Faraday's law of induction

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: The law was discovered by Michael Faraday in 1831 and independently at the same time by Joseph Henry.

Quantitatively, the law takes the following form:


 * $$ \mathcal{E} = - {{d\Phi_B} \over dt}$$.

where


 * $$\mathcal{E}$$ is the electromotive force (EMF) in volts
 * &Phi;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.

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.



Terminology
The phenomenon of electromagnetic induction, connecting 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.

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:


 * $$\nabla\times \mathbf{E}(\mathbf{r},\ t) = -\frac{\partial \mathbf{B}(\mathbf{r},\ t)}{\partial t},$$

where
 * E and B are the electric and magnetic fields,
 * &nabla; × denotes curl
 * $$\begin{matrix} \frac{\part}{\part t}\end{matrix}$$&thinsp; 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 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.

Nonetheless, the Maxwell-Faraday equation often simply is called "Faraday's law". 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
Faraday's law of induction makes use of the magnetic flux ΦB through a surface Σ, defined by an integral over a surface:


 * $$ \Phi_B = \iint_{\Sigma (t)} \mathbf{B}(\mathbf{r},\ t) \cdot d \mathbf{A}\, $$

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. 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.

When the flux changes, Faraday's law of induction says that the work $$\stackrel{\mathcal{E}}{}$$ &thinsp; done (per unit charge) moving a test charge around the closed curve ∂Σ(t), called the electromotive force (EMF), is given by:


 * $$ \mathcal{E} = - {{d\Phi_B} \over dt}\ ,$$

where:


 * $$\mathcal{E}$$ is the electromotive force (emf) in volts
 * &Phi;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 ΦB, Faraday's law of induction states that


 * $$ \mathcal{E} = - N{{d\Phi_B} \over dt}$$

where:


 * N is the number of turns of wire
 * &Phi;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
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 xC satisfies v = dxC / 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( xC − w / 2), and on the right side is B( xC + 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(xC − w / 2) j &thinsp; ( 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(xC − 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(xC + 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. 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


 * $$ \mathcal{E} = v\ell [ B(x_C+w/2) - B(x_C-w/2)] \ . $$

Faraday's law method
At any position of the loop the magnetic flux through the loop is
 * $$\Phi_B = \pm \int_0^{\ell} dy \int_{x_C-w/2}^{x_C+w/2} B(x) dx $$
 * $$= \pm \ell \int_{x_C-w/2}^{x_C+w/2} B(x) dx \ .$$

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):


 * $$\frac {d \Phi_B} {dt} = (-) \frac {d}{dx_C} \left[ \int_0^{\ell}dy \ \int_{x_C-w/2}^{x_C+w/2} dx B(x)\right] \frac {dx_C}{dt} \, $$
 * $$ = (-)  v\ell  [ B(x_C+w/2) - B(x_C-w/2)] \, $$

(where v = dxC / dt is the rate of motion of the loop in the x-direction ) leading to:
 * $$ \mathcal{E} = -\frac {d\Phi_B} {dt} = v\ell [ B(x_C+w/2) - B(x_C-w/2)] \, $$

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 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


 * $$F = q\ B \ v\ . $$

Consequently the EMF is


 * $$ \mathcal {E} = B v \ell \ = B r \ell \omega \, $$

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
An intuitively appealing but mistaken approach to using the flux rule would say the flux through the circuit was just ΦB = 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.

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:


 * $$ \Phi_B = -B r \theta \ell \, $$

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


 * $$ \mathcal{E} = -\frac {d \Phi_B} {dt} = B r \ell \frac {d \theta} {dt} $$
 * $$ = B r \ell \omega \, $$

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
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. 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.

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:


 * $$ \mathcal{E} = - {{d\Phi_B} \over dt} = B v \ell \ .$$

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
A changing magnetic field creates an electric field; this phenomenon is described by the Maxwell-Faraday equation:


 * $$\nabla \times \mathbf{E}( \mathbf{r},\ t) = -\frac{\partial \mathbf{B}( \mathbf{r},\ t)} {\partial t}$$

where:
 * $$\nabla\times$$ 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.

