Telephoto lenses allow one to "zoom in" on distant objects, making them bigger. A particular telephoto lens consists of a combination of two thin lenses having focal lengths of f1=20 cm and f2=−8 cm, respectively. The lenses are separated by a distance of d=15 cm. Determine the image size in cm on the film plane, located at the back focal point of the lens system, of a distant object subtending an angle of α=2∘ at the camera.
The image size of a distant object is simply its angular size multiplied by
the lens focal length.
1/f = 1/f₁+ 1/f₂−d/f₁•f₂.
f₁= 20 cm =0.2 m
f₂= - 8 cm = - 0.08 m
1/f = 1/0.2 – 1/0.08 - 1/0.2•0.08=
=5-12.5-62.5 = - 70
f= -1/70 =-0.0143 m
= -1.43 cm
h= 2•1.43 = 2.86 cm
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http://www.fulviofrisone.com/attachments/article/411/the%20light%20fantastic.pdf
Chapter 4.5.4 page 89
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To determine the image size on the film plane, we need to use the thin lens formula and consider the lens system as a whole.
The formula to calculate the image size formed by a thin lens is given by the magnification equation:
magnification (m) = -image distance (di) / object distance (do)
In this case, since we are interested in the image size on the film plane, we need to calculate the magnification at the back focal point of the lens system.
First, let's calculate the total focal length (ftotal) of the lens system:
1/ftotal = 1/f1 + 1/f2
Where f1 is the focal length of the first lens (+20 cm) and f2 is the focal length of the second lens (-8 cm).
1/ftotal = 1/20 cm + 1/(-8 cm)
1/ftotal = (1/20) - (1/8)
1/ftotal = (2 - 5) / 40
1/ftotal = -3 / 40
ftotal = -40 / 3 cm ≈ -13.33 cm
Since the lens system is composed of two lenses, the total focal length is negative.
Now, let's calculate the magnification at the back focal point of the lens system. Since the distant object subtends an angle α at the camera, the object distance (do) can be calculated using the formula:
do = ftotal * tan(α)
Substituting the values:
do = (-13.33 cm) * tan(2∘)
do ≈ (-13.33 cm) * (0.0349)
do ≈ -0.465 cm
The negative sign indicates that the light from the object is coming from the opposite direction (i.e., it is a distant object).
Now, using the magnification equation:
magnification (m) = -di / do
Since we want to find the image size on the film plane (at the back focal point), the image distance (di) is equal to the focal length of the lens system (ftotal):
di = -13.33 cm
Finally, substituting the values into the magnification equation:
m = -(-13.33 cm) / (-0.465 cm)
m ≈ 28.6
The magnification is approximately 28.6.
The image size on the film plane can be determined by multiplying the magnification by the size of the object:
image size = m * object size
Since we are not given the size of the object, we cannot calculate the exact image size in centimeters without that information. However, with the magnification value of 28.6, we can say that the image on the film plane will be significantly larger than the original object size.