Effects of varying magnetic
fields
Self-inductance and mutual
inductance
The self-inductance of a circuit is
used to describe the reaction of the circuit to a changing current in the
circuit, while the mutual inductance with respect to a second circuit
describes the reaction to a changing current in the second circuit. When a
current i1 flows in circuit 1, i1 produces a magnetic field B1; the magnetic
flux through circuit 1 due to current i1 is 11. Since B1 is proportional to
i1, 11 is as well. The constant of proportionality is the self-inductance L1
of the circuit. It is defined by the equation
Ø11 = L1I1
As indicated earlier, the units of
inductance are henrys. If a second circuit is present, some of the field B1
will pass through circuit 2 and there will be a magnetic flux 21 in circuit
2 due to the current i1. The mutual inductance M21 is given by
Ø21 = M21i1
The magnetic flux in circuit 1 due to a current in circuit 2 is given by 12
= M12i2. An important property of the mutual inductance is that M21 = M12.
It is therefore sufficient to use the label M without subscripts for the
mutual inductance of two circuits.
The value of the mutual inductance of two circuits can range from +L1L2 to
-L1L2, depending on the flux linkage between the circuits. If the two
circuits are very far apart or if the field of one circuit provides no
magnetic flux through the other circuit, the mutual inductance is zero. The
maximum possible value of the mutual inductance of two circuits is
approached as the two circuits produce B fields with increasingly similar
spatial configurations.
If the rate of change with respect to time is taken for the terms on both
sides of equation (44 ), the result is d11/dt = L1di1/dt. According to
Faraday's law, d11/dt is the negative of the induced electromotive force.
The result is the equation frequently used for a single inductor in an AC
circuit--i.e.,
emf
= - L ( di / dt)
The phenomenon of self-induction was first recognized by the American
scientist Joseph Henry. He was able to generate large and spectacular
electric arcs by interrupting the current in a large copper coil with many
turns. While a steady current is flowing in a coil, the energy in the
magnetic field is given by 1/2Li2. If both the inductance L and the current
i are large, the amount of energy is also large. If the current is
interrupted, as, for example, by opening a knife-blade switch, the current
and therefore the magnetic flux through the coil drop quickly. Equation (46
) describes the resulting electromotive force induced in the coil, and a
large potential difference is developed between the two poles of the switch.
The energy stored in the magnetic field of the coil is dissipated as heat
and radiation in an electric arc across the space between the terminals of
the switch. Due to advances in superconducting wires for electromagnets, it
is possible to use large magnets with magnetic fields of several teslas for
temporarily storing electric energy as energy in the magnetic field. This is
done to accommodate short-term fluctuations in the consumption of electric
power.
A transformer is an example of a device that uses circuits with maximum
mutual induction. Figure 5
An AC transformer
Illustrates the configuration of a
typical transformer. Here, coils of insulated conducting wire are wound
around a ring of iron constructed of thin isolated laminations or sheets.
The laminations minimize eddy currents in the iron. Eddy currents are
circulatory currents induced in the metal by the changing magnetic field.
These currents produce an undesirable by-product--heat in the iron. Energy
loss in a transformer can be reduced by using thinner laminations, very
"soft" (low-carbon) iron and wire with a larger cross section, or by winding
the primary and secondary circuits with conductors that have very low
resistance. Unfortunately, reducing the heat loss increases the cost of
transformers. Transformers used to transmit and distribute power are
commonly 98 to 99 percent efficient. While eddy currents are a problem in
transformers, they are useful for heating objects in a vacuum. Eddy currents
are induced in the object to be heated by surrounding a relatively
nonconducting vacuum enclosure with a coil carrying a high-frequency
alternating current.
In a transformer, the iron ensures that nearly all the lines of B passing
through one circuit also pass through the second circuit and that, in fact,
essentially all the magnetic flux is confined to the iron. Each turn of the
conducting coils has the same magnetic flux; thus, the total flux for each
coil is proportional to the number of turns in the coil. As a result, if a
source of sinusoidally varying electromotive force is connected to one coil,
the electromotive force in the second coil is given by
emf2
= emf1 ( N2 / N1 )
Thus, depending on the ratio of N2 to
N1, the transformer can be either a step-up or a step-down device for
alternating voltages. For many reasons, including safety, generation and
consumption of electric power occur at relatively low voltages. Step-up
transformers are used to obtain high voltages before electric power is
transmitted, since for a given amount of power, the current in the
transmission lines is much smaller. This minimizes energy lost by resistive
heating of the conductors.
Faraday's law constitutes the basis for the power industry and for the
transformation of mechanical energy into electric energy. In 1821, a decade
before his discovery of magnetic induction, Faraday conducted experiments
with electric wires rotating around compass needles. This earlier work, in
which a wire carrying a current rotated around a magnetized needle and a
magnetic needle was made to rotate around a wire carrying an electric
current, provided the groundwork for the development of the electric motor.
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