Faraday's law of induction (briefly, Faraday's
law) is a basic law of electromagnetism predicting
how a magnetic field will interact with an
electric circuit to produce an electromotive
force (EMF)—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
(listed as one of Maxwell's equations) describes
the fact that a spatially varying (and also
possibly time-varying, depending on how a
magnetic field varies in time) electric field
always accompanies a time-varying magnetic
field, while Faraday's law states that there
is EMF (electromotive force, defined as electromagnetic
work done on a unit charge when it has traveled
one round of a conductive loop) 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 (transformer EMF) was formulated
as the Maxwell–Faraday equation later. The
equation of Faraday's law can be derived by
the Maxwell–Faraday equation (describing
transformer EMF) and the Lorentz force (describing
motional EMF). 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 (August 29, 1831), he wrapped two
wires around opposite sides of an iron ring
(torus) (an arrangement similar to a modern
toroidal transformer). 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 (which he called
a "wave of electricity") 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
(DC) current by rotating a copper disk near
the bar magnet with a sliding electrical lead
("Faraday's disk").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 (see Maxwell–Faraday
equation below) 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
(elaborated upon in the examples below).
== Faraday's law ==
The most widespread version of Faraday's law
states:
The electromotive force around a closed path
is equal to the negative of the time rate
of change of the magnetic flux enclosed by
the path.
The closed path here is, in fact, conductive.
=== Mathematical statement ===
For a loop of wire in a magnetic field, the
magnetic flux ΦB is defined for any surface
Σ whose boundary is the given loop. Since
the wire loop may be moving, we write Σ(t)
for the surface. The magnetic flux is the
surface integral:
Φ
B
=
∬
Σ
(
t
)
B
(
t
)
⋅
d
A
,
{\displaystyle \Phi _{B}=\iint \limits _{\Sigma
(t)}\mathbf {B} (t)\cdot \mathrm {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 representing
the element of flux through dA. 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, E, defined
as the energy available from a unit charge
that has travelled once around the wire loop.
(Note that different textbooks may give different
definitions. The set of equations used throughout
the text was chosen to be compatible with
the special relativity theory.) 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:
E
=
−
d
Φ
B
d
t
,
{\displaystyle {\mathcal {E}}=-{\frac {\mathrm
{d} \Phi _{B}}{\mathrm {d} t}},}
where
E
{\displaystyle {\mathcal {E}}}
is the electromotive force (EMF) and ΦB 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 (EMF) directly from
Faraday’s law, without invoking Lenz's law.
A left hand rule helps doing that, as follows:
Align the curved fingers of the left hand
with the loop (yellow line).
Stretch your thumb. The stretched thumb indicates
the direction of n (brown), the normal to
the area enclosed by the loop.
Find the sign of ΔΦB, the change in flux.
Determine the initial and final fluxes (whose
difference is ΔΦB) with respect to the normal
n, as indicated by the stretched thumb.
If the change in flux, ΔΦB, is positive,
the curved fingers show the direction of the
electromotive force (yellow arrowheads).
If ΔΦB is negative, the direction of the
electromotive force is opposite to the direction
of the curved fingers (opposite to the yellow
arrowheads).For a tightly wound coil of wire,
composed of N identical turns, each with the
same ΦB, Faraday's law of induction states
that
E
=
−
N
d
Φ
B
d
t
{\displaystyle {\mathcal {E}}=-N{\frac {\mathrm
{d} \Phi _{B}}{\mathrm {d} t}}}
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 states that
a time-varying magnetic field always accompanies
a spatially varying (also possibly time-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,
thereby reproducing Faraday's law:
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 surface Σ is not changing in time,
the equation can be rewritten:
∮
∂
Σ
⁡
E
⋅
d
l
=
−
d
d
t
∫
Σ
B
⋅
d
A
.
{\displaystyle \oint _{\partial \Sigma }\mathbf
{E} \cdot \mathrm {d} \mathbf {l} =-{\frac
{\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma
}\mathbf {B} \cdot \mathrm {d} \mathbf {A}
.}
The surface integral at the right-hand side
is the explicit expression for the magnetic
flux ΦB through Σ.
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:
E
s
(
r
)
≈
−
1
4
π
∭
V
(
∂
B
∂
t
d
V
)
×
r
^
′
|
r
′
|
2
{\displaystyle \mathbf {E} _{s}(\mathbf {r}
)\approx -{\frac {1}{4\pi }}\iiint _{V}\ {\frac
{({\frac {\partial \mathbf {B} }{\partial
t}}\,dV)\times \mathbf {{\hat {r}}'} }{|\mathbf
{r} '|^{2}}}}
== Proof ==
The four Maxwell's equations (including the
Maxwell–Faraday equation), 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 Σ (that can move or
be deformed) in space:
d
Φ
B
d
t
=
d
d
t
∫
Σ
(
t
)
B
(
t
)
⋅
d
A
{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm
{d} t}}={\frac {\mathrm {d} }{\mathrm {d}
t}}\int _{\Sigma (t)}\mathbf {B} (t)\cdot
\mathrm {d} \mathbf {A} }
(by definition). 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:
The result is:
d
Φ
B
d
t
=
−
∮
∂
Σ
⁡
(
E
+
v
l
×
B
)
⋅
d
l
.
