Mathematical Methods for Physics and Engineering - Matematica.NET

21.6 EXERCISES21.18 A sphere of radius a and thermal conductivity k 1 is surrounded by an infinitemedium of conductivity k 2 in which far away the temperature tends to T ∞ .A distribution of heat sources q(θ) embedded in the sphere’s surface establishsteady temperature fields T 1 (r, θ) inside the sphere and T 2 (r, θ) outside it. It canbe shown, by considering the heat flow through a small volume that includespart of the sphere’s surface, that∂T 1k 1∂r − k ∂T 22 = q(θ) on r = a.∂rGiven thatq(θ) = 1 ∞∑q n P n (cos θ),an=0find complete expressions for T 1 (r, θ) andT 2 (r, θ). What is the temperature atthe centre of the sphere?21.19 Using result (21.74) from the worked example in the text, find the generalexpression for the temperature u(x, t) in the bar, given that the temperaturedistribution at time t =0isu(x, 0) = exp(−x 2 /a 2 ).21.20 Working in spherical polar coordinates r =(r, θ, φ), but for a system that hasazimuthal symmetry around the polar axis, consider the following gravitationalproblem.(a) Show that the gravitational potential due to a uniform disc of radius a andmass M, centred at the origin, is given for raby[GM1 − 1 ( a) 2P2 (cos θ)+ 1 ( a) 4P4 (cos θ) − ···],r 4 r8 rwhere the polar axis is normal to the plane of the disc.(b) Reconcile the presence of a term P 1 (cos θ), which is odd under θ → π − θ,with the symmetry with respect to the plane of the disc of the physicalsystem.(c) Deduce that the gravitational field near an infinite sheet of matter of constantdensity ρ per unit area is 2πGρ.21.21 In the region −∞