Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.6 EXERCISES

21.18 A sphere of radiusaand thermal conductivityk 1 is surrounded by an infinite
medium of conductivityk 2 in which far away the temperature tends toT∞.
A distribution of heat sourcesq(θ) embedded in the sphere’s surface establish
steady temperature fieldsT 1 (r, θ) inside the sphere andT 2 (r, θ) outside it. It can
be shown, by considering the heat flow through a small volume that includes
part of the sphere’s surface, that

k 1

∂T 1

∂r

−k 2

∂T 2

∂r

=q(θ)onr=a.

Given that

q(θ)=

1

a

∑∞

n=0

qnPn(cosθ),

find complete expressions forT 1 (r, θ)andT 2 (r, θ). What is the temperature at
the centre of the sphere?
21.19 Using result (21.74) from the worked example in the text, find the general
expression for the temperatureu(x, t) in the bar, given that the temperature
distribution at timet=0isu(x,0) = exp(−x^2 /a^2 ).
21.20 Working insphericalpolar coordinatesr=(r, θ, φ), but for a system that has
azimuthal symmetry around the polar axis, consider the following gravitational
problem.

(a) Show that the gravitational potential due to a uniform disc of radiusaand

massM, centred at the origin, is given forr<aby

2 GM

a

[

1 −

r

a

P 1 (cosθ)+

1

2

(r

a

) 2

P 2 (cosθ)−

1

8

(r

a

) 4

P 4 (cosθ)+···

]

,

and forr>aby

GM

r

[

1 −

1

4

(a

r

) 2

P 2 (cosθ)+

1

8

(a

r

) 4

P 4 (cosθ)−···

]

,

where the polar axis is normal to the plane of the disc.

(b) Reconcile the presence of a termP 1 (cosθ), which is odd underθ→π−θ,

with the symmetry with respect to the plane of the disc of the physical

system.

(c) Deduce that the gravitational field near an infinite sheet of matter of constant

densityρper unit area is 2πGρ.

21.21 In the region−∞<x,y<∞and−t≤z≤t, a charge-density waveρ(r)=
Acosqx,inthex-direction, is represented by

ρ(r)=

eiqx

√

2 π

∫∞

−∞

̃ρ(α)eiαzdα.

The resulting potential is represented by

V(r)=

eiqx

√

2 π

∫∞

−∞

V ̃(α)eiαzdα.

Determine the relationship betweenV ̃(α)and ̃ρ(α), and hence show that the

potential at the point (0, 0 ,0) is

A

π 0

∫∞

−∞

sinkt

k(k^2 +q^2 )

dk.