Faraday's law of induction is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators.^{[1]} The law states that:
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.^{[1]}
Or alternatively:
The EMF generated is proportional to the rate at which flux is linked.
Electromagnetic induction was discovered independently by Michael Faraday and Joseph Henry in 1831, however Faraday was the first to publish the results of his experiments. Expanding on Faraday’s work, James Clerk Maxwell formulated the law quantitatively in the form: ^{[2]}^{[3]}^{[4]}
where:
The direction of the electromotive force 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). In the case of motional EMF, the idea is to move all or part of the circuit through the magnetic field.
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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. James Clerk Maxwell drew attention to this fact in his 1861 paper On Physical Lines of Force. In the latter half of part II of that paper, Maxwell gives a separate physical explanation for each of the two phenomena. A reference to these two aspects of electromagnetic induction is made in some modern textbooks.^{[7]} As Richard Feynman states:^{[8]}
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
Reflection on this apparent dichotomy was one of the principal paths that led Einstein to develop special relativity:
It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.
– Albert Einstein, On the Electrodynamics of Moving Bodies^{[9]}
Faraday's law of induction makes use of the magnetic flux Φ_{B} through a surface Σ, defined by an integral over a surface:
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 surface is considered to have a "mouth" outlined by a closed curve denoted ∂Σ(t). 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:
where is the magnitude of the electromotive force (EMF) in volts and Φ_{B} is the magnetic flux in webers. The direction of the electromotive force is given by Lenz's law.
For a tightlywound coil of wire, composed of N identical loops, each with the same Φ_{B}, Faraday's law of induction states that
where N is the number of turns of wire and 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 tdependence 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.
Consider the case in Figure 3 of a closed rectangular loop of wire in the xyplane translated in the xdirection at velocity v. Thus, the center of the loop at x_{C} satisfies v = dx_{C} / dt. The loop has length ℓ in the ydirection and width w in the xdirection. A timeindependent but spatially varying magnetic field B(x) points in the zdirection. 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 by using either the Lorentz force law or equivalently by using Faraday's induction law above.
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 zdirections; 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 Bfield increases with increase in x, the force on the right side is largest, and the current will be clockwise: using the righthand rule, the Bfield generated by the current opposes the impressed field.^{[10]} 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
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 Bfield 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):
(where v = dx_{C} / dt is the rate of motion of the loop in the xdirection ) 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.
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 radial Bfield lies in the plane of the collecting loop, so the collecting loop contributes no flux to the circuit. The electromotive force is to be found directly and by using Faraday's law above.
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 Bfield 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.
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 timeindependent, 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 Bfield, and so contributes to the flux an amount:
where the sign is negative because the righthand rule suggests the Bfield generated by the current loop is opposite in direction to the applied B field. As this is the only timedependent 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 righthand rule would suggest the current loop added to the applied Bfield, 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.
The use of a closed path to find EMF as done above appears to depend upon details of the path geometry. In contrast, the Lorentzlaw 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 pathrelated analysis works, but a simplification is suggested. The timeindependent aspects of the circuit cannot affect the timerateofchange 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 Bfield swept out by the moving loop. This suggestion amounts to finding the rate at which flux is cut by the circuit.^{[11]} That notion provides direct evaluation of the rate of change of flux, without concern over the timeindependent 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 rateofchange 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:^{[12]}
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.
A changing magnetic field creates an electric field; this phenomenon is described by the MaxwellFaraday equation:^{[13]}
where:
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 KelvinStokes theorem:^{[14]}
where the movement of the derivative before the integration requires a timeindependent surface Σ (considered in this context to be part of the interpretation of the partial derivative), and as indicated in Figure 6:
Both dℓ and dA have a sign ambiguity; to get the correct sign, the righthand rule is used, as explained in the article KelvinStokes theorem. For a planar surface Σ, a positive path element dℓ of curve ∂Σ is defined by the righthand 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 righthand side of the MaxwellFaraday 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 chargegenerated Efield 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 timeindependent. 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.
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 MaxwellFaraday equation is (see 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.
