Page 290 - Engineering Electromagnetics, 8th Edition

272 ENGINEERING ELECTROMAGNETICS

8.15 A solid conducting filament extends from x =−b to x = b along the line
y = 2, z = 0. This filament carries a current of3Ainthe a x direction. An
infinite filament on the z axis carries5Ainthe a z direction. Obtain an
expression for the torque exerted on the finite conductor about an origin
located at (0, 2, 0).
8.16 Assume that an electron is describing a circular orbit of radius a about a
positively charged nucleus. (a)By selecting an appropriate current and area,
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show that the equivalent orbital dipole moment is ea ω/2, where ω is the
electron’s angular velocity. (b) Show that the torque produced by a magnetic
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field parallel to the plane of the orbit is ea ωB/2. (c)By equating the
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2 −1/2
Coulomb and centrifugal forces, show that ω is (4π 0 m e a /e ) , where
m e is the electron mass. (d) Find values for the angular velocity, torque,
and the orbital magnetic moment for a hydrogen atom, where a is about
6 × 10 −11 m; let B = 0.5T.
8.17 The hydrogen atom described in Problem 8.16 is now subjected to a
magnetic field having the same direction as that of the atom. Show that the
forces caused by B result in a decrease of the angular velocity by eB/(2m e )
2 2
and a decrease in the orbital moment by e a B/(4m e ). What are these
decreases for the hydrogen atom in parts per million for an external magnetic
flux density of 0.5 T?
8.18 Calculate the vector torque on the square loop shown in Figure 8.15 about
an origin at A in the field B, given (a) A(0, 0, 0) and B = 100a y mT;
(b) A(0, 0, 0) and B = 200a x + 100a y mT; (c) A(1, 2, 3) and B = 200a x +
100a y − 300a z mT; (d) A(1, 2, 3) and B = 200a x + 100a y − 300a z mT
for x ≥ 2 and B = 0 elsewhere.
8.19 Given a material for which χ m = 3.1 and within which B = 0.4ya z T, find
(a)H;(b) µ;(c) µ r ;(d) M;(e) J;( f ) J B ;(g) J T .
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8.20 Find H in a material where (a) µ r = 4.2, there are 2.7 × 10 29 atoms/m , and
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each atom has a dipole moment of 2.6 × 10 −30 a y A · m ;(b) M = 270a z A/m
and µ = 2µ H/m; (c) χ m = 0.7 and B = 2a z T. (d) Find M in a material
where bound surface current densities of 12a z A/m and −9a z A/m exist at
ρ = 0.3m and 0.4 m, respectively.
8.21 Find the magnitude of the magnetization in a material for which (a) the
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magnetic flux density is 0.02 Wb/m ;(b) the magnetic field intensity is
1200 A/m and the relative permeability is 1.005; (c) there are 7.2 × 10 28
atoms per cubic meter, each having a dipole moment of 4 × 10 −30 A·m 2
in the same direction, and the magnetic susceptibility is 0.003.
8.22 Under some conditions, it is possible to approximate the effects of
ferromagnetic materials by assuming linearity in the relationship of B and
H. Let µ r = 1000 for a certain material of which a cylindrical wire of
radius 1 mm is made. If I = 1A and the current distribution is uniform,
find (a) B,(b) H,(c) M,(d) J, and (e) J B within the wire.