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123. (a) The binomial theorem (Appendix E) allows us to write
kx
k
xx x
k
x
k
11
28
3
48
2
23
+=
+
+
+
+
F
H
G
I
K
J
≈+
bg
"
for
x
±
1. Thus, the end result from the solution of Problem 3575 yields
rR
m
m
Rm
m
m
=+
F
H
G
I
K
J
λλ
λ
1
1
2
1
4
and
m
m
m
m
+
F
H
G
I
K
J
1
1
3
2
3
4
λ
for very large values of
m
. Subtracting these, we obtain
∆
r
m
m
R
m
=−=
3
4
1
4
1
2
λ
.
(b) We take the differential of the area:
dA = d
(
π
r
2
) = 2
π
r dr
, and replace
dr
with
∆
r
in
anticipation of using the result from part (a). Thus, the area between adjacent rings for
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 Spring '08
 Any
 Physics

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