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 crossproduct):
so the change in magnetic flux ΔΦ_{B} through the 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.
Revisiting the example of Figure 3 in a moving frame of reference brings out the close connection between E and Bfields, and between motional and induced EMF's.^{[15]} 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 Bfield varies with position x, the moving observer sees a timevarying magnetic field, namely:
where k is a unit vector pointing in the zdirection.^{[16]}
The MaxwellFaraday equation says the moving observer sees an electric field E_{y} in the ydirection given by (see Curl (mathematics)):
Here the chain rule is used:
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.
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 Bfield. 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 xderivative.
The stationary observer thought the EMF was a motional EMF, while the moving observer thought it was an induced EMF.^{[17]}
Faraday's law was originally an experimental law based upon observation.^{[18]}^{[19]} 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 versions of Maxwell's equations as expressed by Oliver Heaviside and Hermann Helmholtz.
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.^{[20]}^{[21]}
See Maxwell's original discussion of induced electromotive force.^{[22]}
Lenz's law, formulated by Baltic German physicist Heinrich Lenz in 1834, gives the direction of the induced electromotive force and current resulting from electromagnetic induction.
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, the disc is rotated in a uniform magnetic field perpendicular to the disc, causing a current to flow in the radial arm due to the Lorentz force. It is interesting to understand how it arises that mechanical work is necessary to drive this current. When the generated current flows through the conducting rim, a magnetic field is generated by this current through Ampere's circuital law (labeled "induced B" in Figure 8). The rim thus becomes an electromagnet that resists rotation of the disc (an example of Lenz's law). On the far side of the figure, the return current flows from the rotating arm through the far side of the rim to the bottom brush. The Bfield induced by this return current opposes the applied Bfield, tending to decrease the flux through that side of the circuit, opposing the increase in flux due to rotation. On the near side of the figure, the return current flows from the rotating arm through the near side of the rim to the bottom brush. The induced Bfield increases the flux on this side of the circuit, opposing the decrease in flux due to rotation. Thus, both sides of the circuit generate an emf opposing the rotation. 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 MaxwellFaraday 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.)^{[23]}
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.
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.
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 ε generated in the magnetic field B due to a conductive liquid moving at velocity v is thus given by:
where ℓ is the distance between electrodes in the magnetic flow meter.
All metal objects moving in relation to a static magnetic field will experience inductive power flow, as do all stationary metal objects in relation to a moving magnetic field. These power flows are occasionally undesirable, resulting in flowing electric current at very low voltage and heating of the metal.
There are a number of methods employed to control these undesirable inductive effects.
Eddy currents occur when a solid metallic mass is rotated in a magnetic field, because the outer portion of the metal cuts more lines of force than the inner portion, hence the induced electromotive force not being uniform, tends to set up currents between the points of greatest and least potential. Eddy currents consume a considerable amount of energy and often cause a harmful rise in temperature.^{[24]}
Only five laminations or plates are shown in this example, so as to show the subdivision of the eddy currents. In practical use, the number of laminations or punchings ranges from 40 to 66 per inch, and brings the eddy current loss down to about one percent. While the plates can be separated by insulation, the voltage is so low that the natural rust/oxide coating of the plates is enough to prevent current flow across the laminations.^{[25]}
This is a rotor approximately 20mm in diameter from a DC motor used in a CD player. Note the laminations of the electromagnet pole pieces, used to limit parasitic inductive losses.
In this illustration, a solid copper bar inductor on a rotating armature is just passing under the tip of the pole piece N of the field magnet. Note the uneven distribution of the lines of force across the bar inductor. The magnetic field is more concentrated and thus stronger on the left edge of the copper bar (a,b) while the field is weaker on the right edge (c,d). Since the two edges of the bar move with the same velocity, this difference in field strength across the bar creates whirls or current eddies within the copper bar.^{[26]}
This is one of the reasons why high voltage devices tend to be more efficient than low voltage devices. High voltage devices use many turns of smallgauge wire in motors, generators, and transformers. These many small turns of inductor wire in the electromagnet break up the eddy flows that can form within the large, thick inductors of low voltage, high current devices.


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