Page 567 - Advanced_Engineering_Mathematics o'neil

15.3 Bessel Functions 547

The quantity of interest is the real part of z(r,t), denoted
x(r,t) = Re(z(r,t)).

We can also think of the current as the real part of the complex exponential function
Ce iωt = C cos(ωt) + iC sin(ωt).
In terms of z, equation (15.17) is
∂
∂z ∂z
ρ r = 4πμr . (15.18)
∂r ∂r ∂t
iωt
Attempt a solution of the form z(r,t) = f (r)e . Substitute this into equation (15.18) to obtain
∂
iωt
ρ (rf (r))e iωt = 4πμrf (r)iωe .

∂r
Upon dividing by e iωt we have
1
2
f (r) + f (r) − b f (r) = 0,
r
in which
4πμω
2
b = i.
ρ
From Section 15.3.6, this equation has general solution
f (r) = c 1 I 0 (br) + c 2 K 0 (br),
in which

4πμω 1 + i
b = √ .
ρ 2
The logarithm term in K 0 (br) is unbounded as r → 0, the center of the wire, so we must choose
c 2 = 0. Then
iωt
f (r) = c 1 I 0 (br) and z(r,t) = c 1 I 0 (br)e .
To determine c 1 , use the fact that the current C cos(ωt) is the real part of Ce . Using equation
iωt
(15.9),
a
2πac 1
C = 2πc 1 rI 0 (br)dr = I (ba).
0
b
0
Therefore,
bC 1
c 1 = .
2πa I (ba)

0
Now
bC 1
iωt
z(r,t) = I 0 (br)e .
2πa I (ba)

0
Then
x(r,t) = Re(z(r,t))
is the current density in the wire. The magnetic intensity is

2C iωt
H(r,t) = Re I 0 (br)e .

aI (ba)
0
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October 14, 2010 15:20 THM/NEIL Page-547 27410_15_ch15_p505-562