Physics 315:
Oscillations and Waves
Homework 10: Due in class on Wednesday, Dec. 2nd
1.
(a) Consider the geometric series
S
=
summationdisplay
n
=0
,N

1
z
n
,
where
z
is a complex number. Demonstrate that
S
=
1

z
N
1

z
.
(b) Suppose that
z
= e
i
θ
, where
θ
is real.
Employing the wellknown
identity
sin
θ
≡
1
2 i
(
e
i
θ

e

i
θ
)
,
show that
S
= e
i(
N

1)
θ/
2
sin(
N θ/
2)
sin(
θ/
2)
.
(c) Finally, making use of de Moivre’s theorem,
e
i
n θ
≡
cos(
n θ
) + i sin(
n θ
)
,
demonstrate that
C
=
summationdisplay
n
=1
,N
cos(
α y
n
)
,
where
y
n
= [
n

(
N
+ 1)
/
2]
d,
evaluates to
C
=
sin(
N α d/
2)
sin(
α d/
2)
.
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2. Suppose that a monochromatic laser of wavelength 632
.
8 nm emits a diffrac
tionlimited beam of initial diameter 2 mm. Estimate how large a light spot
the beam would produce on the surface of the Moon (which is a mean
distance 3
.
76
×
10
5
km from the surface of the Earth). Neglect any effects
of the Earth’s atmosphere.
3. Estimate how far away an automobile is when you can only just barely
resolve the two headlights with your eyes.
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 Spring '10
 Fitzpatrick
 Work, Wavelength, Telescope, radio telescope, National Radio Astronomy Observatory, farﬁeld interference pattern, C= sin

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