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Chapter 26: Geometrical Optics, Summaries of Optics

Stanford UniversityOptics

A concave mirror produces a real, inverted, and magnified image of an object. Use the magnification equation (equation 26-8) to calculate the image distance.

Typology: Summaries

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Chapter 26: Geometrical Optics
8.
The image shows a light beam that is oriented 15.0° from the vertical as it
reflects back and forth between two mirrors.
Let d equal the horizontal distance traveled by the light between reflections
off either mirror. Calculate the distance d by multiplying the separation
distance by the tangent of the beam angle. Divide the total distance (168
cm) by d to calculate the total number of reflections. From this result
calculate the number of reflections off each mirror, where the first
reflection is off the top mirror and the reflections alternate between the
mirrors.
1. (a) Calculate the distance d:
( ) ( )
68.0 cm tan 15 18.2 cmd= °=
2. Divide the horizontal distance by d:
168 cm 9.23 9 reflections
18.2 cm
x
Nd
= = = ⇒
3. Because all of the odd-numbered reflections are off the top mirror (1, 3, 5, 7, and 9) the light will reflect 5 times off
of the top mirror.
4. (b) All of the even-numbered reflections are off the bottom mirror (2, 4, 6, and 8), so the light will reflect 4 times off
of the bottom mirror.
34.
The image shows a 1.7-m-tall person standing 0.60 m from a reflecting
globe with diameter 0.16 m.
Use equation 26-2 to calculate the focal length of the globe, where the
radius is one half of the diameter. Then use equation 26-6 to calculate the
image distance from the focal length and object distance. Finally, use
equation 26-4 to calculate the height of the image.
1. (a) Calculate the
focal length:
( )
11
22
1
2
2
0.18 m 2
0.045 m
fR D
f
=−=−
= −
= −
2. Use equation 26-4 to calculate
the image distance:
The image is 4.2 cm behind the surface of the globe.
3. (b) Calculate the image height:
( )
i
io
o
0.042 m 1.7 m 11 cm
0.66 m
d
hh
d
−
=−=− =
38.
A concave mirror produces a real, inverted, and magnified image of an object.
Use the magnification equation (equation 26-8) to calculate the image distance
from the mirror. Then solve equation 26-6 for the focal length.
1. (a) Calculate the image distance:
( )( )
io
3 22 cm 66 cmd md=− =−− =
pf3

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Chapter 26: Geometrical Optics

  1. (^) The image shows a light beam that is oriented 15.0° from the vertical as it reflects back and forth between two mirrors.

Let d equal the horizontal distance traveled by the light between reflections off either mirror. Calculate the distance d by multiplying the separation distance by the tangent of the beam angle. Divide the total distance ( cm) by d to calculate the total number of reflections. From this result calculate the number of reflections off each mirror, where the first reflection is off the top mirror and the reflections alternate between the mirrors.

1. (a) Calculate the distance d : d = ( 68.0 cm tan 15) ( °) =18.2 cm

2. Divide the horizontal distance by d :

168 cm 9.23 9 reflections 18.2 cm

x N d

3. Because all of the odd-numbered reflections are off the top mirror (1, 3, 5, 7, and 9) the light will reflect 5 times off of the top mirror. 4. (b) All of the even-numbered reflections are off the bottom mirror (2, 4, 6, and 8), so the light will reflect 4 times off of the bottom mirror.

  1. The image shows a 1.7-m-tall person standing 0.60 m from a reflecting globe with diameter 0.16 m.

Use equation 26-2 to calculate the focal length of the globe, where the radius is one half of the diameter. Then use equation 26-6 to calculate the image distance from the focal length and object distance. Finally, use equation 26-4 to calculate the height of the image.

1. (a) Calculate the focal length:

1 2

0.18 m 2 0.045 m

f R D

f

2. Use equation 26-4 to calculate the image distance:

1 1 i o

4.2 cm 0.045 m 0.66 m

d f d

− −     = (^)  − (^)  = (^)  − (^)  = −    − 

The image is 4.2 cm behind the surface of the globe.

3. (b) Calculate the image height: i i o ( )

o

0.042 m 1.7 m 11 cm 0.66 m

d h h d

  1. A concave mirror produces a real, inverted, and magnified image of an object.

Use the magnification equation (equation 26-8) to calculate the image distance from the mirror. Then solve equation 26-6 for the focal length.

1. (a) Calculate the image distance: d i = − md o = − −( 3 )( 22 cm )= 66 cm

2. (b) Calculate the focal length from equation 26-6:

(^1 )

o i

17 cm 22 cm 66 cm

f d d

− (^) −   (^)   = (^)  + (^)  = (^)  + (^)  =   ^ 

  1. The figure shows a tree located 23 m from a mirror and its real, inverted 3.8-cm-tall image located 7.0 cm in front of the mirror.

Use equation 26-4 to calculate the object height from the image height and the object and image distances.

Solve equation 26-4 for the object height:

o i^ (^ )

o i

23 m 3.8 cm 12 m 7.0 cm

d h h d

  1. The image shows a pond of total thickness 3.25 meters that has a 0.38 m layer of ice on the surface. Light travels from the top of the pond to the bottom.

Calculate the depth of the water by subtracting the depth of the ice from the total depth. Calculate the time for light to travel through the ice (and then through the water) by dividing the distance by the speed of light in that medium. Use equation 26-10 to calculate the speed of light in each medium, using the indices of refraction given in Table 26-2.

1. Calculate the water depth:

w total ice 3.25 m 0.38 m = 2.87 m

d = d − d = −

2. Calculate the time to cross the ice:

ice ice^ (^ )

ice (^8)

1.31 0.38 m 1.7 ns 3.00 10 m/s

c d vt t n n d t c

×

3. Calculate the time to cross the water:

w w^ (^ )

w (^8)

1.33 2.87 m 12.7 ns 3.00 10 m/s

n d t c

×

4. Add the two times together: (^) t total (^) = t ice (^) + t w = 1.7 ns + 12.7 ns =14.4 ns

  1. The image shows a person looking into an empty glass at an angle that allows her to barely see the bottom of the glass. When looking at the same angle after filling the glass with water, she can see the center of the bottom of the glass.

Use the empty glass to calculate the sine of the angle of refraction θ 2 in terms of the height H of the glass and the width of the bottom W. With the glass full, calculate sine of the angle of incidence θ 1 in terms of H and W 2.Then use Snell’s Law (equation 26-11) to calculate the height of the glass.

1. Calculate sin θ 2 : sin^2 2

W

W H

2. Calculate sin θ 1 :

(^1 2 22 )

sin 2 4

W W

W H W^ H

θ = =