Page 262 - Engineering Electromagnetics, 8th Edition

244 ENGINEERING ELECTROMAGNETICS

D8.5. A conducting filamentary triangle joins points A(3, 1, 1), B(5, 4, 2),
and C(1, 2, 4). The segment AB carries a current of 0.2 A in the a AB direction.
There is present a magnetic field B = 0.2a x − 0.1a y + 0.3a z T. Find: (a) the
force on segment BC;(b) the force on the triangular loop; (c) the torque on the
loop about an origin at A;(d) the torque on the loop about an origin at C.

Ans. −0.08a x + 0.32a y + 0.16a z N; 0; −0.16a x − 0.08a y + 0.08a z N · m; −0.16a x −
0.08a y + 0.08a z N · m

8.5 THE NATURE OF MAGNETIC MATERIALS

We are now in a position to combine our knowledge of the action of a magnetic field
on a current loop with a simple model of an atom and obtain some appreciation of
the difference in behavior of various types of materials in magnetic fields.
Although accurate quantitative results can only be predicted through the use
of quantum theory, the simple atomic model, which assumes that there is a central
positive nucleus surrounded by electrons in various circular orbits, yields reasonable
quantitative results and provides a satisfactory qualitative theory. An electron in an
orbit is analogous to a small current loop (in which the current is directed oppositely
to the direction of electron travel) and, as such, experiences a torque in an external
magnetic field, the torque tending to align the magnetic field produced by the orbiting
electron with the external magnetic field. If there were no other magnetic moments to
consider, we would then conclude that all the orbiting electrons in the material would
shift in such a way as to add their magnetic fields to the applied field, and thus that
the resultant magnetic field at any point in the material would be greater than it would
be at that point if the material were not present.
A second moment, however, is attributed to electron spin. Although it is tempting
to model this phenomenon by considering the electron as spinning about its own axis
and thus generating a magnetic dipole moment, satisfactory quantitative results are
not obtained from such a theory. Instead, it is necessary to digest the mathematics of
relativistic quantum theory to show that an electron may have a spin magnetic moment
2
of about ±9 × 10 −24 A · m ; the plus and minus signs indicate that alignment aiding
or opposing an external magnetic field is possible. In an atom with many electrons
present, only the spins of those electrons in shells which are not completely filled will
contribute to a magnetic moment for the atom.
A third contribution to the moment of an atom is caused by nuclear spin. Although
this factor provides a negligible effect on the overall magnetic properties of materials,
it is the basis of the nuclear magnetic resonance imaging (MRI) procedure provided
by many of the larger hospitals.
Thus each atom contains many different component moments, and their com-
bination determines the magnetic characteristics of the material and provides its
general magnetic classification. We describe briefly six different types of material:
diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic, and
superparamagnetic.