Faraday's law of induction


Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force —a phenomenon known as electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.
The Maxwell–Faraday equation describes the fact that a spatially varying electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is EMF on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.
Faraday's law had been discovered and one aspect of it was formulated as the Maxwell–Faraday equation later. The equation of Faraday's law can be derived by the Maxwell–Faraday equation and the Lorentz force. The integral form of the Maxwell–Faraday equation describes only the transformer EMF, while the equation of Faraday's law describes both the transformer EMF and the [|motional EMF].

History

Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Faraday was the first to publish the results of his experiments. In Faraday's first experimental demonstration of electromagnetic induction, he wrapped two wires around opposite sides of an iron ring . Based on his assessment of recently discovered properties of electromagnets, he expected that when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. He plugged one wire into a galvanometer, and watched it as he connected the other wire to a battery. Indeed, he saw a transient current when he connected the wire to the battery, and another when he disconnected it. This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. Within two months, Faraday had found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady current by rotating a copper disk near the bar magnet with a sliding electrical lead.
Michael Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically. An exception was James Clerk Maxwell, who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory. In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law even though it is different from the original version of Faraday's law, and does not describe motional EMF. Heaviside's version is the form recognized today in the group of equations known as Maxwell's equations.
Lenz's law, formulated by Emil Lenz in 1834, describes "flux through the circuit", and gives the direction of the induced EMF and current resulting from electromagnetic induction.

Faraday's law

The most widespread version of Faraday's law states:
The closed path here is, in fact, conductive.

Mathematical statement

For a loop of wire in a magnetic field, the magnetic flux is defined for any surface whose boundary is the given loop. Since the wire loop may be moving, we write for the surface. The magnetic flux is the surface integral:
where is an element of surface area of the moving surface, is the magnetic field, and is a vector dot product representing the element of flux through. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic field lines that pass through the loop.
When the flux changes—because changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF, defined as the energy available from a unit charge that has traveled once around the wire loop. Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.
Faraday's law states that the EMF is also given by the rate of change of the magnetic flux:
where is the electromotive force and is the magnetic flux.
The direction of the electromotive force is given by Lenz's law.
The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845.
Faraday's law contains the information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula.
It is possible to find out the direction of the electromotive force directly from Faraday’s law, without invoking Lenz's law. A left hand rule helps doing that, as follows:
For a tightly wound coil of wire, composed of identical turns, each with the same, Faraday's law of induction states that
where is the number of turns of wire and is the magnetic flux through a single loop.

Maxwell–Faraday equation

The Maxwell–Faraday equation states that a time-varying magnetic field always accompanies a spatially varying, non-conservative electric field, and vice versa. The Maxwell–Faraday equation is
where is the curl operator and again is the electric field and is the magnetic field. These fields can generally be functions of position and time.
The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It can also be written in an integral form by the Kelvin–Stokes theorem, thereby reproducing Faraday's law:
where, as indicated in the figure:
Both and 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, a positive path element of curve 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 to the surface.
The integral around is called a path integral or line integral.
Notice that a nonzero path integral for is different from the behavior of the electric field generated by charges. A charge-generated -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 through space, and any surface for which that path is a boundary.
If the surface is not changing in time, the equation can be rewritten:
The surface integral at the right-hand side is the explicit expression for the magnetic flux through.
The electric vector field induced by a changing magnetic flux, the solenoidal component of the overall electric field, can be approximated in the non-relativistic limit by the following volume integral equation:

Proof

The four Maxwell's equations, along with Lorentz force law, are a sufficient foundation to derive everything in classical electromagnetism. Therefore, it is possible to "prove" Faraday's law starting with these equations.
The starting point is the time-derivative of flux through an arbitrary surface in space:
. This total time derivative can be evaluated and simplified with the help of the Maxwell–Faraday equation and some vector identities; the details are in the box below:
Consider the time-derivative of magnetic flux through a closed boundary, then the time-derivative can be expressed as
The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore:
where is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF. The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation:
Next, we analyze the second term on the right-hand side:
The proof of this is a little more difficult than the first term; more details and alternate approaches for the proof can be found in the references. As the loop moves and/or deforms, it sweeps out a surface. As a small part of the loop moves with velocity over a short time, it sweeps out an area which vector is . Therefore, the change of the magnetic flux through the loop due to the deformation or movement of the loop over the time is
Here, identities of triple scalar products are used. Therefore,
where is the velocity of a part of the loop.
Putting these together results in,
The result is:
where is the boundary of the surface, and is the velocity of a part of the boundary.
In the case of a conductive loop, EMF is the electromagnetic work done on a unit charge when it has traveled around the loop once, and this work is done by the Lorentz force. Therefore, EMF is expressed as
where is EMF and is the unit charge velocity.
In a macroscopic view, for charges on a segment of the loop, consists of two components in average; one is the velocity of the charge along the segment, and the other is the velocity of the segment . does not contribute to the work done on the charge since the direction of is same to the direction of. Mathematically,
since is perpendicular to as and are along the same direction. Now we can see that, for the conductive loop, EMF is same to the time-derivative of the magnetic flux through the loop except for the sign on it. Therefore, we now reach the equation of Faraday's law as
where. With breaking this integral, is for the transformer EMF and is for the motional EMF.

EMF for non-thin-wire circuits

It is tempting to generalize Faraday's law to state: If ' is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through ' equals the EMF around . This statement, however, is not always true and the reason is not just from the obvious reason that EMF is undefined in empty space when no conductor is present. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve matches the actual velocity of the material conducting the electricity. The two examples illustrated below show that one often obtains incorrect results when the motion of is divorced from the motion of the material.
One can analyze examples like these by taking care that the path moves with the same velocity as the material. Alternatively, one can always correctly calculate the EMF by combining Lorentz force law with the Maxwell–Faraday equation:
where "it is very important to notice that is the velocity of the conductor... not the velocity of the path element and in general, the partial derivative with respect to time cannot be moved outside the integral since the area is a function of time."

Faraday's law and relativity

Two phenomena

Faraday's law is a single equation describing two different phenomena: the motional EMF generated by a magnetic force on a moving wire, and the transformer EMF generated by an electric force due to a changing magnetic field.
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. As Richard Feynman states:

Einstein's view

Reflection on this apparent dichotomy was one of the principal paths that led Albert Einstein to develop special relativity:
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