor. For example, with the Faraday disc, suppose a DC current is driven through the conducting radial arm by a voltage. Then by the Lorentz force law, this traveling charge experiences a force in the magnetic field B that will turn the disc in a direction given by Fleming's left hand rule. In the absence of irreversible effects, like friction or Joule heating, the disc turns at the rate necessary to make d Φ_B / dt equal to the voltage driving the current. Electric motors of various sizes. ... Flemings left hand rule Alternate representation of Flemings LHR Flemings left hand rule (for electric motors) shows the direction of the thrust on a conductor carrying a current in a magnetic field. ...

Electrical transformer

Main article: transformer

The EMF predicted by Faraday's law is also responsible for electrical transformers. When the electric current in a loop of wire changes, the changing current creates a changing magnetic field. A second wire in reach of this magnetic field will experience this change in magnetic field as a change in its coupled magnetic flux, a d Φ_B / d t. Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF. If the two ends of this loop are connected through an electrical load, current will flow. For other uses, see Transformer (disambiguation). ...

Magnetic flow meter

Main article: magnetic flow meter

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

,

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

See also

Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... For the indie-pop band, see The Magnetic Fields. ... Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... An electric current produces a magnetic field. ... Lenzs law (pronounced (IPA) ) gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction. ... Lorentz force. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... Conductor moving in a magnetic field. ... An electric current i flowing around a circuit produces a magnetic field and hence a magnetic flux Φ through the circuit. ... In telecommunication, the term crosstalk (XT) has the following meanings: 1. ... A homopolar generator, also known as a unipolar generator, acyclic generator, or disk dynamo, is an electrical generator in which the magnetic field has the same polarity at every point, so that the armature passes through the magnetic field lines of force continually in the same direction. ... In 1832 famous scientist Michael Faraday performed some very interesting experiments with magnets and conducting disks. ...

References

In chemistry, Le Chateliers principle, also called the Le Chatelier-Braun principle, can be used to predict the effect of a change in conditions on a chemical equilibrium. ... Lenzs law (pronounced (IPA) ) gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction. ... In physics, the Lorentz transformation converts between two different observers measurements of space and time, where one observer is in constant motion with respect to the other. ... Conductor moving in a magnetic field. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 334th day of the year (335th in leap years) in the Gregorian calendar. ...

Further reading

A clear discussion of the different uses of the term Faraday's law is found in Tankersley and Mosca: Introducing Faraday's law

External links

• A simple interactive Java tutorial on electromagnetic induction National High Magnetic Field Laboratory
• R. Vega Induction: Faraday's law and Lenz's law - Highly animated lecture