Physics 208, Assignment 9, Solutions, 3/30/09, RT
Problem 1.
The figure shows a conjectured “reflection” for which, in order
to set up equations, the outgoing angle
β
is visibly different from the
incoming angle
α
. By definition of a “wavefront” the incident rays labeled
1
2
x
cos
α
x
cos
β
α
α
β
1’
2’
G
F’
x
F
G’
β
1 and 2 are “in phase” at the points F and G, points on the same wavefront.
According to Huyghens, when the outgoing rays 1’ and 2’ are viewed in
the distance (or, as one says, at infinity) they will add constructively if
they are in phase at the points F’ and G’. This will be true
if
the distances
GG’ and FF’ are equal, or
x
cos
β
=
x
cos
α,
or
β
=
α.
(1)
Strictly speaking we also have to prove an
only if
statement as well.
Eq. (1) is also satisfied when distances GG’ and FF’ differ by an inte
gral number of wavelengths. This can happen for various discrete values
of
x
. But, for constructive interference of
all
rays on the wavefront F’G’
requires Eq. (1) to be satisfied for
all
values of
x
. So there is reflection if
and only if
α
=
β
.
This reflection is referred to as “specular” which, I think, means, “the
same for all colors”. The argument above showing independence from
x
can be rephrased to prove independence of wavelength
λ
.
1
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Problem 2.
The problem states that two slits are separated by 25 wave
lengths. It would not be wrong to work in units of length such that the
wavelength is 1 unit. But that might be confusing. Hence we introduce
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 Spring '08
 AMADEURI
 Physics, Visible spectrum, 0.030 cm, 1 70 L

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