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


 * $$ \oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = - \   \iint_{\Sigma}  { \partial \over {\partial t} } \mathbf{B} \cdot d\mathbf{A}  $$


 * $$=- \ { \partial \over {\partial t} }  \iint_{\Sigma}   \mathbf{B} \cdot d\mathbf{A}  $$

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.

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 ΦB 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.

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 $$\overset{ \mathcal{E}}{} $$,&thinsp; and indeed cannot do so without introduction of the Lorentz force law to enable a calculation of work.

Using the complete Lorentz force to calculate the EMF,


 * $$\mathcal{E} = \oint_{\partial \Sigma (t)}\left( \mathbf{E}( \mathbf{r},\ t) +\mathbf{ v \times B}(\mathbf{r},\ t)\right) \cdot d\boldsymbol{\ell}\ ,$$

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


 * $$ \oint_{\partial \Sigma (t)}\left( \mathbf{E}( \mathbf{r},\ t) +\mathbf{ v \times B}(\mathbf{r},\ t)\right) \cdot d\boldsymbol{\ell}\ $$  $$ \ =-\frac {d} {dt}  \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf {B}(\mathbf{r},\ t) \, $$

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.

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):


 * $$ d\mathbf{A} = d \boldsymbol{\ell \times v } dt \, $$

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


 * $$\frac {d \Delta \Phi_B} {dt} = \mathbf{B} \cdot \ d \boldsymbol{\ell \times v } \ = \mathbf{v} \times \mathbf{B} \cdot \ d \boldsymbol{\ell} \, $$

and if we add these ΔΦB-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
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. 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:


 * $$ \mathbf{B} = \mathbf{k}{B}(x+vt) \ ,$$

where  k  is a unit vector pointing in the z-direction.

Lorentz force law version
The Maxwell-Faraday equation says the moving observer sees an electric field Ey in the y-direction given by (see Curl (mathematics)):


 * $$ \nabla \times \mathbf{E} = \mathbf{k}\ \frac {dE_y}{dx} $$


 * $$=- \frac { \partial \mathbf{B}}{\partial t}=-\mathbf{k}\frac {d B(x+vt)} {dt} = -\mathbf{k}\frac {dB}{dx} v \ \, $$

Here the chain rule is used:


 * $$ \frac {dB}{dt} = \frac {dB}{d(x+vt)} \frac {d(x+vt)}{dt} =\frac {dB} {dx} v \ . $$

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


 * $$ E_y (x,\ t) = -B(x+vt) \ v \ .$$

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:


 * $$ \mathcal{E} = -\ell [ E_y (x_C+w/2,\ t) - E_y(x_C-w/2,\ t)] $$
 * $$ = v\ell  [ B(x_C+w/2+v t) - B(x_C-w/2+vt)] \, $$

which is exactly the same result found by the stationary observer, who sees the centroid xC has advanced to a position xC + 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 xC sees a changing magnetic flux, but the loop does not appear to move: the center of the loop xC is fixed because the moving observer is moving with the loop. The flux is then:


 * $$\Phi_B =-\int_0^{\ell} dy \int_{x_C-w/2}^{x_C+w/2} B(x+vt) dx \ ,$$

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:


 * $$ \mathcal{E} = -\frac {d\Phi_B} {dt} = \int_0^{\ell} dy \int_{x_C-w/2}^{x_C+w/2} \frac{d}{dt}B(x+vt) dx$$
 * $$ = \int_0^{\ell} dy \int_{x_C-w/2}^{x_C+w/2} \frac{d}{dx}B(x+vt)\ v\  dx$$


 * $$=v\ell \ [ B(x_C+w/2+vt) - B(x_C-w/2+vt)] \, $$

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.

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:

A similar statement is made in Griffiths.

History
Faraday's law was originally an experimental law based upon observation. 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.

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.

See Maxwell's original discussion of induced electromotive force.

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.

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.

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 Lenz's law). 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.

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 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 .)

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 ΦB / dt equal to the voltage driving the current.

Electrical 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 ΦB / d t. Therefore, an electromotive force is set up in the second loop called the induced EMF or transformer EMF. If the two ends of this loop are connected through an electrical load, current will flow.

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 $$\stackrel{ \mathcal{E}}{}$$ generated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by:


 * $$\mathcal{E}= B \ell v$$,

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