{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm
{d} t}}=-\oint _{\partial \Sigma }\left(\mathbf
{E} +\mathbf {v} _{\mathbf {l} }\times \mathbf
{B} \right)\cdot \mathrm {d} \mathbf {l} .}
where ∂Σ is the boundary (loop) of the
surface Σ, and vl is the velocity of a part
of the boundary.
In the case of a conductive loop, EMF (Electromotive
Force) 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
E
=
∮
⁡
(
E
+
v
×
B
)
⋅
d
l
{\displaystyle {\mathcal {E}}=\oint \left(\mathbf
{E} +\mathbf {v} \times \mathbf {B} \right)\cdot
\mathrm {d} \mathbf {l} }
where
E
{\displaystyle {\mathcal {E}}}
is EMF and v is the unit charge velocity.
In a macroscopic view, for charges on a segment
of the loop, v consists of two components
in average; one is the velocity of the charge
along the segment vt, and the other is the
velocity of the segment vl (the loop is deformed
or moved). vt does not contribute to 
the work done on the charge since the direction
of vt is same to the direction of
d
l
{\displaystyle d\mathbf {l} }
. Mathematically,
(
v
×
B
)
⋅
d
l
=
(
(
v
t
+
v
l
)
×
B
)
⋅
d
l
=
(
v
t
×
B
+
v
l
×
B
)
⋅
d
l
=
(
v
l
×
B
)
⋅
d
l
{\displaystyle (\mathbf {v} \times B)\cdot
d\mathbf {l} =((\mathbf {v} _{t}+\mathbf {v}
_{l})\times B)\cdot d\mathbf {l} =(\mathbf
{v} _{t}\times B+\mathbf {v} _{l}\times B)\cdot
d\mathbf {l} =(\mathbf {v} _{l}\times B)\cdot
d\mathbf {l} }
since
(
v
t
×
B
)
{\displaystyle (\mathbf {v} _{t}\times B)}
is perpendicular to
d
l
{\displaystyle d\mathbf {l} }
as
v
t
{\displaystyle \mathbf {v} _{t}}
and
d
l
{\displaystyle d\mathbf {l} }
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 (for the conductive loop) as
d
Φ
B
d
t
=
−
E
{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm
{d} t}}=-{\mathcal {E}}}
where
E
=
∮
⁡
(
E
+
v
l
×
B
)
⋅
d
l
{\displaystyle {\mathcal {E}}=\oint \left(\mathbf
{E} +\mathbf {v} _{l}\times \mathbf {B} \right)\cdot
\mathrm {d} \mathbf {l} }
. With breaking this integral,
∮
⁡
E
⋅
d
l
{\displaystyle \oint \mathbf {E} \cdot \mathrm
{d} \mathbf {l} }
is for the transformer EMF (due to a time-varying
magnetic field) and
∮
⁡
(
v
l
×
B
)
⋅
d
l
{\displaystyle \oint \left(\mathbf {v} _{l}\times
\mathbf {B} \right)\cdot \mathrm {d} \mathbf
{l} }
is for the motional EMF (due to the magnetic
Lorentz force on charges by the motion or
deformation of the loop in the magnetic field).
== EMF for non-thin-wire circuits ==
It is tempting to generalize Faraday's law
to state: If ∂Σ is any arbitrary closed
loop in space whatsoever, then the total time
derivative of magnetic flux through Σ equals
the EMF around ∂Σ. 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 ∂Σ 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 ∂Σ is divorced from the motion
of the material.
One can analyze examples like these by taking
care that the path ∂Σ 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:
E
=
∫
∂
Σ
(
E
+
v
m
×
B
)
⋅
d
l
=
−
∫
Σ
∂
B
∂
t
⋅
d
Σ
+
∮
∂
Σ
⁡
(
v
m
×
B
)
⋅
d
l
{\displaystyle {\mathcal {E}}=\int _{\partial
\Sigma }(\mathbf {E} +\mathbf {v} _{m}\times
\mathbf {B} )\cdot \mathrm {d} \mathbf {l}
=-\int _{\Sigma }{\frac {\partial \mathbf
{B} }{\partial t}}\cdot \mathrm {d} \Sigma
+\oint _{\partial \Sigma }(\mathbf {v} _{m}\times
\mathbf {B} )\cdot \mathrm {d} \mathbf {l}
}
where "it is very important to notice that
(1) [vm] is the velocity of the conductor
... not the velocity of the path element dl
and (2) 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 (see the Lorentz force), and the transformer
EMF generated by an electric force due to
a changing magnetic field (described by the
Maxwell–Faraday equation).
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 (or both) ...
Yet in our explanation of the rule we have
used two completely distinct laws for the
two cases – v × B for "circuit moves"
and ∇ × E = −∂tB 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.
=== 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.
== See also
