Electromagnetic induction
Electromagnetic induction is the production
of an electromotive force across a conductor
when it is exposed to a varying magnetic field.
It is described mathematically by Faraday's
law of induction, named after Michael Faraday
who is generally credited with the discovery
of induction in 1831.
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 (August 29, 1831), he wrapped two
wires around opposite sides of an iron ring
or "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").
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 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 slightly different in form 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 Heinrich 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 and the Maxwell–Faraday equation
The law of physics describing the process
of electromagnetic induction is known as Faraday's
law of induction and the most widespread version
of this law states that the induced electromotive
force in any closed circuit is equal to the
rate of change of the magnetic flux through
the circuit. Or mathematically,
where is the electromotive force (EMF) and
ΦB is the magnetic flux. The direction of
the electromotive force is given by Lenz's
law. This version of Faraday's law strictly
holds only when the closed circuit is a loop
of infinitely thin wire, and is invalid in
some other circumstances. A different version,
the Maxwell–Faraday equation (discussed
below), is valid in all circumstances.
For a tightly wound coil of wire, composed
of N identical turns, each with the same magnetic
flux going through them, the resulting EMF
is given by
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 (the
infinitesimal amount of magnetic flux). 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.
According to the Lorentz force law (in SI
units),
the EMF on a wire loop is:
where E is the electric field, B is the magnetic
field (aka magnetic flux density, magnetic
induction), dℓ is an infinitesimal arc length
along the wire, and the line integral is evaluated
along the wire (along the curve the conincident
with the shape of the wire).
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:
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.
Applications
The principles of electromagnetic induction
are applied in many devices and systems, including:
Current clamp
Electrical generators
Electromagnetic forming
Graphics tablet
Hall effect meters
Induction cookers
Induction motors
Induction sealing
Induction welding
Inductive charging
Inductors
Magnetic flow meters
Mechanically powered flashlight
Pickups
Rowland ring
Transcranial magnetic stimulation
Transformers
Wireless energy transfer
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
the figure to the right. A different implementation
of this idea is the Faraday's disc, shown
in simplified form on the right.
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 Ampère's
circuital law (labeled "induced B" in the
figure). 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 B-field induced by this
return current opposes the applied B-field,
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 B-field
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.
Electrical transformer
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, 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 ℇ 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.
Eddy currents
Conductors (of finite dimensions) moving through
a uniform magnetic field, or stationary within
a changing magnetic field, will have currents
induced within them. These induced eddy currents
can be undesirable, since they dissipate energy
in the resistance of the conductor. There
are a number of methods employed to control
these undesirable inductive effects.
Electromagnets in electric motors, generators,
and transformers do not use solid metal, but
instead use thin sheets of metal plate, called
laminations. These thin plates reduce the
parasitic eddy currents, as described below.
Inductive coils in electronics typically use
magnetic cores to minimize parasitic current
flow. They are a mixture of metal powder plus
a resin binder that can hold any shape. The
binder prevents parasitic current flow through
the powdered metal.
Electromagnet laminations
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.
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.
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.
Parasitic induction within inductors
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 whorls or current eddies
within the copper bar.
High current power-frequency devices such
as electric motors, generators and transformers
use multiple small conductors in parallel
to break up the eddy flows that can form within
large solid conductors. The same principle
is applied to transformers used at higher
than power frequency, for example, those used
in switch-mode power supplies and the intermediate
frequency coupling transformers of radio receivers.
