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
called electromagnetic induction. It is the
fundamental operating principle of transformers,
inductors, and many types of electrical motors,
generators and solenoids.
The Maxwell–Faraday equation is a generalisation
of Faraday's law, and forms one of Maxwell's
equations.
History
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. In Faraday's
first experimental demonstration of electromagnetic
induction, he wrapped two wires around opposite
sides of an iron ring or "torus". 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
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 Heinrich 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
Qualitative statement
The most widespread version of Faraday's law
states:
The induced electromotive force in any closed
circuit is equal to the negative of the time
rate of change of the magnetic flux through
the circuit.
This version of Faraday's law strictly holds
only when the closed circuit is a loop of
infinitely thin wire, and is invalid in other
circumstances as discussed below. A different
version, the Maxwell–Faraday equation, is
valid in all circumstances.
Quantitative
Faraday's law of induction makes use of the
magnetic flux ΦB through a hypothetical surface
Σ whose boundary is a wire loop. Since the
wire loop may be moving, we write Σ(t) for
the surface. The magnetic flux is defined
by a surface integral:
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.
In more visual terms, the magnetic flux through
the wire loop is proportional to the number
of magnetic flux lines that pass through the
loop.
When the flux changes—because B changes,
or because the wire loop is moved or deformed,
or both—Faraday's law of induction says
that the wire loop acquires an EMF, , defined
as the energy available from a unit charge
that has travelled 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 ΦB is
the magnetic flux. The direction of the electromotive
force is given by Lenz's law.
For a tightly wound coil of wire, composed
of N identical turns, each with the same ΦB,
Faraday's law of induction states that
where N is the number of turns of wire and
ΦB is the magnetic flux through a single
loop.
Maxwell–Faraday equation
The Maxwell–Faraday equation is a generalisation
of Faraday's law that states that a time-varying
magnetic field is always accompanied by a
spatially-varying, non-conservative electric
field, and vice-versa. The Maxwell–Faraday
equation is
(in SI units) where is the curl operator and
again E(r, t) is the electric field and B(r,
t) is the magnetic field. These fields can
generally be functions of position r and time
t.
The Maxwell–Faraday equation is one of the
four Maxwell's equations, and therefore plays
a fundamental role in the theory of classical
electromagnetism. It can also be written in
an integral form by the Kelvin-Stokes theorem:
where, as indicated in the figure:
Σ is a surface bounded by the closed contour
∂Σ,
E is the electric field, B is the magnetic
field.
dℓ is an infinitesimal vector element of
the contour ∂Σ,
dA is an infinitesimal vector element of surface
Σ. If its direction is orthogonal to that
surface patch, the magnitude is the area of
an infinitesimal patch of surface.
Both dℓ and dA have a sign ambiguity; to
get the correct sign, the right-hand rule
is used, as explained in the article Kelvin-Stokes
theorem. For a planar surface Σ, a positive
path element dℓ of curve ∂Σ is defined
by the right-hand rule as one that points
with the fingers of the right hand when the
thumb points in the direction of the normal
n to the surface Σ.
The integral around ∂Σ is called a path
integral or line integral.
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.
If the path Σ 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 ΦB through Σ.
Proof of Faraday's law
The four Maxwell's equations, along with the
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. Click "show"
in the box below for an outline of this proof.
"Counterexamples" to Faraday's law
Although Faraday's law is always true for
loops of thin wire, it can give the wrong
result if naively extrapolated to other contexts.
One example is the homopolar generator: A
spinning circular metal disc in a homogeneous
magnetic field generates a DC EMF. In Faraday's
law, EMF is the time-derivative of flux, so
a DC EMF is only possible if the magnetic
flux is getting uniformly larger and larger
perpetually. But in the generator, the magnetic
field is constant and the disc stays in the
same position, so no magnetic fluxes are growing
larger and larger. So this example cannot
be analyzed directly with Faraday's law.
Another example, due to Feynman, has a dramatic
change in flux through a circuit, even though
the EMF is arbitrarily small. See figure and
caption above right.
In both these examples, the changes in the
current path are different from the motion
of the material making up the circuit. The
electrons in a material tend to follow the
motion of the atoms that make up the material,
due to scattering in the bulk and work function
confinement at the edges. Therefore, motional
EMF is generated when a material's atoms are
moving through a magnetic field, dragging
the electrons with them, thus subjecting the
electrons to the Lorentz force. In the homopolar
generator, the material's atoms are moving,
even though the overall geometry of the circuit
is staying the same. In the second example,
the material's atoms are almost stationary,
even though the overall geometry of the circuit
is changing dramatically. On the other hand,
Faraday's law always holds for thin wires,
because there the geometry of the circuit
always changes in a direct relationship to
the motion of the material's atoms.
Although Faraday's law does not apply to all
situations, the Maxwell–Faraday equation
and Lorentz force law are always correct and
can always be used directly.
Both of the above examples can be correctly
worked by choosing the appropriate path of
integration for Faraday's law. Outside of
context of thin wires, the path must never
be chosen to go through the conductor in the
shortest direct path. This is explained in
detail in "The Electromagnetodynamics of Fluid"
by W. F. Hughes and F. J. Young, John Wiley
Inc..
Faraday's law and relativity
Two 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.
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:
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 ...
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
Einstein's view
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.
Examples of this sort, together with unsuccessful
attempts to discover any motion of the earth
relative to the "light medium," suggest that
the phenomena of electrodynamics as well as
of mechanics possess no properties corresponding
to the idea of absolute rest.
— Albert Einstein, On the Electrodynamics
of Moving Bodies
See also
References
Further reading
Maxwell, James Clerk, A treatise on electricity
and magnetism, Vol. II, Chapter III, §530,
p. 178. Oxford, UK: Clarendon Press. ISBN
0-486-60637-6.
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
Notes from Physics and Astronomy HyperPhysics
at Georgia State University
Faraday's Law for EMC Engineers
Tankersley and Mosca: Introducing Faraday's
law
Lenz's Law at work.
A free java simulation on motional EMF
Two videos demonstrating Faraday's and Lenz's
laws at EduMation
