334
Chapter 33
33.14.
I
DENTIFY
:
Apply the law of reflection.
S
ET
U
P
:
The mirror in its original position and after being rotated by an angle
θ
are shown in Figure 33.14.
α
is
the angle through which the reflected ray rotates when the mirror rotates. The two angles labeled
φ
are equal and
the two angles labeled
φ
′
are equal because of the law of reflection. The two angles labeled
θ
are equal because the
lines forming one angle are perpendicular to the lines forming the other angle.
E
XECUTE
:
From the diagram,
2
2
2(
)
α
φ
φ
φ
φ
′
′
=
−
=
−
and
θ
φ
φ
′
=
−
.
2
α
θ
=
, as was to be shown.
E
VALUATE
:
This result is independent of the initial angle of incidence.
Figure 33.14
33.15.
I
DENTIFY
:
Apply
sin
sin
a
a
b
b
n
n
θ
θ
=
.
S
ET
U
P
:
The light refracts from the liquid into the glass, so
1.70
a
n
=
,
62.0
a
θ
=
°
.
1.58
b
n
=
.
E
XECUTE
:
1.70
sin
sin
sin62.0
0.950
1.58
a
b
a
b
n
n
θ
θ
⎛
⎞
⎛
⎞
=
=
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
°
and
71.8
b
θ
=
°
.
E
VALUATE
:
The ray refracts into a material of smaller
n
, so it is bent away from the normal.
33.16.
I
DENTIFY
:
Apply Snell’s law.
S
ET
U
P
:
a
θ
and
b
θ
are measured relative to the normal to the surface of the interface.
60.0
15.0
45.0
a
θ
=
−
=
°
°
°
.
E
XECUTE
:
1.33
arcsin
sin
arcsin
sin45.0
38.2
1.52
a
b
a
b
n
n
θ
θ
⎛
⎞
⎛
⎞
=
=
° =
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
°
. But this is the angle from the normal to the surface,
so the angle from the vertical is an additional 15° because of the tilt of the surface. Therefore, the angle is 53.2°.
E
VALUATE
:
Compared to Example 33.1,
a
θ
is shifted by
15
°
but the shift in
b
θ
is only
53.2
49.3
3.9
−
=
°
°
°
.
33.17.
I
DENTIFY
:
The critical angle for total internal reflection is
a
θ
that gives
90
b
θ
=
°
in Snell’s law.
S
ET
U
P
:
In Figure 33.17 the angle of incidence
a
θ
is related to angle
θ
by
90
a
θ
θ
+
=
°
.
E
XECUTE
:
(a)
Calculate
a
θ
that gives
90
b
θ
=
°
.
1.60
a
n
=
,
1.00
b
n
=
so
sin
sin
a
a
b
b
n
n
θ
θ
=
gives
(1.60)sin
(1.00)sin90
a
θ
=
°
.
1.00
sin
1.60
a
θ
=
and
38.7
a
θ
=
°
.
90
51.3
a
θ
θ
=
=
°−
°
.
(b)
1.60
a
n
=
,
1.333
b
n
=
.
(1.60)sin
(1.333)sin90
a
θ
=
°
.
1.333
sin
1.60
a
θ
=
and
56.4
a
θ
=
°
.
90
33.6
a
θ
θ
=
=
°−
°
.
E
VALUATE
:
The critical angle increases when the ratio
a
b
n
n
increases.
Figure 33.17
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The Nature and Propagation of Light
335
33.18.
I
DENTIFY
:
Since the refractive index of the glass is greater than that of air or water, total internal reflection will
occur at the cube surface if the angle of incidence is greater than or equal to the critical angle.
S
ET
U
P
:
At the critical angle
θ
c
, Snell±s law gives
n
glass
sin
θ
c
=
n
air
sin 90° and likewise for water.
E
XECUTE
:
(a)
At the critical angle
θ
c
,
n
glass
sin
θ
c
=
n
air
sin 90°. 1.53 sin
θ
c
= (1.00)(1) and
θ
c
= 40.8°.
(b)
Using the same procedure as in part (a), we have 1.53 sin
θ
c
= 1.333 sin 90° and
θ
c
= 60.6°.
E
VALUATE
:
Since the refractive index of water is closer to the refractive index of glass than the refractive index
of air is, the critical angle for glasstowater is greater than for glasstoair.
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 Spring '06
 Buchler
 Physics, Snell's Law, Sin, refractive index, Total internal reflection

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