Page 503 - Electromagnetics

Of course we could use exponentials or a combination of exponentials and trigonometric
2
functions instead. Rearranging (A.119) and multiplying through by ρ , we obtain
2
2
2
1 d # 2 2 2 ρ dP ρ d P
− = k − k z ρ + + .
# dφ 2 P dρ P dρ 2
The left and right sides depend only on φ and ρ, respectively; both must equal some
2
constant k :
φ
2
1 d # 2
− = k , (A.121)
φ
# dφ 2
2
2
ρ dP ρ d P
2 2 2 2
k − k z ρ + + = k . (A.122)
φ
P dρ P dρ 2
The variables ρ and φ are thus separated, and harmonic equation (A.121) has solutions

A φ sin k φ φ + B φ cos k φ φ, k φ = 0,
#(φ) = (A.123)
a φ φ + b φ , k φ = 0.
Equation (A.122) is a bit more involved. In rearranged form it is
2
2
d P 1 dP 2 k φ
+ + k − P = 0 (A.124)
c
dρ 2 ρ dρ ρ 2
where
2
2
2
k = k − k .
c
z
The solution depends on whether any of k z , k φ ,or k c are zero. If k c = k φ = 0, then
2
d P 1 dP
+ = 0
dρ 2 ρ dρ
so that
P(ρ) = a ρ ln ρ + b ρ .
If k c = 0 but k φ = 0, we have
2
d P 1 dP k 2 φ
+ − P = 0
dρ 2 ρ dρ ρ 2
so that
P(ρ) = a ρ ρ −k φ + b ρ ρ . (A.125)
k φ
This includes the case k = k z = 0 (Laplace’s equation). If k c = 0 then (A.124) is Bessel’s
(z)
differential equation. For noninteger k φ the two independent solutions are denoted J k φ
(z), where J ν (z) is the ordinary Bessel function of the first kind of order ν.For
and J −k φ
k φ an integer n, J n (z) and J −n (z) are not independent and a second independent solution
denoted N n (z) must be introduced. This is the ordinary Bessel function of the second
kind, order n. As it is also independent when the order is noninteger, J ν (z) and N ν (z)
are often chosen as solutions whether ν is integer or not. Linear combinations of these
independent solutions may be used to produce new independent solutions. The functions

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