PHYS 434: Assignment 3 solutions
Problems 4.3
Use the paraxial approximation (Equation 5.8). Proceed as in Example 4.1:
θ
−
sin
θ
sin
θ
=
e
where
e
is the error, in this case
e
=
0.001
. Thus
θ
=
1
+
e
(
)
sin
θ
Trial and error will suffice, but it is better to use a Taylor series:
θ
≈
1
+
e
(
)
θ
−
θ
3
3!
Thus
θ
=
3! 1
−
1
1
+
e
≈
6
e
=
0.775
rad
Use the small angle approximation again, this time with the tangent:
tan
θ
=
D
2
f
≈
θ
⇒
D
≈
2
f
θ
=
7.75
mm
The above approximations yield 3digit accuracy – more than sufficient for a 1%
tolerance window.
Problem 4.4
Ray diagrams for problem 4.4. (a)
s
o
=
2.75
f
. (b)
s
o
=
0.750
f
.
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Proceed as in example 4.2. Let
s
o
=
2.75
f
. The image distance is
s
i
=
f s
o
s
o
−
f
=
2.75
f
2
1.75
f
=
1.57
f
The image distance is positive, so the image is real. The magnification is
M
=
−
s
i
s
o
=
−
0.571
M
<
0
so the image is inverted, with a height of 0.571 cm. See ray diagram figure.
Let
s
o
=
0.75
f
. The image distance is
s
i
=
−
3.00
f
.
s
i
<
0
so the image is virtual.
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 Winter '09
 Kilfoil

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