Page 676 - Advanced_Engineering_Mathematics o'neil

656 CHAPTER 18 The Potential Equation

z

ϕ
ρ
y
θ

x
FIGURE 18.3 Spher-
ical coordinates.

Assuming symmetry of the temperature function about the z-axis, the solution is inde-
pendent of θ and depends only on ρ and ϕ. Laplace’s equation in spherical coordinates (with
independence from θ)is

2
2
∂ u 2 ∂u 1 ∂ u cot(ϕ) ∂u
2
∇ u(ρ,ϕ) = + + + = 0.
∂ρ 2 ρ ∂ρ ρ ∂ϕ 2 ρ 2 ∂ϕ
2
The temperature on the surface is u(R,ϕ) = f (ϕ) with f being given.
Let u(ρ,ϕ) = X(ρ)
(ϕ) to obtain
2 1 cot(ϕ)

X
+ X
+ X
+ X
= 0.
ρ ρ 2 ρ 2
Upon dividing this equation by X
, we can separate the variables, obtaining

X X
+ cot(ϕ) =−ρ 2 − 2ρ =−λ.

X X
Then
2

ρ X + 2ρX − λX = 0 and
+ cot(ϕ)
+ λ
= 0.

To solve this equation for
, write it as
1
[
sin(ϕ)] + λ
= 0. (18.5)

sin(ϕ)
Change variables by putting x = cos(ϕ). Then ϕ = arccos(x).Let
G(x) =
(arccos(x)).

Since 0 ≤ ϕ ≤ π, then −1 ≤ x ≤ 1. Compute
d
dx

(ϕ)sin(ϕ) = sin(ϕ)

dx dϕ

= sin(ϕ)G (x)[−sin(ϕ)]
2
2

=−sin (ϕ)G (x) =−[1 − cos (x)]G (x)

2

=−(1 − x )G (x).
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:27 THM/NEIL Page-656 27410_18_ch18_p641-666