e/Example 24: without the
iron core, the field is so weak that
it barely deflects the compass.
With it, the deflection is nearly
90 ◦.f/A transformer with a lami-
nated iron core. The input and
output coils are inside the paper
wrapper. The iron core is the
black part that passes through
the coils at the center, and
also wraps around them on the
outside.
g/Example 25: ferrite beads.
The top panel shows a clip-on
type, while the bottom shows one
built into a cable.
An iron-core electromagnet example 24
.A solenoid has 1000 turns of wire wound along a cylindrical
core with a length of 10 cm. If a current of 1.0 A is used, find the
magnetic field inside the solenoid if the core is air, and if the core
is made of iron withμ/μo= 4, 000.
.Air has essentially the same permability as vacuum, so using
the result of example 13 on page 701, we find that the field is
0.013 T.
We now consider the case where the core is filled with iron. The
original derivation in example 13 started from Ampere’s law, which`
we now rewrite asΓH=Ithrough, free. As argued previously, the only
significant contributions to the circulation come from line segment
AB. This segment lies inside the iron, whereH= B/μ. TheH
field is the same as in the air-core case, since the new form of
Ampere’s law only relates` Hto the current in the wires (the free
current). This means thatB=μHis greater by a factor of 4,000
than in the air-core case, or 52 T. This is an extremely intense
field — so intense, in fact, that the iron’s magnetic polarization
would probably become saturated before we could actually get
the field that high.
The electromagnet of example 24 could also be used as an in-
ductor, and its inductance would be proportional to the permittivity
of the core. This makes it possible to construct high-value induc-
tors that are relatively compact. Permeable cores are also used in
transformers.
A transformer or inductor with a permeable core does have
some disadvantages, however, in certain applications. The oscil-
lating magnetic field induces an electric field, and because the core
is typically a metal, these currents dissipate energy strongly as heat.
This behaves like a fairly large resistance in series with the coil. Fig-
ure f shows a method for reducing this effect. The iron core of this
transformer has been constructed out of laminated layers, which has
the effect of blocking the conduction of the eddy currents.A ferrite bead example 25
Cables designed to carry audio signals are typically made with
two adjacent conductors, such that the current flowing out through
one conductor comes back through the other one. Computer ca-
bles are similar, but usually have several such pairs bundled in-
side the insulator. This paired arrangement is known as differen-
tial mode, and has the advantage of cutting down on the reception
and transmission of interference. In terms of transmission, the
magnetic field created by the outgoing current is almost exactly
canceled by the field from the return current, so electromagnetic
waves are only weakly induced. In reception, both conductors are
bathed in the same electric and magnetic fields, so an emf that
adds current on one side subtracts current from the other side,738 Chapter 11 Electromagnetism