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HW 3 Phys5406 S03 due 2/10/03
1)Recall the solution for the potential, Φ(
r
), inside a conducting box of sides (a,b,c) in the
(x,y,z) directions when all sides are grounded except for the side at z = c that is maintained
at constant potential V,
Φ(
r
) =
16
V
π
2
Σ
j,k
=1
,
3
,
5
...
[
sin
jπx/a
j
sin
kπy/b
k
sinh
γ
jk
z
sinh
γ
jk
c
]
,
(1)
where
γ
jk
= [(
jπ/a
)
2
+(
kπ/b
)
2
]
1
/
2
. We will use this solution to get the Green’s function
G
D
for all problems of this type.
a) First expand the sinh function in the range (0,c) in a sine series,
sinh
γ
jk
z
sinh
γ
jk
c
= Σ
m
[
A
m
sin
mπz/c
]
,
(2)
and using the orthogonality relation for such a series obtain the coe±cients
A
m
.
b) To see the connection between
∇
0
2
G
D
(
r
,
r
0
) =

4
πδ
(
r

r
0
) and a sine series, expand a
one dimensional delta function in the range (0,a) as below,
δ
(
x

x
0
) = Σ
j
[
A
j
sin
jπx/a
sin
jπx
0
/a
]
,
(3)
and ﬁnd the coe±cients
A
j
. This is called the completeness relation for the sine series,
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 Spring '10
 Blecher
 Magnetism

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