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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Chapter 4.5.4 page 89
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To determine the image size on the film plane, we first need to find the effective focal length of the lens system. The effective focal length, denoted as f, can be calculated using the lens formula for thin lenses in combination:
1/f = 1/f1 + 1/f2 - d/(f1*f2),
where f1 and f2 are the focal lengths of the individual lenses, and d is the distance between them. Plugging in the given values:
1/f = 1/20 + 1/(-8) - 15/(20*(-8)).
Simplifying this equation will give us the effective focal length.
Once we have the effective focal length, we can use the magnification formula to find the magnification produced by the lens system:
m = -f/F,
where m is the magnification and F is the object distance. In this case, the object distance is considered very large (since it is a distant object), so it can be approximated as infinity.
Finally, the image size on the film plane can be calculated using the formula:
image size = object size * magnification.
Given that the object is subtending an angle of α=2∘ at the camera, you can determine the object size using trigonometry:
object size = 2 * tan(α/2) * F,
where F is again the object distance (approximated as infinity).
By plugging in the values into the formulas and performing the necessary calculations, you should be able to determine the image size on the film plane.