PDES: SEPARATION OF VARIABLES AND OTHER METHODS
21.22 Point chargesqand−qa/b(witha<b) are placed, respectively, at a pointP,a
distancebfrom the originO, and a pointQbetweenOandP,adistancea^2 /b
fromO. Show, by considering similar trianglesQOSandSOP,whereSis any
point on the surface of the sphere centred atOand of radiusa, that the net
potential anywhere on the sphere due to the two charges is zero.
Use this result (backed up by the uniqueness theorem) to find the force with
which a point chargeqplaced a distancebfrom the centre of a spherical
conductor of radiusa(<b) is attracted to the sphere (i) if the sphere is earthed,
and (ii) if the sphere is uncharged and insulated.
21.23 Find the Green’s functionG(r,r 0 ) in the half-spacez>0 for the solution of
∇^2 Φ = 0 with Φ specified in cylindrical polar coordinates (ρ, φ, z) on the plane
z=0by
Φ(ρ, φ, z)={
1forρ≤ 1 ,
1 /ρ forρ> 1.Determine the variation of Φ(0, 0 ,z)alongthez-axis.
21.24 Electrostatic charge is distributed in a sphere of radiusRcentred on the origin.
Determine the form of the resultant potentialφ(r) at distances much greater than
R, as follows.
(a) Express in the form of an integral over all space the solution of
∇^2 φ=−ρ(r)
0.
(b) Show that, forrr′,|r−r′|=r−r·r′
r+O
(
1
r)
.
(c) Use results (a) and (b) to show thatφ(r)hastheformφ(r)=M
r+
d·r
r^3+O
(
1
r^3)
.
Find expressions forMandd, and identify them physically.21.25 Find, in the form of an infinite series, the Green’s function of the∇^2 operator for
the Dirichlet problem in the region−∞<x<∞,−∞<y<∞,−c≤z≤c.
21.26 Find the Green’s function for the three-dimensional Neumann problem
∇^2 φ=0 forz>0and∂φ
∂z=f(x, y)onz=0.Determineφ(x, y, z)iff(x, y)={
δ(y)for|x|<a,
0for|x|≥a.21.27 Determine the Green’s function for the Klein–Gordon equation in a half-space
as follows.
(a) By applying the divergence theorem to the volume integral
∫
V[
φ(∇^2 −m^2 )ψ−ψ(∇^2 −m^2 )φ]
dV ,obtain a Green’s function expression, as the sum of a volume integral and a
surface integral, for the functionφ(r′) that satisfies
∇^2 φ−m^2 φ=ρ