Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

(b) Find the principal axes and verify that they are orthogonal.

26.17 A rigid body consists of eight particles, each of massm, held together by light
rods. In a certain coordinate frame the particles are at positions

±a(3, 1 ,−1), ±a(1,− 1 ,3), ±a(1, 3 ,−1), ±a(− 1 , 1 ,3).

Show that, when the body rotates about an axis through the origin, if the angular

velocity and angular momentum vectors are parallel then their ratio must be

40 ma^2 ,64ma^2 or 72ma^2.

26.18 The paramagnetic tensorχijof a body placed in a magnetic field, in which its
energy density is−^12 μ 0 M·HwithMi=

∑

jχijHj,is





2 k 00

03 kk

0 k 3 k



.

Assuming depolarizing effects are negligible, find how the body will orientate

itself if the field is horizontal, in the following circumstances:

(a) the body can rotate freely;

(b) the body is suspended with the (1, 0, 0) axis vertical;

(c) the body is suspended with the (0, 1, 0) axis vertical.

26.19 A block of wood contains a number of thin soft-iron nails (of constant perme-
ability). A unit magnetic field directed eastwards induces a magnetic moment in
the block having components (3, 1 ,−2), and similar fields directed northwards
and vertically upwards induce moments (1, 3 ,−2) and (− 2 ,− 2 ,2) respectively.
Show that all the nails lie in parallel planes.

26.20 For tin, the conductivity tensor is diagonal, with entriesa, a,andbwhen referred
to its crystal axes. A single crystal is grown in the shape of a long wire of lengthL
and radiusr, the axis of the wire making polar angleθwith respect to the crystal’s
3-axis. Show that the resistance of the wire isL(πr^2 ab)−^1

(

acos^2 θ+bsin^2 θ

)

.

26.21 By considering an isotropic body subjected to a uniform hydrostatic pressure
(no shearing stress), show that the bulk modulusk, defined by the ratio of the
pressure to the fractional decrease in volume, is given byk=E/[3(1− 2 σ)] where
Eis Young’s modulus andσis Poisson’s ratio.

26.22 For an isotropic elastic medium under dynamic stress, at timetthe displacement
uiand the stress tensorpijsatisfy

pij=cijkl

(

∂uk

∂xl

+

∂ul

∂xk

)

and

∂pij

∂xj

=ρ

∂^2 ui

∂t^2

,

wherecijklis the isotropic tensor given in equation (26.47) andρis a constant.

Show that both∇·uand∇×usatisfy wave equations and find the corresponding

wave speeds.